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Page 1 of 8 + +Abstract—The domain wall-magnetic tunnel junction (DW- +MTJ) is a versatile device that can simultaneously store data and +perform computations. These three-terminal devices are +promising for digital logic due to their nonvolatility, low-energy +operation, and radiation hardness. Here, we augment the DW- +MTJ logic gate with voltage controlled magnetic anisotropy +(VCMA) to improve the reliability of logical concatenation in the +presence of realistic process variations. VCMA creates potential +wells that allow for reliable and repeatable localization of domain +walls. The DW-MTJ logic gate supports different fanouts, allowing +for multiple inputs and outputs for a single device without +affecting area. We simulate a systolic array of DW-MTJ Multiply- +Accumulate (MAC), which uses the nonvolatility of DW-MTJ logic +gates to enable fine-grained pipelining and achieve massive +parallelism. The DW-MTJ 8-bit systolic array provides +comparable throughput and efficiency to state-of-the-art CMOS +systolic arrays while being radiation-hard. These results improve +the feasibility of using domain wall-based processors, especially +for extreme-environment applications such as space. + +Index Terms—Domain Wall, Magnetic Tunnel Junction, +VCMA, Logic, In-Memory Computing, Magnetism, Spintronics +I. INTRODUCTION +HERE is a pressing need for new computational methods +arising from the demand for fast computation and efficient +processing of data, which is currently hindered by the +bottlenecks of the von Neumann architecture. Even with +continuous +improvements +in +transistor +technology, +computations are fundamentally hindered by the speed of +accessing data, leading to an energy-inefficient architecture [1]. +To address these bottlenecks, processing digital logic using +non-volatile memory presents new opportunities to improve the +efficiency and reliability of digital computation. By engineering +non-volatile memory for logic, novel digital processors may +operate at low voltages, dissipate nearly zero power in static +operation, and operate more reliably as compared to +conventional CMOS logic. Improvements in energy efficiency +are important for enabling the deployment of more +sophisticated applications, such as artificial intelligence and +machine learning algorithms, to edge or remote computing +systems that have a limited power budget. + +This work was supported by the Laboratory Directed Research and +Development Program at Sandia National Laboratories, a multimission +laboratory managed and operated by National Technology & Engineering +Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell +International Inc., for the U.S. Department of Energy’s National Nuclear +Security Administration under contract DE-NA0003525. This work was also +supported by the National Science Foundation Graduate Research Fellowship +Program under Grant No. 2021311125 (S. Liu). +Spintronics is a unique platform for enabling dense and +reliable non-volatile memory, digital and analog in-memory +computing, neuromorphic computing, normally-off computing, +and new, efficient implementations of digital logic [2-8]. These +varying compute architectures derive their energy and +performance benefits from a shared foundation: spintronic +devices that provide non-volatile data storage, low read and +write energy, high write endurance, and back-end-of-the-line +compatibility with a CMOS fabrication process [9]. +Additionally, because spintronic devices encode information in +magnetization rather than charge, they are known to be more +resilient to the effects of radiation (especially ionizing +radiation), which makes spintronics particularly attractive for +edge computing in space and defense applications [10-14]. + Several spintronic architectures have been proposed for +accelerating digital logic operations. These include logic +circuits that use a mixture of CMOS and magnetic tunnel +junction (MTJ) devices, spin wave majority gates, and multiple +logic gates implemented using magnetic domain walls (DWs) +[14-24]. A practical magnetics-based digital processor that +avoids significant issues with extreme edge computing should +be: (1) all-magnetic, with no accessory CMOS inside each logic +gate, (2) all-electrical, requiring no external magnetic fields or +optical excitation to read and write, (3) cascadable to form large +circuits, having current-in/current-out or voltage-in/voltage-out +operation without needing to convert data between logic gates. +Of the proposed architectures, the three-terminal domain +wall-magnetic tunnel junction (DW-MTJ) logic gate fulfills all +three of the key requirements, while compactly implementing +each logic gate (e.g. inverter, NAND) within a single nano- +device. In this device, a logical bit is encoded in the position of +a domain wall (DW) along a ferromagnetic track, which forms +the free layer of an MTJ. The logic state is read out and +transmitted to the next logic gate through the MTJ. In our +previous work, DW-MTJ prototypes have been experimentally +shown to operate with both spin-transfer torque (STT) and spin- +orbit torque (SOT) current input, to function as logical inverters +with fanout > 1, to have electrically controllable operation of +the DW with < 10% cycling variation, and to build circuits [14, +15]. More complex DW-MTJ circuits were benchmarked using +N. Zogbi, S. Liu, and J. A. C. Incorvia are with the Department of Electrical +and Computer Engineering, University of Texas at Austin, Austin, TX 78712 +USA (e-mail: incorvia@austin.utexas.edu). +M. J. Marinella is with Arizona State University, Tempe, AZ 85287 USA. +C. H. Bennett, S. Agarwal, and T. P. Xiao are with Sandia National +Laboratories, Albuquerque, NM 87123 USA (email: txiao@sandia.gov). + +Nicholas Zogbi, Samuel Liu, Christopher H. Bennett, Sapan Agarwal, Matthew J. Marinella, Jean Anne C. Incorvia, +and T. Patrick Xiao +Massively Parallel Matrix Multiplication Using Voltage +Controlled Magnetic Anisotropy Domain Wall Logic +T + + +Zogbi, Liu, et al. Page 2 of 8 +micromagnetic simulations and compact device models [23- +25]. Nevertheless, challenges remain for the reliability of DW- +MTJ logic operations. Prior modeling has relied on DW inertia +to sustain the DW motion across a perfectly smooth magnetic +track after the cessation of an applied current. However, in +realistic tracks fabricated in scaled process nodes, this inertial +motion can be rapidly halted by edge roughness or local +material defects. Furthermore, the energy and performance of +DW-MTJ logic has not been evaluated for the multiply- +accumulate (MAC) kernel, which dominates modern edge +computing workloads such as machine learning. +In this paper, we demonstrate that voltage-controlled +magnetic anisotropy (VCMA) ensures reliable DW-MTJ logic +concatenation along magnetic tracks with realistic roughness +instead of relying on DW inertial motion. We show using +micromagnetic simulations that VCMA can be used to +electrically pin the DW at set locations, enabling deterministic +and robust switching. We also demonstrate how to implement +different logic functions and fanouts with minimal changes to +device geometry. These device-level results are used to +benchmark the energy and performance of a systolic array of +DW-MTJ 8-bit MAC units, a highly parallel spatial architecture +that computes matrix multiplications. The non-volatility of +DW-MTJ logic enables pipelined processing inside the MAC +units, greatly increasing parallelism and enabling the +processing throughput of DW-MTJ MAC units to approach that +of CMOS MAC units, despite the much slower switching speed +of a DW-MTJ logic gate. We discuss the scaling requirements +on spintronic devices and material properties to enable reliable +operation and energy-efficient extreme edge computing. +II. DW-MTJ DEVICE DESIGN +The structure and operation of the three-terminal DW-MTJ +device is shown in Fig 1. The DW is the transition region +between two oppositely oriented magnetic domains in a +ferromagnetic wire, part of which also forms the free layer of +an overlying MTJ. The ‘1’ and ‘0’ states are encoded by the +position of the DW along with the MTJ’s fixed layer +orientation. In Fig 1(a), the DW is on the left and the MTJ is in +a high resistance state, resulting in a low output current when a +voltage is applied from CLK to OUT to read the device, +interpreted as a ‘0’. The two resistance states of the MTJ are +anti-parallel (AP) RAP and parallel (P) RP, and the tunnel +magnetoresistance is defined as TMR = (RAP − RP) / RP. +To write the device, current is pulsed through the IN terminal +while CLK is grounded. If this current surpasses the threshold +current for DW motion, the DW will move in the same direction +as the electron flow which moves the DW to the opposite (right) +side of the device as shown in Fig 1(b). This results in a high +output current when a voltage is applied from CLK to OUT, +interpreted as a ‘1’. The DW moves due to a combination of the +spin transfer torque (STT) exerted by the spin-polarized current +through the magnetic track and by the spin orbit torque (SOT) +caused by current flow in the underlying heavy metal layer. The +SOT effect is more dominant due to the lower resistivity of the +heavy metal which contains most of the current flow. +DW-MTJ logic gates are concatenated by connecting the +OUT terminal of one device to the IN terminal of the next +device as shown in Fig 1(d). To transmit a stored bit from +Device 1 to Device 2, a voltage pulse is applied to the CLK +terminal of Device 1 and current flows through its OUT +terminal. The magnitude of this current depends on the DW +position, and thus this current is used to pass an input bit to +Device 2. If the current exceeds Device 2’s threshold, its DW +moves from left to right. Concurrently, current also flows from +the CLK to the IN terminal of Device 1 to reset its DW to the +left, causing a current to flow from the OUT to CLK terminal +of Device 0. This current does not affect the magnetic state of +Device 0. The combined read-reset scheme is desirable because +by not being required to hold its state, the logic gate can be +immediately re-used to process another operation [15, 23]. This +enables extremely fine-grained pipelining and massively +parallel computation as will be described in Section VI. +The current that is transmitted to Device 2 corresponds to the +output bit that was stored in Device 1 until the moment that the +DW passes under the MTJ. After this point, Device 1 has been +reset and its output current no longer transmits the correct +information. Earlier versions of the DW-MTJ used a very short +current pulse to transmit the output bit to Device 2 before +Device 1 was reset and relied on DW inertia to settle the DWs +in both devices to their correct positions after the end of the +current pulse. On a smooth ferromagnetic track, a DW can be +modeled as a massive particle and it can be shown using +micromagnetic simulations that the DW continues to move after +current is turned off [26]. However, in a physical magnetic +track, DWs can be easily pinned by local defects or edge +roughness, bringing the inertial motion to a premature halt, and +causing bit errors. To overcome this challenge and make DW- +MTJ logic more feasible for implementation, VCMA terminals, +labeled as VCMA, are introduced in Fig 1(a-c) to pull the DW +to one of two locations on either side of the OUT terminal after +the current pulse is turned off. + +Figure 1: DW-MTJ logical buffer. (a) Side-view of the device when +the DW is on the left and the MTJ is in a high resistance state. +Orange/blue are oppositely magnetized regions. Electron flow +direction during a write is indicated. (b) After a write, the DW is +on the right and the MTJ is in a low resistance state. (c) Top-down +view of the logic device with varying fanouts. For a fanout of 0.5, +the MTJ length LMTJ is set to the minimum feature size F = 15 nm. +For a fanout of 1, LMTJ is set to 3F = 45 nm. For a fanout of 2, LMTJ +is set to 9F = 135 nm. (d) Side-view of a 1×3 chain of DW-MTJ +buffers. Device 1 is being read/reset and Device 2 is being written. + + +Device 0 +Device 1 +Device 2 +(d) +IN +OUT +CLK +MgO +Heavy Metal +Free Layer +e– +Fixed Layer +Insulator +VCMA +(a) Logical ‘0’ +VCMA +LMTJ +IN +255 nm +F +CLK +VCMA +OUT +(c) +F +F +VCMA +IN +OUT +CLK +(b) Logical ‘1’ +VCMA +VCMA + + +Zogbi, Liu, et al. Page 3 of 8 +Variable logic gate fanout is important for implementing +practical logic circuits. To allow the design of multiple fanouts +while keeping the threshold current and device footprint +constant, the device width is set to the minimum feature size (F += 15 nm) and the magnetic wire length is 17F = 255 nm. These +parameters are fixed for all logic gate types (buffer, inverter, +NAND, etc.). The only parameter that is varied with fanout is +the length of the MTJ, LMTJ, shown in Fig 1(c), which is set to +F for fanout of 0.5, 3F for a fanout of 1, and 9F for fanout of 2. +A fanout of 0.5 means that the output goes to one of the two +input ports of a logic gate such as NAND, thus requiring less +current per device. A fanout of 2 means a single device is +concatenated to two devices at its output, thus requiring greater +current flow because it needs to supply current to both devices. +Table I shows the parameters used for micromagnetic +modeling of the device. For the parameters shown, the threshold +current and threshold current density is simulated to be 2.1 μA +and 7×1010 A/m2, respectively, when these devices are modeled +with edge roughness, by removing nanometer pieces from the +free layer track, and random grain anisotropy to the +ferromagnetic track. This is a reasonable value for SOT-driven +DW motion in CoFeB with perpendicular magnetic anisotropy +(PMA) [14]. Additionally, the sign of the applied voltage to the +CLK terminal and the signs of currents that result from the +voltage are modified to allow for the correct direction of +electron flow to propagate the DW. +III. VCMA-BASED DW-MTJ CONCATENATION +Fig 2(a) shows the effect of VCMA when a voltage +VB = 2.5 V is applied to the VCMA terminals, which are +separated from the ferromagnetic wire by an insulator. This +causes a reduction in the local PMA of the ferromagnetic layer +under the terminals, creating an energy well that attracts the +DW to the minimum PMA point. The VCMA voltage is applied +before and after the read/reset current pulse, as shown in Fig. +2(b). The current pulse, if sufficiently high, initiates the DW +motion, while the VCMA voltage pulse reliably pins the DW +under one of the two VCMA terminals. After the VCMA pulse +has been applied for a time trest, called the rest time, the DW +position can be considered stable. +To model how the applied VCMA voltage affects the DW, +we compute the electric field at the insulator/CoFeB interface, +which changes the number of electrons in the out-of-plane 3d- +orbitals of the Fe atoms at the interface [27]. This changes the +PMA of the ferromagnetic free layer, which is expressed by +𝐾!(𝑉") = 𝐾!(0) − 𝜉𝐸 +𝑡#$ +(1) +where 𝐾!(𝑉") is the PMA at an applied voltage VB, 𝐾!(0) is the +PMA when there is no applied voltage to the ferromagnetic free +layer, 𝐸 is the electric field being applied to the DW wire, 𝜉 is +the VCMA coefficient, and 𝑡#$ is the thickness of the +ferromagnetic track where all the parameters are shown in +Table I [28, 29]. +To allow the realization of the electric field at the +insulator/ferromagnetic interface, calculations of the electric +field were conducted by solving Poisson’s equation using the +Jacobi method [30]. The strength of the voltage VB determines +the minimum PMA from VCMA. Thus, the higher the voltage, +the greater the change in the minimum PMA because there is a +stronger electric field. This results in a difference between the +minimum and maximum PMA, denoted as ∆Ku. The resulting +∆Ku in Fig 2(a) is 25 kJ/m3. +Fig 2(c) shows DW position as a function of time for a device +that is driven by a current through its IN terminal, which comes +from another logic gate, with three different values of the +applied VCMA voltage VB. The vertical dashed lines show the +TABLE II +BUFFER CONFIGURATIONS & ENERGIES +Initial DW1 +Initial MTJ0 +Initial MTJ1 +Reset Energy (fJ) +Right +P +P +2.2 +Right +AP +P +2.0 +Left +P +AP +2.0 +Left +AP +AP +1.8 + + +Figure 2: (a) Distribution of perpendicular magnetic anisotropy +along the ferromagnetic free layer when both VCMA terminals +have an applied voltage of VB = 2.5 V. The collection of charges on +the edges of the ferromagnet are not displayed. (b) Diagram of +VCLK and VCMA pulses with time of tpulse (2 ns) and trest (2 ns), +respectively. (c) Simulated DW position vs time for three voltage +values when the TMR of the simulated logic devices is 115%. +IN +VCMA +CLK +OUT +(a) +(c) +(b) +VCMA +Magnetic +anisotropy (kJ/m3) +TABLE I +PARAMETERS USED IN THE MODEL +Parameter +Value +Gilbert damping α +0.05 +Saturation magnetization MS +8 ´ 105 A/m +Exchange stiffness Aex +1.3 ´ 10-11 J/m +Uniaxial anisotropy constant KS(0) +5 ´ 105 J/m3 +Spin polarization P +0.7 +VCMA coefficient ξ +10 pJ/Vm +Clock voltage VCLK +0.04 V +Clock pulse time tpulse +2 ns +Device rest time trest +2 ns +Track length Lwire +255 nm +Track width W +15 nm +Ferromagnet thickness tFL +3 nm +Ferromagnet resistivity ρFL +500 µΩ⸱cm +Heavy metal thickness tHM +7 nm +Heavy metal resistivity ρHM +40 µΩ⸱cm +Insulator dielectric constant kINS +7 +MTJ RA product +0.675 Ω*µm2 +MTJ parallel resistance (Fanout (FO) 0.5) +3 kΩ +MTJ length LMTJ +FO 0.5: LMTJ = 15 nm +FO 1: LMTJ = 45 nm +FO 2: LMTJ = 135 nm +MTJ width WMTJ +15 nm + + + +Zogbi, Liu, et al. Page 4 of 8 +periods for the sequence, rest/pulse/rest, shown in Fig 2(b) +where the duration of the pulses is in Table I. The horizontal +dashed lines show the region for the location of the MTJ, 105- +150 nm, and the regions at which VCMA terminals are located, +30-45 nm and 210-225 nm. When the VCMA is not applied to +the device, the DW stays in approximately the same position +after the CLK pulse ceases. Since this DW position is +precariously close to the MTJ, the resistance of the MTJ is +sensitive to edge roughness variations, noise, and timing +imprecision in the CLK pulse, all of which can lead to bit errors. +When VCMA is applied after the CLK pulse, the DW reliably +moves to a position that is far to the right of the MTJ, so long +as the current pulse moves the DW past the midpoint of the +MTJ. This makes the concatenation of logic devices +significantly more robust to process variations and noise. +IV. SINGLE INPUT/OUTPUT LOGIC +Reliable logical concatenation of DW-MTJ devices is +verified using the micromagnetic simulation software MuMax3 +with parameters from Table I [31]. The circuit shown in Fig +1(d) is used to demonstrate the concatenation of devices with +single input, single output, and fanout of 1. A SPICE simulation +is used to calculate the Thevenin resistance of the three +concatenated DW-MTJ devices, which is then used to calculate +the current density through the logic device. By sweeping the +amplitude of the voltage pulse, the threshold current density of +the logic device was found along with its resulting voltage. +Fig 3(a) shows the DW position vs. time for Device 1 when +Device 1 is reset and its state is transmitted to Device 2, for the +three-device circuit in Fig 1(d). The colored curves on the plot +show 25 independent simulations with random grain anisotropy +on the ferromagnetic track to emulate device-to-device +variations. Fig 3(b) shows the DW position as a function of time +for Device 2 during the same voltage pulse and its DW is driven +by current through its IN terminal. For the same voltage pulse, +the current that goes through the OUT of Device 1 to the IN of +Device 2 (to its CLK, grounded) is shown in Fig 3(c). This +result shows the successful transfer of a ‘1’ bit from Device 1 +to Device 2, which is robust to device-to-device variations. +Fig 1(d) represents one of eight possible configurations that +are possible for fanout of 1. This is because there are two +configurations for the starting positions for Device 1’s DW; two +fixed layer orientations for Device 1, which decides whether the +device acts as an inverter or a buffer; and two MTJ resistances +for Device 0, which affects the current through Device 1. Table +II shows the configurations that result when Device 1 is a buffer +and their energies from the applied voltage pulse to the CLK +terminal, denoted as reset energy. Due to different +configurations that are possible that perform the same logic +function, these configurations dictate whether DWs are being +propagated by the pulse and what the Thevenin resistance is +within the circuit. This leads to some configurations being more +prone to errors compared to other configurations. Therefore, it +is important to make sure that all of the eight configurations can +perform their logic functions correctly. +Fig 4(a-b) shows the functionality of the logic devices by +testing the 8 configurations of the buffer/inverter 25 times each +for a total of 200 tests per data point to calculate the correctness. +The correctness was calculated by dividing the number of +successful tests by the total number of tests for each data point. +Each of the devices was simulated with random variations in +edge roughness and grain anisotropy on the ferromagnetic +track, with a maximum anisotropy variation of 7.5 kJ/m3. Fig +4(a) shows that successful circuit operation strongly depends on +TMR. When the TMR of the MTJs is less than 55%, there is a +steep drop-off in correctness: the difference between RP and +RAP is too small, which reduces the difference between a ‘high’ +and a ‘low’ current seen by Device 2. The reduced difference +makes it more likely that the DW in Device 2 will erroneously +move in response to a ‘low’ current. +There is a range of TMR = 75% - 115% for which there are +no errors, but, surprisingly, for TMR > 115% there is a +reduction in correctness across the eight configurations. For +145% < TMR < 235% the correctness is 87.5%, which +decreases with increasing TMR until correctness reaches 75%. +For high TMR, the MTJ draws very little current in its AP state. +From these lower currents in the AP state, DW propagation in +the concatenated device is slowed down. This can cause the DW +to not propagate fully during a logical ‘1’ write, leading some +logic outputs to incorrectly be ‘0’. While high TMR is often +thought of as the holy grail of MTJs used as memory devices +because it makes it easier to distinguish the two MTJ states, here +an optimal TMR may need to be tuned as a function of circuit +design decisions and can no longer be treated as a linear figure +of merit. The range of TMR for 100% correctness is wide +enough to be achievable with today’s MTJ processes. + +Figure 4: (a) Correctness for all 8 possible configurations of the 3- +device circuit vs. TMR. Yellow highlights the range with 100% +correctness. (b) Correctness vs. ∆Ku. +(b) +(a) + +Figure 3: Simulation results for the three-device circuit in Fig 1(d) +with TMR = 115%. DW position vs. time of (a) Device 1 during its +reset pulse and (b) simultaneous DW position vs. time of Device 2. +(c) Output current density from Device 1 being pulsed to Device 2. +(c) +(a) +(b) + + +Zogbi, Liu, et al. Page 5 of 8 +The ∆Ku that is induced by VCMA also affects the logic +function correctness, shown in Fig 4(b). The logic is 100% +correct for ∆Ku = 25 - 30 kJ/m3, which corresponds to +VB = 2.5 - 3V. Higher values of VB increase the velocity of the +DW propagation as shown in Fig 2(c). This reduces the length +of the VCMA pulse but requires a higher VCMA voltage. Thus, +the choice of VCMA voltage within the range of 100% +correctness can be used to trade off energy consumption and +speed. While not shown here, there is a limit to maximum ∆Ku +as well, when eventually VCMA would make the magnetization +in-plane. These results show adding VCMA to the DW-MTJ +devices results in reliable and robust concatenation into circuits, +and that both TMR and VCMA can be reasonably optimized to +ensure 100% correctness of the circuit operation. +V. MULTIPLE INPUT/OUTPUT LOGIC +As stated previously, to extend logic functions for different +fanouts, all that is required is to change the length of the MTJ +while no other modifications to the design are necessary. This +way, the supply voltage is constant throughout the circuit, and +the DW threshold current is also constant for all specified +fanouts. Table III shows the calculated reset energy for the three +fanouts depicted in Fig 1. These ranges for reset energies for the +different fanouts are scaled based on the RA product of the MTJ +which results in a varying amount of current flow through the +device depending on the fanout of the device. +To support multiple input logic functions, a fanout of 1 and +0.5 can be used for the device that is being reset. If each reset +device has a fanout of 1, a logical OR/NOR is realized because +this function only requires one input current to be high to +propagate the DW. However, for a logical AND/NAND the +reset devices each have a fanout of 0.5 because both input +currents must be high to propagate the DW. Fig 5(a-b) shows a +logical AND/NAND when the two buffer devices connected to +the IN terminal are a logical ‘1’ and a logical ‘0’ that results in +the DW not propagating to the opposite side. Fig 5(a) shows the +DW position as a function of time for the AND/NAND logic +device that is driven by a current through its IN terminal. Fig +5(b) shows the two currents that are supplied by the buffers, and +total current that is passed through the IN terminal of the +AND/NAND device. If both buffers connected to the IN +terminal of the AND/NAND device had a logical ‘1’, Fig 5(c- +d), the current is strong enough to propagate the DW to the other +side of the ferromagnetic track. Fig 5(c) shows the DW position +in an AND/NAND device that is driven by a current through its +IN terminal, and Fig 5(d) shows the current that is being driven +through the IN terminal of the device. +DW-MTJ logic devices also support multiple fanout, which +allows the output bit to be passed to multiple output devices. To +obtain a fanout of 2, the operation is fairly similar to when the +fanout of the device is 0.5, but instead of modifying the input +current, the output current is divided in half. Greater fanouts can +be achieved through the process of increasing the MTJ length, +but the circuit design becomes more challenging since the total +length of the logic devices must be increased: e.g. a fanout of 4 +for F = 15 nm requires device length of 27F (1.335 μm). +Additionally, if the device length is too long, VCMA could no +longer be used effectively since VB applied to the VCMA +terminals would cause little to no change in PMA values at the +middle of the wire. If the DW were to not propagate fully, it +may not be drawn to one of the two sides, causing errors. +Alternative solutions to obtaining larger fanouts can be +achieved by increasing the width of the wire to adjust the RA +product or adding resistors to the OUT pin of the device. +However, this would cause a change in the threshold current for +the DW or require resistors for devices with a small fanout. +Thus, here we limit the circuit design to a maximum fanout of +2 because we can achieve greater fanouts by using a cascade of +buffers with a fanout of 2 as described below. +VI. PIPELINED DW-MTJ MATRIX MULTIPLICATION +The ability of a DW-MTJ device to function both as a logic +gate and as a non-volatile memory element allows the device to +efficiently implement a form of sequential logic that results in +far greater data parallelism than the conventional combinatorial +logic used in CMOS processors. Fig 6 shows how DW-MTJ +buffers, inverters, and NAND gates can be connected to form a +full adder. Because data is passed between gates only during +clock pulses, buffers are used to keep the data aligned in time. +Moreover, a logic gate cannot simultaneously receive and +transmit data. To avoid interference, it also cannot receive data +while its output gate is being reset. Therefore, at any moment, +TABLE III +FANOUT ENERGIES +Fanout +Reset Energy (fJ) +0.5 +1.3-1.8 +1 +1.8-2.2 +2 +2.6-3.1 + + +Figure 5: (a) Simulated DW position vs time for an AND/NAND +device when the input currents are ‘10’ and the TMR of the logic +devices is 115%. (b) Input currents going into an AND/NAND +device showing a logical ‘1’ input current (blue), logical ‘0’ input +current (orange), and the total current being pulsed into the device +(black). (c) Simulated DW position vs time for an AND/NAND +device when the input currents are ‘11’. (d) Input currents going +into an AND/NAND device showing the two separate logical ‘1’ +input currents (blue, orange) and the total current (black). +(b) +(a) +(d) +(c) +10 +11 +1 +0 +11 + + +Zogbi, Liu, et al. Page 6 of 8 +each gate operates in one of three modes: receive, transmit, or +standby. Gates that are aligned in their logical depth operate in +the same mode, shown by the yellow, blue, and green bands in +Fig 6. A three-phase clock (Φ1, Φ2, and Φ3) switches each band +of gates from one operational mode to the next. Each mode lasts +for one phase of the clock period (4 ns). All gates cycle through +all three modes within one full clock period (12 ns). +Due to its non-volatility, a DW-MTJ logic gate does not need +to hold onto its state after it has transmitted its output to the next +gate. Each gate can therefore receive and process a new set of +inputs in the very next clock phase. This allows a single logic +circuit to concurrently process many independent operations, +pipelined through the stages of computation on each clock +cycle. The data buffering that is needed for pipelining is +implemented by the DW-MTJ logic gates themselves, and the +size of the pipeline stage is a window of logical depth three as +shown in Fig 6. This fine-grained pipelining allows concurrent +computation on much more data than standard instruction-level +pipelining. It enables high throughput while only fitting a small +number of sequential operations into a clock cycle, which +compensates for the relatively slow switching speed of a DW- +MTJ gate. Pipelining at this granularity is possible in CMOS, +but would be impractical due to the area and power +consumption of the many dedicated pipeline buffers required +[32, 33]. However, it comes at no cost in a DW-MTJ processor. +A disadvantage of this fine-grained pipelining is that the cost +of flushing the pipeline upon an interrupt is high, because a +large amount of data is stored by the DW-MTJ gates in the +pipeline. This makes the logic scheme a poor fit for general- +purpose microprocessors where pipeline flushes are common +(e.g. due to branch mispredictions). However, fine-grained +pipelining benefits hardware accelerators that perform +specialized computation and process data in large blocks. In this +scenario, pipeline flushes are non-existent or extremely rare by +design. A notable example is accelerators for matrix vector +multiplication (MVM), which are growing in importance and +popularity due to the growth of workloads implementing +machine learning algorithms [34]. The building block of these +accelerators is a unit for computing the Multiply-Accumulate +(MAC) operation: D = A×B + C. Here, we focus on MACs +where A, B, C, and D are multi-bit integers. Integer MACs, with +a typical resolution of 8 bits, are the main computational +primitive for deep neural network inference [35]. +Fig 7(a) shows a DW-MTJ MAC unit. The schematic for a +MAC with 4-bit multiply and 8-bit accumulate is shown for +simplicity. This circuit implements a pipelined array multiplier, +rather than a bit serial multiplier, to ensure that a full MAC is +completed on every clock cycle. In a given cycle, one bit of one +MAC output is available at every output bit position (e.g. D8 of +MAC 1, D7 of MAC 2, …, D0 of MAC 9). This format allows +the output terminal “D” of one MAC to be directly connected +to the input terminal “C” of another MAC, which has the same +format. The 4-bit multiplier has 13 pipeline stages and +processes 13 MACs in parallel. Data is buffered by one full +clock cycle using a buffer chain, which is a linear chain of three +buffers or a tree of buffers with depth three, depending on the +needed fanout (1 to 8). The AND chain is the same as the buffer +chain, but the first element is an AND gate. +Fig 7(b) shows how MAC units can be concatenated to +process MVMs Wx. We evaluate an 8-bit systolic array +architecture to process neural network inference, where the +matrix W is usually fixed [36, 37]. In a systolic array, inputs are +broadcast horizontally while partial sums are accumulated +vertically, and processing is pipelined at the MAC level. We +use DW-MTJs to further pipeline the MACs themselves, +increasing the MVM throughput. The MAC unit is augmented +so that it stores a fixed operand (an element of W). +The DW-MTJ pipelined systolic array operation is shown in +Fig 7(b). The first element of the input vector, x1 (8 bits), arrives +on the top left MAC unit which stores W11 (8 bits). The product +W11x1 (16 bits) is passed downward to the “C” input port of the +next MAC unit. This unit computes W21x2, where x& is passed + +Figure 6: Illustration of pipelined logic in a DW-MTJ full adder. +Data moves through the circuit one logical stage at a time, and +gates in the same stage are driven by the same signal from a three- +phase clock. Data is pipelined through each set of three stages, +which takes one clock cycle to traverse. DW-MTJs serve both as +logic gates and as storage elements for pipelining. +Φ1 +Φ2 +Φ3 +Φ1 +Φ2 +Φ3 +Pipeline stage 1 +Pipeline stage 2 +A1 +B1 +A2 +B2 +Cin1 +S1 +Cout1 +Computing FA(A1, B1, Cin1) +Computing FA(A1, B1, Cin1) +Computing FA(A2, B2, Cin2) +Clock +cycle 1: +Clock +cycle 2: + +Figure 7: (a) DW-MTJ MAC unit with 4-bit multiply, 8-bit +accumulate. (b) DW-MTJ 8-bit MAC systolic array, with 8-bit +multiply and 24-bit accumulate to support a 256×256 array. +Pipeline stages +& +& +& +& +& +& +& +& +& +& +& +& +& +& +A0 +B0 +A1 +B0 +A0 +B1 +A1 +B1 +A2 +B0 +A0 +B2 +A2 +B1 +A3 +B0 +A1 +B2 +& +& +A2 +B2 +A3 +B1 +A0 +B3 +A1 +B3 +A3 +B2 +A2 +B3 +A3 +B3 +Full +adder +A +B +Cin +Cout +S +Half +adder +A +B +Cout +S +AND +chain +A +B & +Buffer +chain +0 +D0 +C1 +C2 +C3 +C4 +C5 +C6 +C7 +D1 +D2 +D3 +D4 +D5 +D6 +D7 +D8 +C0 +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 13 14 +4-bit MAC +D = A×B + C +(a) +(b) +1 +0 +1 +0 +1 +0 +x1 +y1 +z1 +6 +6 +6 +7 +7 +7 +… +… +… +MAC +W21 +MAC +W22 +W11z1 + +W21z2 +W11x1 + +W21x2 +W12z1 + +W22z2 +W12x1 + +W22x2 +t2 +t3 +t7 +t8 +MAC +B (8b) +A (8b) +D (24b) +C (24b) +A (8b) +MAC +W11 +MAC +W12 +W11x1 +W11z1 +W12x1 +W12z1 +t0 +t1 +t5 +t6 +2 +1 +0 +2 +1 +0 +2 +1 +0 +23 +22 +23 +22 +23 +22 +… +1 +0 +1 +0 +1 +0 +x1 +y1 +z1 +6 +6 +6 +7 +7 +7 +… +… +… +1 +0 +1 +0 +1 +0 +x2 +y2 +z2 +6 +6 +6 +7 +7 +7 +… +… +… +1 +0 +1 +0 +1 +0 +x2 +y2 +z2 +6 +6 +6 +7 +7 +7 +… +… +… +2 +1 +0 +2 +1 +0 +2 +1 +0 +23 +22 +23 +22 +23 +22 +… +2 +1 +0 +2 +1 +0 +2 +1 +0 +23 +22 +23 +22 +23 +22 +… +2 +1 +0 +2 +1 +0 +2 +1 +0 +23 +22 +23 +22 +23 +22 +… + + +Zogbi, Liu, et al. Page 7 of 8 +in with a 2-cycle delay relative to the first MAC unit, then +performs the addition W11x1 + W21x2 (17 bits). Fine-grained +pipelining allows this addition to begin on the less significant +bits while the more significant bits of W11x1 and W21x2 are still +being computed by their respective multipliers. This partial sum +is again passed downward, so that the output of the bottom +MAC unit on the column is the dot product ∑ Wi1xi. To support +a 256-element dot product, we include a 24-bit accumulator +inside each 8-bit MAC unit. Each MAC unit also passes the +input xi to the right so that it is broadcast to all units on row i. +Thus, each column computes an independent dot product, with +a 1-cycle delay between columns, to implement an MVM. +VII. PERFORMANCE AND ENERGY +We evaluate a 256×256, 8-bit DW-MTJ systolic array to +match the size and MAC precision of the systolic array in the +Google TPUv1, which operated at a faster clock frequency [38]. +Both systolic arrays complete a new MVM on every clock +cycle. The peak throughput of the TPUv1 is 92 TOPS +(TeraOperations/s), while that of the DW-MTJ systolic array is +10.9 TOPS. Though the DW-MTJ throughput is lower by 9×, +fine-grained pipelining allows the DW-MTJ system to make up +for a much larger speed deficit at the device level. The +switching delay of a DW-MTJ device (4 ns) is ~1000× larger +than that of a CMOS logic gate (~1 ps) at a similar 15 nm +process node [39]. +The device-level energy is a function of both the supply +voltage needed to reset and write the logic gates and the VCMA +voltage needed to create a PMA minimum and attract the DW +to one side. The dependence of the reset energy, energy +resulting from the voltage pulse, on TMR is shown in Fig 8(a) +where the energy is calculated as an average between its 8 +different configurations. Higher TMR causes the resistance +values to be higher for the AP state of the MTJ; thus, there is +lower current in the device, lowering the average energy. The +range of TMR values with 100% correctness for the 3-device +circuit is highlighted in yellow, showing that the optimal TMR +for the device parameters modeled is TMR = 115%. Device +switching energy also increases as ∆Ku increases, shown in Fig +8(b); the optimal 100% correctness ∆Ku = 25 kJ/m3 results in +an applied voltage of VB = 2.5 V. +We developed a simulator to model the functionality and +efficiency of digital circuits constructed from DW-MTJ gates. +Circuit-level energies are based on logic gate concatenation and +VCMA energies from micromagnetic simulations, accounting +for energy differences due to device state (P vs AP) and fanout. +Fig 8(c) shows the energy per 8-bit MAC of a 256×256 DW- +MTJ systolic array, using the device parameters in Table I. At +TMR = 115%, which guarantees reliability, the systolic array’s +efficiency is 4.5 pJ/MAC, or about 0.44 TOPS/W. This is +comparable to state-of-the-art digital CMOS accelerators for +neural network inference [34, 40]. +Fig 8(d) projects how the DW-MTJ systolic array’s +efficiency will scale with improvements in device, notably in +the threshold current density Jth to move the DW. A 10× +reduction in Jth relative to the simulated device improves the +efficiency to 192 fJ/MAC (10.4 TOPS/W), limited by the +VCMA energy. Reduction in the DW threshold current density +below 1010 A/m2 can be accomplished by optimization of the +SOT device geometry and current injection mechanism, as well +as through tighter control of edge roughness and defects in the +thin ferromagnetic strip [41]. +VIII. CONCLUSION +We have demonstrated the ability to perform reliable and +robust concatenation of logic using the three-terminal DW-MTJ +device with the addition of VCMA. This work shows that TMR +and VCMA can be optimized to realistic values to ensure 100% +correctness of logic circuit operations. These logic devices have +shown the ability to function with multiple inputs and outputs +to a single device with no change to the device footprint. Along +with this, there is a wide enough range in TMR that is +achievable in current MTJ processes. Additionally, we have +demonstrated that an 8-bit pipelined MAC can be created from +a 256×256 DW-MTJ systolic array. The resulting energy per +MAC operation is on par with current CMOS accelerators, +while providing a high MAC throughput (10.9 TOPS) despite +the slower switching speed of DW-MTJ devices. These results +show that non-volatile spintronic logic devices can be used to +effectively accelerate edge computing applications while being +able to offer robustness under extreme environments. +ACKNOWLEDGMENT +The authors acknowledge the Texas Advanced Computing +Center (TACC) at The University of Texas at Austin for +providing HPC resources that have contributed to the research +results +reported +within +this +paper. +URL: +http://www.tacc.utexas.edu. The authors acknowledge support +from the National Science Foundation Graduate Research + +Figure 8: (a) Reset energy vs. TMR for a DW-MTJ buffer, +averaged over possible DW-MTJ initial states. Yellow highlight +shows values that have 100% correctness. (b) Energy vs. minimum +PMA of the applied VCMA voltage per logic gate. (c) Energy per +8-bit MAC vs. TMR. The accumulation is 24 bits. Each point is +averaged over 100 random test MACs. (d) Projected energy per 8- +bit MAC vs DW threshold current density at TMR = 115%. The +dot marks the Jth value derived from the parameters in Table I. +(a) +(b) +(d) +TMR (%) +8-bit MAC energy (pJ) +(c) +8-bit MAC energy (fJ) +Jth (A/m2) +109 +1010 +1011 +10 +102 +103 +104 +TMR = 115% +MAC +8b + + +Zogbi, Liu, et al. Page 8 of 8 +Fellowship under Grant No. 2021311125 (S.L.). +This article has been authored by employees of National +Technology & Engineering Solutions of Sandia, LLC under +Contract No. DE-NA0003525 with the U.S. Department of +Energy (DOE). 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Khvalkovskiy et al., "Matching domain-wall configuration and +spin-orbit torques for efficient domain-wall motion," Physical Review B, +vol. 87, no. 2, 2013. + diff --git a/1NFIT4oBgHgl3EQf3ivh/content/tmp_files/load_file.txt b/1NFIT4oBgHgl3EQf3ivh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f8892ad844bf0efa8e84e4859401f11efe956d22 --- /dev/null +++ b/1NFIT4oBgHgl3EQf3ivh/content/tmp_files/load_file.txt @@ -0,0 +1,944 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf,len=943 +page_content='Zogbi, Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Page 1 of 8 Abstract—The domain wall-magnetic tunnel junction (DW- MTJ) is a versatile device that can simultaneously store data and perform computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' These three-terminal devices are promising for digital logic due to their nonvolatility, low-energy operation, and radiation hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Here, we augment the DW- MTJ logic gate with voltage controlled magnetic anisotropy (VCMA) to improve the reliability of logical concatenation in the presence of realistic process variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' VCMA creates potential wells that allow for reliable and repeatable localization of domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The DW-MTJ logic gate supports different fanouts, allowing for multiple inputs and outputs for a single device without affecting area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' We simulate a systolic array of DW-MTJ Multiply- Accumulate (MAC), which uses the nonvolatility of DW-MTJ logic gates to enable fine-grained pipelining and achieve massive parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The DW-MTJ 8-bit systolic array provides comparable throughput and efficiency to state-of-the-art CMOS systolic arrays while being radiation-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' These results improve the feasibility of using domain wall-based processors, especially for extreme-environment applications such as space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Index Terms—Domain Wall, Magnetic Tunnel Junction, VCMA, Logic, In-Memory Computing, Magnetism, Spintronics I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' INTRODUCTION HERE is a pressing need for new computational methods arising from the demand for fast computation and efficient processing of data, which is currently hindered by the bottlenecks of the von Neumann architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Even with continuous improvements in transistor technology, computations are fundamentally hindered by the speed of accessing data, leading to an energy-inefficient architecture [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' To address these bottlenecks, processing digital logic using non-volatile memory presents new opportunities to improve the efficiency and reliability of digital computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' By engineering non-volatile memory for logic, novel digital processors may operate at low voltages, dissipate nearly zero power in static operation, and operate more reliably as compared to conventional CMOS logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Improvements in energy efficiency are important for enabling the deployment of more sophisticated applications, such as artificial intelligence and machine learning algorithms, to edge or remote computing systems that have a limited power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This work was supported by the Laboratory Directed Research and Development Program at Sandia National Laboratories, a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=', for the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This work was also supported by the National Science Foundation Graduate Research Fellowship Program under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' 2021311125 (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Liu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Spintronics is a unique platform for enabling dense and reliable non-volatile memory, digital and analog in-memory computing, neuromorphic computing, normally-off computing, and new, efficient implementations of digital logic [2-8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' These varying compute architectures derive their energy and performance benefits from a shared foundation: spintronic devices that provide non-volatile data storage, low read and write energy, high write endurance, and back-end-of-the-line compatibility with a CMOS fabrication process [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Additionally, because spintronic devices encode information in magnetization rather than charge, they are known to be more resilient to the effects of radiation (especially ionizing radiation), which makes spintronics particularly attractive for edge computing in space and defense applications [10-14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Several spintronic architectures have been proposed for accelerating digital logic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' These include logic circuits that use a mixture of CMOS and magnetic tunnel junction (MTJ) devices, spin wave majority gates, and multiple logic gates implemented using magnetic domain walls (DWs) [14-24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' A practical magnetics-based digital processor that avoids significant issues with extreme edge computing should be: (1) all-magnetic, with no accessory CMOS inside each logic gate, (2) all-electrical, requiring no external magnetic fields or optical excitation to read and write, (3) cascadable to form large circuits, having current-in/current-out or voltage-in/voltage-out operation without needing to convert data between logic gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Of the proposed architectures, the three-terminal domain wall-magnetic tunnel junction (DW-MTJ) logic gate fulfills all three of the key requirements, while compactly implementing each logic gate (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' inverter, NAND) within a single nano- device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' In this device, a logical bit is encoded in the position of a domain wall (DW) along a ferromagnetic track, which forms the free layer of an MTJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The logic state is read out and transmitted to the next logic gate through the MTJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' In our previous work, DW-MTJ prototypes have been experimentally shown to operate with both spin-transfer torque (STT) and spin- orbit torque (SOT) current input, to function as logical inverters with fanout > 1, to have electrically controllable operation of the DW with < 10% cycling variation, and to build circuits [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' More complex DW-MTJ circuits were benchmarked using N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Zogbi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Liu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Incorvia are with the Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX 78712 USA (e-mail: incorvia@austin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Marinella is with Arizona State University, Tempe, AZ 85287 USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Bennett, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Agarwal, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Xiao are with Sandia National Laboratories, Albuquerque, NM 87123 USA (email: txiao@sandia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='gov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Nicholas Zogbi, Samuel Liu, Christopher H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Bennett, Sapan Agarwal, Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Marinella, Jean Anne C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Incorvia, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Patrick Xiao Massively Parallel Matrix Multiplication Using Voltage Controlled Magnetic Anisotropy Domain Wall Logic T Zogbi, Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Page 2 of 8 micromagnetic simulations and compact device models [23- 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Nevertheless, challenges remain for the reliability of DW- MTJ logic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Prior modeling has relied on DW inertia to sustain the DW motion across a perfectly smooth magnetic track after the cessation of an applied current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' However, in realistic tracks fabricated in scaled process nodes, this inertial motion can be rapidly halted by edge roughness or local material defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Furthermore, the energy and performance of DW-MTJ logic has not been evaluated for the multiply- accumulate (MAC) kernel, which dominates modern edge computing workloads such as machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' In this paper, we demonstrate that voltage-controlled magnetic anisotropy (VCMA) ensures reliable DW-MTJ logic concatenation along magnetic tracks with realistic roughness instead of relying on DW inertial motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' We show using micromagnetic simulations that VCMA can be used to electrically pin the DW at set locations, enabling deterministic and robust switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' We also demonstrate how to implement different logic functions and fanouts with minimal changes to device geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' These device-level results are used to benchmark the energy and performance of a systolic array of DW-MTJ 8-bit MAC units, a highly parallel spatial architecture that computes matrix multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The non-volatility of DW-MTJ logic enables pipelined processing inside the MAC units, greatly increasing parallelism and enabling the processing throughput of DW-MTJ MAC units to approach that of CMOS MAC units, despite the much slower switching speed of a DW-MTJ logic gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' We discuss the scaling requirements on spintronic devices and material properties to enable reliable operation and energy-efficient extreme edge computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' DW-MTJ DEVICE DESIGN The structure and operation of the three-terminal DW-MTJ device is shown in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The DW is the transition region between two oppositely oriented magnetic domains in a ferromagnetic wire, part of which also forms the free layer of an overlying MTJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The ‘1’ and ‘0’ states are encoded by the position of the DW along with the MTJ’s fixed layer orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' In Fig 1(a), the DW is on the left and the MTJ is in a high resistance state, resulting in a low output current when a voltage is applied from CLK to OUT to read the device, interpreted as a ‘0’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The two resistance states of the MTJ are anti-parallel (AP) RAP and parallel (P) RP, and the tunnel magnetoresistance is defined as TMR = (RAP − RP) / RP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' To write the device, current is pulsed through the IN terminal while CLK is grounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' If this current surpasses the threshold current for DW motion, the DW will move in the same direction as the electron flow which moves the DW to the opposite (right) side of the device as shown in Fig 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This results in a high output current when a voltage is applied from CLK to OUT, interpreted as a ‘1’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The DW moves due to a combination of the spin transfer torque (STT) exerted by the spin-polarized current through the magnetic track and by the spin orbit torque (SOT) caused by current flow in the underlying heavy metal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The SOT effect is more dominant due to the lower resistivity of the heavy metal which contains most of the current flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' DW-MTJ logic gates are concatenated by connecting the OUT terminal of one device to the IN terminal of the next device as shown in Fig 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' To transmit a stored bit from Device 1 to Device 2, a voltage pulse is applied to the CLK terminal of Device 1 and current flows through its OUT terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The magnitude of this current depends on the DW position, and thus this current is used to pass an input bit to Device 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' If the current exceeds Device 2’s threshold, its DW moves from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Concurrently, current also flows from the CLK to the IN terminal of Device 1 to reset its DW to the left, causing a current to flow from the OUT to CLK terminal of Device 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This current does not affect the magnetic state of Device 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The combined read-reset scheme is desirable because by not being required to hold its state, the logic gate can be immediately re-used to process another operation [15, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This enables extremely fine-grained pipelining and massively parallel computation as will be described in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The current that is transmitted to Device 2 corresponds to the output bit that was stored in Device 1 until the moment that the DW passes under the MTJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' After this point, Device 1 has been reset and its output current no longer transmits the correct information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Earlier versions of the DW-MTJ used a very short current pulse to transmit the output bit to Device 2 before Device 1 was reset and relied on DW inertia to settle the DWs in both devices to their correct positions after the end of the current pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' On a smooth ferromagnetic track, a DW can be modeled as a massive particle and it can be shown using micromagnetic simulations that the DW continues to move after current is turned off [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' However, in a physical magnetic track, DWs can be easily pinned by local defects or edge roughness, bringing the inertial motion to a premature halt, and causing bit errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' To overcome this challenge and make DW- MTJ logic more feasible for implementation, VCMA terminals, labeled as VCMA, are introduced in Fig 1(a-c) to pull the DW to one of two locations on either side of the OUT terminal after the current pulse is turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Figure 1: DW-MTJ logical buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (a) Side-view of the device when the DW is on the left and the MTJ is in a high resistance state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Orange/blue are oppositely magnetized regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Electron flow direction during a write is indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (b) After a write, the DW is on the right and the MTJ is in a low resistance state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (c) Top-down view of the logic device with varying fanouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' For a fanout of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5, the MTJ length LMTJ is set to the minimum feature size F = 15 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' For a fanout of 1, LMTJ is set to 3F = 45 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' For a fanout of 2, LMTJ is set to 9F = 135 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (d) Side-view of a 1×3 chain of DW-MTJ buffers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Device 1 is being read/reset and Device 2 is being written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Device 0 Device 1 Device 2 (d) IN OUT CLK MgO Heavy Metal Free Layer e– Fixed Layer Insulator VCMA (a) Logical ‘0’ VCMA LMTJ IN 255 nm F CLK VCMA OUT (c) F F VCMA IN OUT CLK (b) Logical ‘1’ VCMA VCMA Zogbi, Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Page 3 of 8 Variable logic gate fanout is important for implementing practical logic circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' To allow the design of multiple fanouts while keeping the threshold current and device footprint constant, the device width is set to the minimum feature size (F = 15 nm) and the magnetic wire length is 17F = 255 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' These parameters are fixed for all logic gate types (buffer, inverter, NAND, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The only parameter that is varied with fanout is the length of the MTJ, LMTJ, shown in Fig 1(c), which is set to F for fanout of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5, 3F for a fanout of 1, and 9F for fanout of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' A fanout of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5 means that the output goes to one of the two input ports of a logic gate such as NAND, thus requiring less current per device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' A fanout of 2 means a single device is concatenated to two devices at its output, thus requiring greater current flow because it needs to supply current to both devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Table I shows the parameters used for micromagnetic modeling of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' For the parameters shown, the threshold current and threshold current density is simulated to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='1 μA and 7×1010 A/m2, respectively, when these devices are modeled with edge roughness, by removing nanometer pieces from the free layer track, and random grain anisotropy to the ferromagnetic track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This is a reasonable value for SOT-driven DW motion in CoFeB with perpendicular magnetic anisotropy (PMA) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Additionally, the sign of the applied voltage to the CLK terminal and the signs of currents that result from the voltage are modified to allow for the correct direction of electron flow to propagate the DW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' VCMA-BASED DW-MTJ CONCATENATION Fig 2(a) shows the effect of VCMA when a voltage VB = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5 V is applied to the VCMA terminals, which are separated from the ferromagnetic wire by an insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This causes a reduction in the local PMA of the ferromagnetic layer under the terminals, creating an energy well that attracts the DW to the minimum PMA point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The VCMA voltage is applied before and after the read/reset current pulse, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The current pulse, if sufficiently high, initiates the DW motion, while the VCMA voltage pulse reliably pins the DW under one of the two VCMA terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' After the VCMA pulse has been applied for a time trest, called the rest time, the DW position can be considered stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' To model how the applied VCMA voltage affects the DW, we compute the electric field at the insulator/CoFeB interface, which changes the number of electrons in the out-of-plane 3d- orbitals of the Fe atoms at the interface [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This changes the PMA of the ferromagnetic free layer, which is expressed by 𝐾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (𝑉") = 𝐾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (0) − 𝜉𝐸 𝑡#$ (1) where 𝐾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (𝑉") is the PMA at an applied voltage VB, 𝐾!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (0) is the PMA when there is no applied voltage to the ferromagnetic free layer, 𝐸 is the electric field being applied to the DW wire, 𝜉 is the VCMA coefficient, and 𝑡#$ is the thickness of the ferromagnetic track where all the parameters are shown in Table I [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' To allow the realization of the electric field at the insulator/ferromagnetic interface, calculations of the electric field were conducted by solving Poisson’s equation using the Jacobi method [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The strength of the voltage VB determines the minimum PMA from VCMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Thus, the higher the voltage, the greater the change in the minimum PMA because there is a stronger electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This results in a difference between the minimum and maximum PMA, denoted as ∆Ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The resulting ∆Ku in Fig 2(a) is 25 kJ/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 2(c) shows DW position as a function of time for a device that is driven by a current through its IN terminal, which comes from another logic gate, with three different values of the applied VCMA voltage VB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The vertical dashed lines show the TABLE II BUFFER CONFIGURATIONS & ENERGIES Initial DW1 Initial MTJ0 Initial MTJ1 Reset Energy (fJ) Right P P 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='2 Right AP P 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='0 Left P AP 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='0 Left AP AP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='8 Figure 2: (a) Distribution of perpendicular magnetic anisotropy along the ferromagnetic free layer when both VCMA terminals have an applied voltage of VB = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The collection of charges on the edges of the ferromagnet are not displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (b) Diagram of VCLK and VCMA pulses with time of tpulse (2 ns) and trest (2 ns), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (c) Simulated DW position vs time for three voltage values when the TMR of the simulated logic devices is 115%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' IN VCMA CLK OUT (a) (c) (b) VCMA Magnetic anisotropy (kJ/m3) TABLE I PARAMETERS USED IN THE MODEL Parameter Value Gilbert damping α 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='05 Saturation magnetization MS 8 ´ 105 A/m Exchange stiffness Aex 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='3 ´ 10-11 J/m Uniaxial anisotropy constant KS(0) 5 ´ 105 J/m3 Spin polarization P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='7 VCMA coefficient ξ 10 pJ/Vm Clock voltage VCLK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='04 V Clock pulse time tpulse 2 ns Device rest time trest 2 ns Track length Lwire 255 nm Track width W 15 nm Ferromagnet thickness tFL 3 nm Ferromagnet resistivity ρFL 500 µΩ⸱cm Heavy metal thickness tHM 7 nm Heavy metal resistivity ρHM 40 µΩ⸱cm Insulator dielectric constant kINS 7 MTJ RA product 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='675 Ω*µm2 MTJ parallel resistance (Fanout (FO) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5) 3 kΩ MTJ length LMTJ FO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5: LMTJ = 15 nm FO 1: LMTJ = 45 nm FO 2: LMTJ = 135 nm MTJ width WMTJ 15 nm Zogbi, Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Page 4 of 8 periods for the sequence, rest/pulse/rest, shown in Fig 2(b) where the duration of the pulses is in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The horizontal dashed lines show the region for the location of the MTJ, 105- 150 nm, and the regions at which VCMA terminals are located, 30-45 nm and 210-225 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' When the VCMA is not applied to the device, the DW stays in approximately the same position after the CLK pulse ceases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Since this DW position is precariously close to the MTJ, the resistance of the MTJ is sensitive to edge roughness variations, noise, and timing imprecision in the CLK pulse, all of which can lead to bit errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' When VCMA is applied after the CLK pulse, the DW reliably moves to a position that is far to the right of the MTJ, so long as the current pulse moves the DW past the midpoint of the MTJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This makes the concatenation of logic devices significantly more robust to process variations and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' SINGLE INPUT/OUTPUT LOGIC Reliable logical concatenation of DW-MTJ devices is verified using the micromagnetic simulation software MuMax3 with parameters from Table I [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The circuit shown in Fig 1(d) is used to demonstrate the concatenation of devices with single input, single output, and fanout of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' A SPICE simulation is used to calculate the Thevenin resistance of the three concatenated DW-MTJ devices, which is then used to calculate the current density through the logic device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' By sweeping the amplitude of the voltage pulse, the threshold current density of the logic device was found along with its resulting voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 3(a) shows the DW position vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' time for Device 1 when Device 1 is reset and its state is transmitted to Device 2, for the three-device circuit in Fig 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The colored curves on the plot show 25 independent simulations with random grain anisotropy on the ferromagnetic track to emulate device-to-device variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 3(b) shows the DW position as a function of time for Device 2 during the same voltage pulse and its DW is driven by current through its IN terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' For the same voltage pulse, the current that goes through the OUT of Device 1 to the IN of Device 2 (to its CLK, grounded) is shown in Fig 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This result shows the successful transfer of a ‘1’ bit from Device 1 to Device 2, which is robust to device-to-device variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 1(d) represents one of eight possible configurations that are possible for fanout of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This is because there are two configurations for the starting positions for Device 1’s DW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' two fixed layer orientations for Device 1, which decides whether the device acts as an inverter or a buffer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' and two MTJ resistances for Device 0, which affects the current through Device 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Table II shows the configurations that result when Device 1 is a buffer and their energies from the applied voltage pulse to the CLK terminal, denoted as reset energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Due to different configurations that are possible that perform the same logic function, these configurations dictate whether DWs are being propagated by the pulse and what the Thevenin resistance is within the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This leads to some configurations being more prone to errors compared to other configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Therefore, it is important to make sure that all of the eight configurations can perform their logic functions correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 4(a-b) shows the functionality of the logic devices by testing the 8 configurations of the buffer/inverter 25 times each for a total of 200 tests per data point to calculate the correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The correctness was calculated by dividing the number of successful tests by the total number of tests for each data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Each of the devices was simulated with random variations in edge roughness and grain anisotropy on the ferromagnetic track, with a maximum anisotropy variation of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5 kJ/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 4(a) shows that successful circuit operation strongly depends on TMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' When the TMR of the MTJs is less than 55%, there is a steep drop-off in correctness: the difference between RP and RAP is too small, which reduces the difference between a ‘high’ and a ‘low’ current seen by Device 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The reduced difference makes it more likely that the DW in Device 2 will erroneously move in response to a ‘low’ current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' There is a range of TMR = 75% - 115% for which there are no errors, but, surprisingly, for TMR > 115% there is a reduction in correctness across the eight configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' For 145% < TMR < 235% the correctness is 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5%, which decreases with increasing TMR until correctness reaches 75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' For high TMR, the MTJ draws very little current in its AP state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' From these lower currents in the AP state, DW propagation in the concatenated device is slowed down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This can cause the DW to not propagate fully during a logical ‘1’ write, leading some logic outputs to incorrectly be ‘0’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' While high TMR is often thought of as the holy grail of MTJs used as memory devices because it makes it easier to distinguish the two MTJ states, here an optimal TMR may need to be tuned as a function of circuit design decisions and can no longer be treated as a linear figure of merit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The range of TMR for 100% correctness is wide enough to be achievable with today’s MTJ processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Figure 4: (a) Correctness for all 8 possible configurations of the 3- device circuit vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' TMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Yellow highlights the range with 100% correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (b) Correctness vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' ∆Ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (b) (a) Figure 3: Simulation results for the three-device circuit in Fig 1(d) with TMR = 115%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' DW position vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' time of (a) Device 1 during its reset pulse and (b) simultaneous DW position vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' time of Device 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (c) Output current density from Device 1 being pulsed to Device 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (c) (a) (b) Zogbi, Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Page 5 of 8 The ∆Ku that is induced by VCMA also affects the logic function correctness, shown in Fig 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The logic is 100% correct for ∆Ku = 25 - 30 kJ/m3, which corresponds to VB = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5 - 3V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Higher values of VB increase the velocity of the DW propagation as shown in Fig 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This reduces the length of the VCMA pulse but requires a higher VCMA voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Thus, the choice of VCMA voltage within the range of 100% correctness can be used to trade off energy consumption and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' While not shown here, there is a limit to maximum ∆Ku as well, when eventually VCMA would make the magnetization in-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' These results show adding VCMA to the DW-MTJ devices results in reliable and robust concatenation into circuits, and that both TMR and VCMA can be reasonably optimized to ensure 100% correctness of the circuit operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' MULTIPLE INPUT/OUTPUT LOGIC As stated previously, to extend logic functions for different fanouts, all that is required is to change the length of the MTJ while no other modifications to the design are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This way, the supply voltage is constant throughout the circuit, and the DW threshold current is also constant for all specified fanouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Table III shows the calculated reset energy for the three fanouts depicted in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' These ranges for reset energies for the different fanouts are scaled based on the RA product of the MTJ which results in a varying amount of current flow through the device depending on the fanout of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' To support multiple input logic functions, a fanout of 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5 can be used for the device that is being reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' If each reset device has a fanout of 1, a logical OR/NOR is realized because this function only requires one input current to be high to propagate the DW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' However, for a logical AND/NAND the reset devices each have a fanout of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5 because both input currents must be high to propagate the DW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 5(a-b) shows a logical AND/NAND when the two buffer devices connected to the IN terminal are a logical ‘1’ and a logical ‘0’ that results in the DW not propagating to the opposite side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 5(a) shows the DW position as a function of time for the AND/NAND logic device that is driven by a current through its IN terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 5(b) shows the two currents that are supplied by the buffers, and total current that is passed through the IN terminal of the AND/NAND device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' If both buffers connected to the IN terminal of the AND/NAND device had a logical ‘1’, Fig 5(c- d), the current is strong enough to propagate the DW to the other side of the ferromagnetic track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 5(c) shows the DW position in an AND/NAND device that is driven by a current through its IN terminal, and Fig 5(d) shows the current that is being driven through the IN terminal of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' DW-MTJ logic devices also support multiple fanout, which allows the output bit to be passed to multiple output devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' To obtain a fanout of 2, the operation is fairly similar to when the fanout of the device is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5, but instead of modifying the input current, the output current is divided in half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Greater fanouts can be achieved through the process of increasing the MTJ length, but the circuit design becomes more challenging since the total length of the logic devices must be increased: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' a fanout of 4 for F = 15 nm requires device length of 27F (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='335 μm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Additionally, if the device length is too long, VCMA could no longer be used effectively since VB applied to the VCMA terminals would cause little to no change in PMA values at the middle of the wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' If the DW were to not propagate fully, it may not be drawn to one of the two sides, causing errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Alternative solutions to obtaining larger fanouts can be achieved by increasing the width of the wire to adjust the RA product or adding resistors to the OUT pin of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' However, this would cause a change in the threshold current for the DW or require resistors for devices with a small fanout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Thus, here we limit the circuit design to a maximum fanout of 2 because we can achieve greater fanouts by using a cascade of buffers with a fanout of 2 as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' PIPELINED DW-MTJ MATRIX MULTIPLICATION The ability of a DW-MTJ device to function both as a logic gate and as a non-volatile memory element allows the device to efficiently implement a form of sequential logic that results in far greater data parallelism than the conventional combinatorial logic used in CMOS processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 6 shows how DW-MTJ buffers, inverters, and NAND gates can be connected to form a full adder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Because data is passed between gates only during clock pulses, buffers are used to keep the data aligned in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Moreover, a logic gate cannot simultaneously receive and transmit data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' To avoid interference, it also cannot receive data while its output gate is being reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Therefore, at any moment, TABLE III FANOUT ENERGIES Fanout Reset Energy (fJ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='3-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='8-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='2 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='6-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='1 Figure 5: (a) Simulated DW position vs time for an AND/NAND device when the input currents are ‘10’ and the TMR of the logic devices is 115%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (b) Input currents going into an AND/NAND device showing a logical ‘1’ input current (blue), logical ‘0’ input current (orange), and the total current being pulsed into the device (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (c) Simulated DW position vs time for an AND/NAND device when the input currents are ‘11’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (d) Input currents going into an AND/NAND device showing the two separate logical ‘1’ input currents (blue, orange) and the total current (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (b) (a) (d) (c) 10 11 1 0 11 Zogbi, Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Page 6 of 8 each gate operates in one of three modes: receive, transmit, or standby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Gates that are aligned in their logical depth operate in the same mode, shown by the yellow, blue, and green bands in Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' A three-phase clock (Φ1, Φ2, and Φ3) switches each band of gates from one operational mode to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Each mode lasts for one phase of the clock period (4 ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' All gates cycle through all three modes within one full clock period (12 ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Due to its non-volatility, a DW-MTJ logic gate does not need to hold onto its state after it has transmitted its output to the next gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Each gate can therefore receive and process a new set of inputs in the very next clock phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This allows a single logic circuit to concurrently process many independent operations, pipelined through the stages of computation on each clock cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The data buffering that is needed for pipelining is implemented by the DW-MTJ logic gates themselves, and the size of the pipeline stage is a window of logical depth three as shown in Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This fine-grained pipelining allows concurrent computation on much more data than standard instruction-level pipelining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' It enables high throughput while only fitting a small number of sequential operations into a clock cycle, which compensates for the relatively slow switching speed of a DW- MTJ gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Pipelining at this granularity is possible in CMOS, but would be impractical due to the area and power consumption of the many dedicated pipeline buffers required [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' However, it comes at no cost in a DW-MTJ processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' A disadvantage of this fine-grained pipelining is that the cost of flushing the pipeline upon an interrupt is high, because a large amount of data is stored by the DW-MTJ gates in the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This makes the logic scheme a poor fit for general- purpose microprocessors where pipeline flushes are common (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' due to branch mispredictions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' However, fine-grained pipelining benefits hardware accelerators that perform specialized computation and process data in large blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' In this scenario, pipeline flushes are non-existent or extremely rare by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' A notable example is accelerators for matrix vector multiplication (MVM), which are growing in importance and popularity due to the growth of workloads implementing machine learning algorithms [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The building block of these accelerators is a unit for computing the Multiply-Accumulate (MAC) operation: D = A×B + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Here, we focus on MACs where A, B, C, and D are multi-bit integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Integer MACs, with a typical resolution of 8 bits, are the main computational primitive for deep neural network inference [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 7(a) shows a DW-MTJ MAC unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The schematic for a MAC with 4-bit multiply and 8-bit accumulate is shown for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This circuit implements a pipelined array multiplier, rather than a bit serial multiplier, to ensure that a full MAC is completed on every clock cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' In a given cycle, one bit of one MAC output is available at every output bit position (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' D8 of MAC 1, D7 of MAC 2, …, D0 of MAC 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This format allows the output terminal “D” of one MAC to be directly connected to the input terminal “C” of another MAC, which has the same format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The 4-bit multiplier has 13 pipeline stages and processes 13 MACs in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Data is buffered by one full clock cycle using a buffer chain, which is a linear chain of three buffers or a tree of buffers with depth three, depending on the needed fanout (1 to 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The AND chain is the same as the buffer chain, but the first element is an AND gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 7(b) shows how MAC units can be concatenated to process MVMs Wx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' We evaluate an 8-bit systolic array architecture to process neural network inference, where the matrix W is usually fixed [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' In a systolic array, inputs are broadcast horizontally while partial sums are accumulated vertically, and processing is pipelined at the MAC level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' We use DW-MTJs to further pipeline the MACs themselves, increasing the MVM throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The MAC unit is augmented so that it stores a fixed operand (an element of W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The DW-MTJ pipelined systolic array operation is shown in Fig 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The first element of the input vector, x1 (8 bits), arrives on the top left MAC unit which stores W11 (8 bits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The product W11x1 (16 bits) is passed downward to the “C” input port of the next MAC unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This unit computes W21x2, where x& is passed Figure 6: Illustration of pipelined logic in a DW-MTJ full adder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Data moves through the circuit one logical stage at a time, and gates in the same stage are driven by the same signal from a three- phase clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Data is pipelined through each set of three stages, which takes one clock cycle to traverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' DW-MTJs serve both as logic gates and as storage elements for pipelining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Φ1 Φ2 Φ3 Φ1 Φ2 Φ3 Pipeline stage 1 Pipeline stage 2 A1 B1 A2 B2 Cin1 S1 Cout1 Computing FA(A1, B1, Cin1) Computing FA(A1, B1, Cin1) Computing FA(A2, B2, Cin2) Clock cycle 1: Clock cycle 2: Figure 7: (a) DW-MTJ MAC unit with 4-bit multiply, 8-bit accumulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (b) DW-MTJ 8-bit MAC systolic array, with 8-bit multiply and 24-bit accumulate to support a 256×256 array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='Pipeline stages ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='& ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='& ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='Zogbi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Page 7 of 8 in with a 2-cycle delay relative to the first MAC unit, then performs the addition W11x1 + W21x2 (17 bits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fine-grained pipelining allows this addition to begin on the less significant bits while the more significant bits of W11x1 and W21x2 are still being computed by their respective multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This partial sum is again passed downward, so that the output of the bottom MAC unit on the column is the dot product ∑ Wi1xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' To support a 256-element dot product, we include a 24-bit accumulator inside each 8-bit MAC unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Each MAC unit also passes the input xi to the right so that it is broadcast to all units on row i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Thus, each column computes an independent dot product, with a 1-cycle delay between columns, to implement an MVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' PERFORMANCE AND ENERGY We evaluate a 256×256, 8-bit DW-MTJ systolic array to match the size and MAC precision of the systolic array in the Google TPUv1, which operated at a faster clock frequency [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Both systolic arrays complete a new MVM on every clock cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The peak throughput of the TPUv1 is 92 TOPS (TeraOperations/s), while that of the DW-MTJ systolic array is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='9 TOPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Though the DW-MTJ throughput is lower by 9×, fine-grained pipelining allows the DW-MTJ system to make up for a much larger speed deficit at the device level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The switching delay of a DW-MTJ device (4 ns) is ~1000× larger than that of a CMOS logic gate (~1 ps) at a similar 15 nm process node [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The device-level energy is a function of both the supply voltage needed to reset and write the logic gates and the VCMA voltage needed to create a PMA minimum and attract the DW to one side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The dependence of the reset energy, energy resulting from the voltage pulse, on TMR is shown in Fig 8(a) where the energy is calculated as an average between its 8 different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Higher TMR causes the resistance values to be higher for the AP state of the MTJ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' thus, there is lower current in the device, lowering the average energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The range of TMR values with 100% correctness for the 3-device circuit is highlighted in yellow, showing that the optimal TMR for the device parameters modeled is TMR = 115%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Device switching energy also increases as ∆Ku increases, shown in Fig 8(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' the optimal 100% correctness ∆Ku = 25 kJ/m3 results in an applied voltage of VB = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' We developed a simulator to model the functionality and efficiency of digital circuits constructed from DW-MTJ gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Circuit-level energies are based on logic gate concatenation and VCMA energies from micromagnetic simulations, accounting for energy differences due to device state (P vs AP) and fanout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 8(c) shows the energy per 8-bit MAC of a 256×256 DW- MTJ systolic array, using the device parameters in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' At TMR = 115%, which guarantees reliability, the systolic array’s efficiency is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='5 pJ/MAC, or about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='44 TOPS/W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This is comparable to state-of-the-art digital CMOS accelerators for neural network inference [34, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Fig 8(d) projects how the DW-MTJ systolic array’s efficiency will scale with improvements in device, notably in the threshold current density Jth to move the DW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' A 10× reduction in Jth relative to the simulated device improves the efficiency to 192 fJ/MAC (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='4 TOPS/W), limited by the VCMA energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Reduction in the DW threshold current density below 1010 A/m2 can be accomplished by optimization of the SOT device geometry and current injection mechanism, as well as through tighter control of edge roughness and defects in the thin ferromagnetic strip [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' CONCLUSION We have demonstrated the ability to perform reliable and robust concatenation of logic using the three-terminal DW-MTJ device with the addition of VCMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This work shows that TMR and VCMA can be optimized to realistic values to ensure 100% correctness of logic circuit operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' These logic devices have shown the ability to function with multiple inputs and outputs to a single device with no change to the device footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Along with this, there is a wide enough range in TMR that is achievable in current MTJ processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Additionally, we have demonstrated that an 8-bit pipelined MAC can be created from a 256×256 DW-MTJ systolic array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The resulting energy per MAC operation is on par with current CMOS accelerators, while providing a high MAC throughput (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='9 TOPS) despite the slower switching speed of DW-MTJ devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' These results show that non-volatile spintronic logic devices can be used to effectively accelerate edge computing applications while being able to offer robustness under extreme environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' ACKNOWLEDGMENT The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='tacc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The authors acknowledge support from the National Science Foundation Graduate Research Figure 8: (a) Reset energy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' TMR for a DW-MTJ buffer, averaged over possible DW-MTJ initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Yellow highlight shows values that have 100% correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (b) Energy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' minimum PMA of the applied VCMA voltage per logic gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (c) Energy per 8-bit MAC vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' TMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The accumulation is 24 bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Each point is averaged over 100 random test MACs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (d) Projected energy per 8- bit MAC vs DW threshold current density at TMR = 115%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The dot marks the Jth value derived from the parameters in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' (a) (b) (d) TMR (%) 8-bit MAC energy (pJ) (c) 8-bit MAC energy (fJ) Jth (A/m2) 109 1010 1011 10 102 103 104 TMR = 115% MAC 8b Zogbi, Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Page 8 of 8 Fellowship under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' 2021311125 (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' This article has been authored by employees of National Technology & Engineering Solutions of Sandia, LLC under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' DE-NA0003525 with the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' Department of Energy (DOE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The employee owns all right, title and interest in and to the article and is solely responsible for its contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} +page_content=' The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this article or allow others to do so, for United States 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFIT4oBgHgl3EQf3ivh/content/2301.11382v1.pdf'} diff --git a/2tAyT4oBgHgl3EQfovik/content/tmp_files/2301.00513v1.pdf.txt b/2tAyT4oBgHgl3EQfovik/content/tmp_files/2301.00513v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e32a8450e725a5d5e36a6ce168a770314b2e0980 --- /dev/null +++ b/2tAyT4oBgHgl3EQfovik/content/tmp_files/2301.00513v1.pdf.txt @@ -0,0 +1,1604 @@ +arXiv:2301.00513v1 [hep-th] 2 Jan 2023 +Holographic entanglement entropy and complexity for the excited states of +holographic superconductors +Dong Wang1, Xiongying Qiao1, Qiyuan Pan1,3∗, Chuyu Lai2†, and Jiliang Jing1,3‡ +1Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, +Synergetic Innovation Center for Quantum Effects and Applications, +and Department of Physics, Hunan Normal University, Changsha, Hunan 410081, China +2 Center for Astrophysics, School of Physics and Materials Science, +Guangzhou University, Guangzhou 510006, China and +3Center for Gravitation and Cosmology, College of Physical Science and Technology, +Yangzhou University, Yangzhou 225009, China +Abstract +We investigate the holographic entanglement entropy (HEE) and holographic subregion complex- +ity (HSC) for the holographic superconductors in the Einstein gravity and Einstein-Gauss-Bonnet +gravity. For the ground state and excited states, we observe that, in the Einstein gravity, the HSC +decreases as the temperature increases and the normal phase has a smaller HSC than the supercon- +ducting phase, which is the opposite to the behavior of the HEE. And for a given temperature T in +the superconducting phase, the higher excited state leads to a lager value of the HEE but a smaller +value of the HSC. However, the Einstein-Gauss-Bonnet gravity has significantly different effect on +the HSC, while the HEE always increases monotonously with the increase of the temperature and its +value in the normal phase always larger than that in the superconducting phase. The HEE and HSC +provide richer physics in the phase transition and the condensate of the scalar hair for holographic +superconductors with excited states. +PACS numbers: 11.25.Tq, 04.70.Bw, 74.20.-z +∗ panqiyuan@hunnu.edu.cn +† laichuyu@gzhu.edu.cn +‡ jljing@hunnu.edu.cn + +2 +I. +INTRODUCTION +The anti-de Sitter/conformal field theory (AdS/CFT) correspondence, more generally the gauge/gravity +duality [1–3], which relates a weakly coupling gravity theory in a (d + 1)-dimensional spacetime to a strongly +coupling field theory on the d-dimensional boundary, has been widely applied to study the strongly correlated +systems in the theoretical condensed matter physics. One of its most remarkable and successful applications +is providing a holographically dual description of a high temperature superconducting phase transition. Holo- +graphic superconductors can be constructed by coupling an AdS black hole with the charged field and U(1) +gauge fields. When the Hawking temperature is decreased to some critical value, the black hole background +becomes unstable against perturbations and gets hair by condensing some fields. According to the AdS/CFT +duality, this hairy black hole solution can be regarded as a superconducting phase. The first simple model +of the s-wave holographic superconductor was built by Hartnoll et al. [4, 5]. By considering the Yang-Mills +theory/Maxwell complex vector field model or the charged tensor field in the bulk, one can get the p-wave +holographic superconductors with a vector order parameter [6, 7] and d-wave holographic superconductors +with a tensor order parameter [8, 9]. Until now, a lot of holographic superconductor models have been con- +structed and have attracted considerable interest for their potential applications to the condensed matter +physics, see Refs. [10–13] and references therein. +Most of the works of holographic superconductors focus on the ground state, which is the first state to +condense. +Since there are many novel and important properties showing up in the excited states for su- +perconducting materials in condensed matter systems [14–17], it is interesting and significant to explore the +holographic superconductors with excited states. As a first step, Wang et al. constructed a novel family of +solutions of holographic superconductor with excited states in the probe limit where the backreaction of the +matter fields on the spacetime metric is neglected [18], and pointed out that the excited states of the holo- +graphic superconductors could be related to the metastable states of the mesoscopic superconductors [19, 20]. +Subsequently, they built a fully backreaction holographic model of superconductor with excited states [21]. +Qiao et al. developed a general analytic technique by including more higher order terms in the expansion of +the trial function to investigate the excited states of the holographic dual models in the backgrounds of AdS +black hole [22] and AdS soliton [23]. Li et al. investigated the non-equilibrium dynamical transition process +between the excited states of holographic superconductors [24]. Along this line, there have been accumulated + +3 +interest to study various holographic superconductors with excited states [25–29]. +Within the AdS/CFT duality, the quantities in the boundary field theories are related to certain geometric +quantities in the bulk spacetime. Two quantities introduced from the quantum information theory, which play +important roles in investigating the quantum gravity and quantum field theory, are the entanglement entropy +and complexity. The entanglement entropy is a powerful tool to probe the phase transitions and keep track of +the degrees of freedom in a strongly coupled system. Holographically it can be computed by Ryu-Takayanagi +formula [30, 31], which states that the entanglement entropy of CFTs is associated with the minimal area +surface in the gravity side, namely +S = Area(γA) +4GN +, +(1) +where GN is the Newtonian constant in the dual gravity theory, and γA is the Ryu-Takayanagi minimal area +surface in the bulk, which shares the same boundary ∂A with the subregion A. Since this dual description +of the entanglement entropy has been checked for several cases, it can be applied to study the properties of +holographic superconductors. The initial work was done by Albash and Johnson who evaluated the holographic +entanglement entropy (HEE) in the s-wave holographic superconductor [32]. Subsequently, the HEE in various +superconducting phase transition models has also been studied [33–41]. The entanglement entropy turns out +to be a good probe to investigate the critical points and the order of the holographic phase transitions, and +provides us new insights into the quantum structure of spacetime. +However, the entanglement entropy is not enough to understand the rich geometric structures that exist +behind the horizon because it only grows for a very short time. Then the holographic dual of the complexity, +which essentially describes the minimal number of gates of any quantum circuit to obtain a desired target +state from a reference state, has recently been presented by Susskind [42]. The computation of the complexity +in holography is refined into two concrete conjectures. One is known as “complexity=volume” (CV) conjec- +ture [43, 44], which proposes that the holographic complexity is proportional to the volume of the extremal +codimension-one bulk hypersurface which meets the asymptotic boundary on the desired time slice. The other +one is known as “complexity=action” (CA) conjecture [45, 46]. It states that the complexity corresponds to +the on-shell bulk action in the Wheeler-DeWitt (WDW) patch which is the domain of dependence of some +Cauchy surface in the bulk ending on the time slice of the boundary. In this work, we will focus on the holo- +graphic subregion complexity (HSC) proposed by Alishahiha [47], which is another definition of holographic +complexity based on the original CV conjecture. Following Alishahiha’s proposal, we can evaluate the HSC by + +4 +the codimension-one volume of the time-slice of the bulk geometry enclosed by the extremal codimension-two +Ryu-Takayanagi (RT) hypersurface used for the computation of holographic entanglement entropy as +C = V olume(γA) +8πLGN +, +(2) +where γA is the RT surface of the corresponding subregion A, and L is the AdS radius. +Because the complexity measures the difficulty of turning one quantum state into another, it is expected that +the holographic complexity should capture the behavior of phase transition of the boundary field theory, and +there raises an intensive interest in studying the complexity for different types of holographic superconductors. +Many efforts have been made in using the HSC as a probe of phase transition in the s-wave superconductor +[48, 49], p-wave superconductor [50], St¨uckelberg superconductor [51] and superconductor with nonlinear elec- +trodynamics [52, 53]. These works show that the holographic complexity is a good parameter to characterize +the superconducting phase transitions, and behaves in the different way with the entanglement entropy which +means that the two quantities reflect different information of the holographic superconductor systems. +In order to further disclose the properties of the holographic superconductors with excited states, here we are +aiming to study the HEE and HSC for the excited states of holographic superconductors with backreaction +in Einstein gravity. Furthermore, we would like to extend our discussion to the case with the presence of +higher curvature corrections to the Einstein gravity. The Einstein-Gauss-Bonnet gravity is one of the natural +modifications for Einstein gravity by including Gauss-Bonnet term which arises naturally from the low-energy +limit of heterotic string theory [54–57]. Importantly, the presence of Gauss-Bonnet term does not lead to more +than second derivatives of the metric in the corresponding field equations and thus the theory is ghost-free. The +Gauss-Bonnet theory has earned much attentions in holographic studies in the past decades, and the previous +works of the holographic superconductors in the Gauss-Bonnet gravity show that the higher curvature terms +have nontrivial contributions to some universal properties in Einstein gravity, for example see Refs. [58–74]. +Particularly, it was believed that the Gauss-Bonnet term only plays the role in spacetime with the dimension +d ≥ 5 until Glavan and Lin presented a novel 4-dimensional Einstein-Gauss-Bonnet gravity by rescaling the +Gauss-Bonnet coupling constant α → α/(d − 4) and taking the limit d → 4, where the Gauss-Bonnet term +makes an important contribution to the gravitational dynamics [75]. Subsequently, the “regularized” versions +of the 4-dimensional Einstein-Gauss-Bonnet gravity [76–79] and the consistent theory of d → 4 [80] have also +been proposed. In Ref. [81], the authors constructed the (2 + 1)-dimensional Gauss-Bonnet superconductors +in the probe limit, which shows that the critical temperature first decreases then increases as the Gauss- + +5 +Bonnet parameter tends towards the Chern-Simons value in a scalar mass dependent fashion. This subtle +effect of the higher curvature correction on the scalar condensates in the s-wave superconductor in (2 + 1)- +dimensions is different from the findings in the higher-dimensional superconductors [58–60]. In this work, +we will also investigate the HEE and HSC for the excited states of the superconductors in the 4-dimensional +Gauss-Bonnet gravity away from the probe limit, which can present us some interesting details of excited +states of superconductors under the impact of the Gauss-Bonnet curvature correction. +This work is organized as follows. In section II, we investigate the entanglement entropy and complexity +for the excited states of the holographic superconductors with fully backreaction in the Einstein gravity. In +section III, we extend the discussion to the entanglement entropy and complexity of the fully backreacting +holographic superconductors in the 4-dimensional Einstein-Gauss-Bonnet gravity. In section IV, we conclude +our results. +II. +ENTANGLEMENT ENTROPY AND COMPLEXITY FOR EXCITED STATES OF +HOLOGRAPHIC SUPERCONDUCTORS IN EINSTEIN GRAVITY +A. +Holographic model and condensates of the scalar field +In a d-dimensional Einstein gravity, we consider a Maxwell field and a charged complex scalar field coupled +via the action +S = +� +ddx√−g +� 1 +2κ2 (R − 2Λ) − 1 +4FµνF µν − |∇ψ − iqAψ|2 − m2|ψ|2 +� +, +(3) +where κ2 = 8πGd is the gravitational constant and Λ = −(d − 1)(d − 2)/(2L2) is the cosmological constant. +A and ψ respectively represent the gauge field and a scalar field with mass m and charge q. To include the +backreaction, we choose the following ansatz of the metric for the black hole with a planar symmetric horizon +ds2 = −f(r)e−χ(r)dt2 + dr2 +f(r) + r2hijdxidxj. +(4) +The Hawking temperature of the black hole, which gives the temperature of the holographic superconductor, +is expressed as +TH = f ′(r+)e−χ(r+)/2 +4π +, +(5) +with the radius of the event horizon r+. +Considering the ansatz for the matter fields ψ = ψ(r), A = φ(r)dt, we obtain the equations of motion for +the matter and metric +χ′ + 4κ2r +d − 2 +� +ψ′2 + q2eχφ2ψ2 +f 2 +� += 0, +(6) + +6 +f ′ − +�(d − 1)r +L2 +− (d − 3)f +r +� ++ 2κ2r +d − 2 +� +m2ψ2 + eχφ′2 +2 ++ f +� +ψ′2 + q2eχφ2ψ2 +f 2 +�� += 0, +(7) +φ′′ + +�d − 2 +r ++ χ′ +2 +� +φ′ − 2q2ψ2 +f +φ = 0, +(8) +ψ′′ + +�d − 2 +r ++ f ′ +f − χ′ +2 +� +ψ′ + +�q2eχφ2 +f 2 +− m2 +f +� +ψ = 0, +(9) +where the prime denotes the derivative with respect to the coordinate r. Just as in Ref. [82], we will set q = 1 +and keep κ2 finite when the backreaction is taken into account. +For the normal phase, there is no condensate, i.e., ψ(r) = 0. Thus, we find that χ is a constant and the +analytic solutions to Eqs. (7) and (8) lead to the AdS Reissner-Nordstr¨om black holes +f(r) = +r2 +L2 − +1 +rd−3 +�rd−1 ++ +L2 ++ (d − 3)κ2ρ2 +(d − 2)rd−3 ++ +� ++ +(d − 3)κ2ρ2 +(d − 2)r2(d−3) , +φ(r) = µ − +ρ +rd−3 , +(10) +where µ and ρ are the chemical potential and the charge density in the dual field theory respectively. If κ = 0, +the metric coefficient f goes back to the case of the Schwarzschild AdS black hole. +In order to get the solutions corresponding to the superconducting phase, i.e., ψ(r) ̸= 0, we have to impose +the appropriate boundary conditions. At the horizon r = r+, the metric coefficient χ and scalar field ψ are +regular, but the metric coefficient f and gauge field φ obey φ(r+) = 0 and f(r+) = 0, respectively. Near the +boundary r → ∞, the asymptotic behaviors of the solutions are +χ → 0, f ∼ r2 +L2 , φ ∼ µ − +ρ +rd−3 , ψ ∼ ψ− +rλ− + ψ+ +rλ+ , +(11) +where the coefficients ψ+ and ψ− are related to the vacuum expectation value of the boundary operator O +with the conformal dimension λ± = [(d − 1) ± +� +(d − 1)2 + 4m2L2]/2, respectively. When λ− is larger than +the unitarity bound, both the modes are normalizable, and we may impose boundary condition that either +ψ− and ψ+ vanishes [4, 5]. +From the equations of motion (6)-(9), we can get the useful scaling symmetries and the transformation of +the relevant quantities +r → βr, +(t, xi) → 1 +β (t, xi), +(χ, ψ, L) → (χ, ψ, L), +(φ, µ, T ) → β(φ, µ, T ), +ρ → βd−2ρ, +ψ± → βλ±ψ±, +(12) +with a real positive number β. Thus, we can choose r+ = 1 and L = 1. For concreteness, we focus on the +4-dimensional AdS black hole spacetime, and set the backreaction parameter κ = 0.05 and the mass of the + +7 +scalar field m2L2 = −2 above the Breitenlohner-Freedman (BF) bound (m2L2 ≥ −9/4 for d = 4). In the +following, we will transform the coordinate as r → z = r+/r for simplicity. +n=0 +n=1 +n=2 +n=3 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.0 +0.2 +0.4 +0.6 +0.8 +T +� +() +1 +�+ +� +n=0 +n=1 +n=2 +n=3 +0.00 +0.05 +0.10 +0.15 +0.20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +T +ρ + +1 +λ- +ρ +FIG. 1: The condensates of scalar operators O+ (left) and O− (right) with excited states versus temperature for the +fixed mass m2L2 = −2. In each panel, the blue, red, green and black lines denote the ground (n = 0), first (n = 1), +second (n = 2) and third (n = 3) states, respectively. +TABLE I: The critical temperature Tc of scalar operators O+ and O− with excited states for the fixed mass m2L2 = −2. +n +0 +1 +2 +3 +4 +< O+ > +0.117710ρ1/2 +0.076345ρ1/2 +0.058377ρ1/2 +0.046861ρ1/2 +0.038202ρ1/2 +< O− > +0.225271ρ1/2 +0.092168ρ1/2 +0.066290ρ1/2 +0.052181ρ1/2 +0.042283ρ1/2 +In Fig. 1, we exhibit the condensates of scalar operators ⟨O+⟩ (left) and ⟨O−⟩ (right) as a function of +temperature from the ground state to the third excited state. When the temperature drops below the critical +value Tc, the condensates of the scalar field emerge. +The results of the critical temperature Tc for both +operators from the ground state to the fourth excited states are presented in Table I. It is shown that the +critical temperature Tc of excited states is lower than that of the ground state, which means that the higher +excited state makes it harder for the scalar hair to form. The results are consistent with the findings in +Ref. [21]. Fitting the curves in Fig. 1, we have the condensate behavior of operators with respect to the +temperature as ⟨O±⟩ ∼ (1 − T/Tc)1/2 near the critical point, which tells us that for the ground and excited +states, the superconducting phase transition of the 4-dimensional holographic model with the backreactions +belongs to the second order and the critical exponent of the system takes the mean field value 1/2. +B. +HEE and HSC of the holographic model +In this section, we will numerically study the behaviors of the HEE and HSC in the metal/superconductor +phase transition with excited states, which will give more physics about the superconducting phase transition +of excited states. + +8 +Let us consider a subsystem A with a straight strip geometry which is described by −l/2 ≤ x ≤ l/2 and +−R/2 ≤ y ≤ R/2 (R → ∞). Here l is defined as the size of region A, and R is a regulator which can be set +to infinity. The radial minimal surface γA starts from z = ǫ at x = l/2, and extends into the bulk until it +reaches z = z∗, then returns back to the AdS boundary z = ǫ at x = −l/2, where ǫ is a UV cutoff. Therefore, +the induced metric on the minimal surface takes the form +ds2 +induced = r2 ++ +z2 +�� +1 + +1 +z2f +�dz +dx +�2� +dx2 + dy2 +� +, +(13) +By using the Ryu-Takayanagi formula given in Eq. (1), the entanglement entropy in the strip geometry is +S = +R +4G4 +� +l +2 +− l +2 +r2 ++ +z2 +� +1 + +1 +z2f +�dz +dx +�2 +dx. +(14) +The minimality condition implies +dz +dx = 1 +z +� +(z4⋆ − z4)f, +(15) +in which the constant z∗ satisfies the condition dz +dx|z=z∗ = 0. Setting x(z∗) = 0, we integrate the condition +(15) and obtain +x(z) = +� z∗ +z +z +� +(z4⋆ − z4)f +dz, +(16) +with x(ǫ → 0) = l/2. After minimizing the area by Eq. (15), the HEE becomes +S = +R +2G4 +� zǫ +ǫ +z2 +∗ +z3� +(z4∗ − z4)f +dz = +R +2G4 +� +s + 1 +ǫ +� +, +(17) +where s is the finite term and 1/ǫ is the divergent term as ǫ → 0 caused by the pure AdS geometry f → z−2 +near the UV cutoff. We can subtract this divergent term from S in Eq. (17), and analyze the physically +important finite part s of the HEE. +Following the proposal given by Eq. (2), the HSC in the strip geometry is +C = +R +4πLG4 +� z∗ +ǫ +x(z)dz +z4f += +R +4πLG4 +� +c + F(z∗) +ǫ2 +� +, +(18) +which also includes a universal term c and a divergent term in the form of F(z∗)/ǫ2 with a function of z∗. Note +that the function F(z∗) has different forms under different situations, we cannot give the general analytical +form of the HSC divergence term and subtract it off to find the universal part of C. Fortunately, the value of +the universal term c is independent of the UV cutoff. So considering two different values of cutoff ǫ1 and ǫ2, +we can numerically compute the value of F(z∗) by +F(z∗) = 4πLG4[C(ǫ1) − C(ǫ2)] +R(ǫ−2 +1 +− ǫ−2 +2 ) +, +(19) + +9 +n=0 +n=1 +n=2 +n=3 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +-0.715 +-0.710 +-0.705 +-0.700 +T +� +s +� +n=0 +n=1 +n=2 +n=3 +0.05 +0.10 +0.15 +0.20 +0.25 +-0.70 +-0.65 +-0.60 +T +� +s +� +0.02 +0� �� +0.06 + + +-0.716 +- + + +-0.712 +-0.710 +n=0 +n=1 +n=2 +n=3 +0.02 +0.04 +0.06 +  +0.10 +0.12 +- +  +- +  +- +!" #$% +- +&' ()* +T ++ +c +, +n=0 +n=1 +n=2 +n=3 +0.05 +0.10 +0.15 +0.20 +0.25 +-0.84 +-0.83 +-0.82 +-0.81 +-0.80 +-0.79 +-0.78 +T +- +c +. +0.03 0.04 0.05 0.06 0.07 0.08 0.09 +-0.789 +-0.788 +-0.787 +-0.786 +-0.785 +FIG. 2: +The HEE and HSC of scalar operators O+ (left) and O− (right) with excited states versus temperature for +the fixed width l√ρ = 1, which shows that the higher excited state leads to a lager value of the HEE but a smaller +value of the HSC for a given temperature in the superconducting phase. In each panel, the blue, red, green and black +dashed lines denote the ground (n = 0), first (n = 1), second (n = 2) and third (n = 3) states of the superconducting +phase, respectively. The magenta solid line denotes the normal phase. +which can help us to pick up the universal term c of the HSC in every situation. +In Fig. 2, we plot the HEE (top) and HSC (bottom) as a function of temperature T for the operators +O+ and O− with excited states, respectively. In each panel, we can see that the HEE and HSC change +discontinuously at the critical point where the curves of the normal phase (solid) intersect with those of the +superconducting phase (dashed). It characterizes that the system undergoes a phase transition from a normal +phase to a superconducting phase as the temperature decreases below the critical value. Moreover, both for +the operator O+ and operator O−, the discontinuous points of the curves of the HEE and HSC correspond to +the critical temperatures from the ground state to the third excited state given in Table I. It is obvious that +the critical temperature Tc of the phase transition decreases as the number of nodes n increases. On the other +hand, there are some differences between the HEE and HSC. Firstly, the HEE increases as the temperature +increases, and its value in the normal phase is larger than that in the superconducting phase. On the contrary, +the HSC decreases with the increasing temperature and always has a smaller value in the normal phase than +that in the superconducting phase. Secondly, it is interesting to find that, for a fixed temperature T , the + +10 +higher excited state has a larger HEE but a smaller HSC in the superconducting phase. +III. +ENTANGLEMENT ENTROPY AND COMPLEXITY FOR EXCITED STATES OF +HOLOGRAPHIC SUPERCONDUCTORS IN 4-DIMENSIONAL EINSTEIN-GAUSS-BONNET +GRAVITY +A. +Holographic model and condensates of the scalar field +In this section, we extend the study to the backreacting holographic superconductor in the 4-dimensional +Einstein-Gauss-Bonnet gravity. We consider the Gauss-Bonnet-AdS black hole solution by using the consistent +d → 4 Einstein-Gauss-Bonnet gravity [80]. In the ADM formalism, we adopt the metric ansatz +ds2 = gµνdxµdxν = −N 2dt2 + γij(dxi + N idt)(dxj + N jdt), +(20) +where N is the lapse function, γij is the spatial metric and N i is the shift vector. We begin with a Maxwell +field and a charged complex scalar field coupled via the action +S = +� +dtd3xN√γ +� +L4D +EGB − 1 +4FµνF µν − |∇ψ − iqAψ|2 − m2|ψ|2 +� +, +(21) +where the Lagrangian density reads +L4D +EGB = +1 +2κ2 +� +2R + 6 +L2 − M + α +2 +� +8R2 − 4RM − M2 − 8 +3 +� +8RijRij − 4RijMij − MijMij +��� +, +(22) +with the Gauss-Bonnet coupling parameter α and the Ricci tensor of the spatial metric Rij. Here, we have +Mij = Rij + Kκ +κKij − KiκKκ +j , +M ≡ Mi +i, +(23) +where Kij ≡ +� +˙γij − 2D(iNj) − γijD2λGF +� +/(2N) with a dot denoting the derivative with respect to the time +t, and Di being the covariant derivative compatible with the spatial metric. +We simply take the following ansatz for the metric +N = +� +f(r)e−χ(r)/2, +N i = 0, +γij = diag +� 1 +f(r), r2, r2 +� +(24) +and consider the matter fields to be real functions of r, i.e., ψ = |ψ(r)| and At = φ(r). So the equations of +motion are +χ′ + +2κ2r3 +r2 − 2αf +� +ψ′2 + q2eχφ2ψ2 +f 2 +� += 0, +(25) +f ′ − +1 +r2 − 2αf +�3r3 +L2 − rf − αf 2 +r +� ++ +κ2r3 +r2 − 2αf +� +m2ψ2 + eχφ′2 +2 ++ f +� +ψ′2 + q2eχφ2ψ2 +f 2 +�� += 0, +(26) + +11 +φ′′ + +�2 +r + χ′ +2 +� +φ′ − 2q2ψ2 +f +φ = 0, +(27) +ψ′′ + +�2 +r + f ′ +f − χ′ +2 +� +ψ′ + +�q2eχφ2 +f 2 +− m2 +f +� +ψ = 0, +(28) +where the prime denotes differentiation in r. When the Gauss-Bonnet parameter α → 0, Eqs. (25)-(28) will +reduce to Eqs. (6)-(9) with d = 4 for the backreacting holographic superconductors investigated in Ref. [82]. +Here, the Hawking temperature has the same form as in (5), which is interpreted as the temperature of the +dual field theory. +For the normal phase, i.e., ψ(r) = 0, we can get the analytic solutions to the field equations (26) and (27) +f(r) = +r2 +2α +� +1 − +� +1 − 4α +L2 +� +1 − r3 ++ +r3 +� ++ 2ακ2ρ2 +r+r3 +� +1 − r+ +r +�� +, +φ(r) = µ − ρ +r . +(29) +In the limit α → 0, the solutions reduce to the case of the 4-dimensional AdS Reissner-Nordstr¨om black hole. +For the superconducting phase, i.e., ψ(r) ̸= 0, the boundary conditions at the horizon and asymptotic AdS +boundary have to be imposed to solve Eqs. (25)-(28). At the horizon r = r+, we still have the regularity +conditions, just as in section II for the Einstein gravity. Near the asymptotic boundary r → ∞, we have +χ → 0, +f ∼ +r2 +L2 +eff +, +φ ∼ µ − ρ +r , +ψ ∼ ψ− +rλ− + ψ+ +rλ+ , +(30) +where the effective asymptotic AdS scale is defined by +L2 +eff = +2α +1 − +� +1 − 4α +L2 +, +(31) +with the characteristic exponents ∆± = (3 ± +� +9 + 4m2L2 +eff)/2. In order to obtain the correct consistent +influence due to α in various condensates for all dimensions, we will choose the mass by selecting the value +of m2L2 +eff, just as pointed out in Ref. [81]. In the following calculation, we fix the mass by m2L2 +eff = −2 +and the backreaction parameter κ = 0.05. And we take the range α ≤ L2/4, i.e., the so-called Chern-Simons +limit, for the Gauss-Bonnet parameter. For simplicity, here we concentrate on the scalar operator O+ since +the behaviors of the HEE or the HSC both for the scalar operators O+ and O− are the same, just as shown +in Fig. 2 for the Einstein gravity. +In Fig. 3, we present the condensates of the scalar operator O+ as a function of the temperature for some +chosen values of α, i.e., α = 0.0001, 0.10, 0.24 and 0.25, in the ground (n = 0), first (n = 1) and second +(n = 2) states, respectively. It is observed that the condensate occurs for O+ with different values of α and + +12 +n=0 +/=0.0001 +1=0.10 +2= +3 4 5 6 +7= +8 9 : ; +< =>? +0.04 +0.06 +0.08 +0.10 +@ AB C +0.0 +DEF +0.4 +0.6 +0.8 +T +ρ +(< +O>) +1 +λ+ +ρ +n=1 +α=0.0001 +α=0.10 +α=GH I J +α=KL M N +PQ +RS +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.0 +TU +V +0.4 +0.6 +0.8 +T +ρ +(< +W>) +1 +λ+ +ρ +n=X +α=0.0001 +α=Y Z[ \ +α=] ^_ ` +α=a bc d +e fg h +0.03 +0.04 +0.05 +0.06 +0.0 +i jk +0.4 +0.6 +0.8 +T +ρ +(< +l>) +1 +λ+ +ρ +FIG. 3: The condensates of the scalar operator O+ versus temperature with the fixed mass m2L2 +eff = −2 for different +Gauss-Bonnet parameters α, i.e., α = 0.0001 (orange), 0.10 (green), 0.24 (red) and 0.25 (blue). The three panels from +left to right represent the ground (n = 0), first (n = 1) and second (n = 2) states, respectively. +n if T < Tc. For small condensate, we see that there is a square root behavior ⟨O+⟩ ∼ (1 − T/Tc)1/2, which +is shown that, for the ground and excited states, the phase transition of the 4D Gauss-Bonnet holographic +superconductors with the backreactions is typical of second order one with the mean field critical exponent +1/2 for all values of α. +n=0 +n=1 +n= +m +0.00 +0.05 +0.10 +0.15 +nopq +rstu +0.05 +0.10 +0.15 +vwxy +α +Tc +ρ +z{|} +~€ + +‚ +ƒ„…† +‡ˆ‰Š +FIG. 4: The critical temperature Tc of the scalar operator O+ as a function of the Gauss-Bonnet parameter α with the +fixed mass m2L2 +eff = −2 for the ground n = 0 (blue), first n = 1 (red) and second n = 2 (green) states, respectively. +In Fig. 4, we give the critical temperature Tc for the condensate of the operator O+ as a function of the +Gauss-Bonnet parameter α with the fixed mass m2L2 +eff = −2 from the ground state to the second excited +state in order to get the effect of the curvature correction on Tc. An interesting feature we can find is that the +critical temperature Tc decreases as α increases, but slightly increases near the Chern-Simons limit α = 0.25. +Furthermore, this non-monotonic behavior of the critical temperature is more pronounced in the ground state +than that in the excited state, which is in good agreement with the results obtained in Ref. [26]. +B. +HEE and HSC of the holographic model +Now we are ready to study the properties of the HEE and HSC for the backreacting holographic supercon- +ductor in the 4-dimensional Einstein-Gauss-Bonnet gravity. Due to the presence of a Gauss-Bonnet term, we + +13 +should use a general formula to calculate the HEE [83–86] +S = −2π +� +ddy√−g +� +∂L +∂Rµρνσ +εµρενσ − +� +α +� +∂2L +∂Rµ1ρ1ν1σ1∂Rµ2ρ2ν2σ2 +� +α +2Kλ1ρ1σ1Kλ2ρ2σ2 +qα + 1 +× +� +(nµ1µ2nν1ν2 − εµ1µ2εν1ν2)nλ1λ2 + (nµ1µ2εν1ν2 + εµ1µ2nν1ν2)ελ1λ2 +�� +, +(32) +where nµν and εµν reduce to the metric and Levi-Civita tensor in the two orthogonal directions with all +other components vanishing, and qα is treated as “anomaly coefficients”. This will result in the corrections to +expressions (14) and (17). Still, we employ the shooting method to carry out our numerical calculation, and +use s and c to denote the entanglement entropy and complexity of the universal term, respectively. +n=0 +α=0.0001 +0.100 +0.105 +0.110 +0.115 +‹Œ Ž +-0.706 +-0.704 +- ‘’ “” +-0.700 +T +ρ +s +ρ +n=0 +α= +• – — ˜ +0.080 +0.085 +0.090 +0.095 +0.100 +-0.410 +-0.405 +-0.400 +-0.395 +T +ρ +s +ρ +n=0 +α=™š › œ +0.085 +0.090 +0.095 +0.100 +0.105 +-žŸ   +-¡¢£ ¤ +-¥¦§ ¨ +-©ª« ¬ +-­®¯ ° +T +ρ +s +ρ +n=1 +α=0.0001 +0.050 +0.055 +0.060 +0.065 +0.070 +0.075 +-0.716 +-0.715 +-0.714 +-0.713 +T +ρ +s +ρ +n=1 +α=± ² +³ ´ µ +0.030 +0.035 +0.040 +0.045 +0.050 +0.055 +0.060 +- +¶ ·¸ ¹º +-0.390 +-0.388 +-0.386 +-0.384 +- +» ¼½ ¾¿ +-0.380 +-0.378 +T +ρ +s +ρ +n=1 +α= +À Á Âà +Ä ÅÆÇ +0.03 +0.04 +0.05 +0.06 +-0.350 +-0.345 +-0.340 +-0.335 +-0.330 +-ÈÉÊ ËÌ +-ÍÎÏ ÐÑ +T +ρ +s +ρ +n= +Ò +α=0.0001 +0.050 +ÓÔ +ÕÖ × +0.054 +0.056 +0.058 +-0.7156 +-0.7154 +-Ø ÙÚ ÛÜ Ý +-0.7150 +-0.7148 +-0.7146 +T +ρ +s +ρ +n= +Þ +α=ß à áâ ã ä +åæ +çè é +êëì íî +0.030 +0.035 +0.040 +0.045 +-0.386 +-0.384 +- +ï ðñ òó +-0.380 +-0.378 +T +ρ +s +ρ +n= +ô +α=õö ÷ ø +ùú ûü ý +þ ÿ0 �� +0.030 +0.035 +0.040 +0.045 +-0.345 +-0.340 +-0.335 +-0.330 +T +ρ +s +ρ +FIG. 5: +The HEE of the scalar operator O+ versus temperature from the ground state (n = 0) to the second excited +state (n = 2) with a fixed width l√ρ = 1 for different Gauss-Bonnet parameters α, which shows that the HEE always +increases monotonously with the increase of the temperature and its value in the normal phase always larger than that +in the superconducting phase. The solid (blue) lines denote the normal phase and the dashed (red) lines are for the +superconducting phase. +Fig. 5 shows our results from the ground state (n = 0) to the second excited state (n = 2) for the HEE of the +scalar operator O+ as a function of the temperature T , respectively. In each panel, the critical temperature Tc +of the system can be read from the joint point of the solid line for the normal phase and the dashed line for the +superconducting phase. For example, in the ground state (n = 0), we have Tc/√ρ = 0.117705 for α = 0.0001 +(top-left panel), Tc/√ρ = 0.100418 for α = 0.24 (top-middle panel) and Tc/√ρ = 0.105064 for α = 0.25 (top- + +14 +right panel), which are consistent with those in Fig. 4. It is obvious that, for the ground state, the critical +temperature first decreases and then increases as α approaches to the Chern-Simons limit. In addition, for the +ground state and excited states, we can always find that the value of the HEE in the superconducting phase +is less than that in the normal phase when T < Tc, which is independent of the Gauss-Bonnet parameter α. +This behavior of the HEE is due to the fact that the condensate turns on at the critical temperature and the +formation of Cooper pairs makes the degrees of freedom decrease in the superconducting phase. While we fix +the Gauss-Bonnet parameter α, we can see that the value of the HEE becomes larger as the number of nodes +n increases. +n=0 +α=0.0001 +0.110 +� �� �� +0.114 +0.116 +0.118 +��� + +-0.7930 +- +    +-    +-0.7915 +T +ρ +c +ρ +n=0 +α=   +0.085 +0.090 +0.095 +0.100 +-0.3689 +-0.3688 +-0.3687 +-0.3686 +-0.3685 +-0.3684 +T +ρ +c +ρ +n=0 +α=   + ! "# +0.04 +0.06 +0.08 +0.10 +-$%& ' +-()* + +-,-. / +-123 4 +-567 8 +-0.18 +T +ρ +c +ρ +n=1 +α=0.0001 +0.050 +0.055 +0.060 +0.065 +0.070 +0.075 +-0.7880 +-0.7875 +-0.7870 +-0.7865 +-0.7860 +T +ρ +c +ρ +n=1 +α=9 : ;< = +0.030 +0.035 +0.040 +0.045 +0.050 +0.055 +0.060 +- +> ?@ AB C +- +D EF GH I +- +J KL MN O +- +P QR ST U +- +V WX YZ [ +T +ρ +c +ρ +n=1 +α=\ ]^ _ +` abc +0.03 +0.04 +0.05 +0.06 +- +def gh +-ijk lm +- +nop qr +-stu vw +- +xyz {| +-}~ € +T +ρ +c +ρ +n=‚ +α=0.0001 +ƒ „… †‡ +0.030 +0.035 +0.040 +0.045 +0.050 +0.055 +0.060 +-0.7866 +-0.7864 +-ˆ ‰Š ‹Œ  +-0.7860 +-0.7858 +-0.7856 +T +ρ +c +ρ +n=Ž +α=  ‘’ “” +•– +—˜ ™ +š› +œ ž +0.030 +0.035 +0.040 +0.045 +-0.3185 +-0.3184 +-0.3183 +- +Ÿ  ¡ ¢£ ¤ +-0.3181 +T +ρ +c +ρ +n=¥ +α=¦ § ¨© +ª« ¬­ ® +¯° ±² ³ +0.030 +0.035 +0.040 +0.045 +- +´µ¶ ·¸ +- +¹º» ¼½ +-¾¿À Á +-ÃÄÅ ÆÇ +-ÈÉÊ ËÌ +-ÍÎÏ ÐÑ +T +ρ +c +ρ +FIG. 6: +The HSC of the scalar operator O+ versus temperature from the ground state (n = 0) to the second excited +state (n = 2) with a fixed width l√ρ = 1 for different Gauss-Bonnet parameters α, which shows that the Gauss-Bonnet +parameter has a more subtle effect on the HSC when compared to the HEE. The solid (blue) lines denote the normal +phase and the dashed (red) lines are for the superconducting phase. +In Fig. 6, we plot the HSC of the scalar operator O+ as a function of the temperature T , and find that +the curve of the HSC in the normal phase and the one in the superconducting phase intersect at the same +critical temperature as that reflected by the HEE in Fig. 5. It means that the HSC is able to capture the +emergence of the phase transitions in the ground state and excited states. What is noteworthy is that the +Gauss-Bonnet parameter α has an interesting effect on the relation between the HSC and the temperature, +which can be seen in the ground state and excited states. Obviously, there is a threshold αt of the Gauss- + +15 +Bonnet parameter. When α < αt, the value of the HSC decreases as the temperature increases. And the +value of the HSC as n increases for the fixed α, which agrees with the finding shown in the bottom-left panel +of Fig. 2. At the threshold α = αt, with the increasing T/√ρ, the value of the HSC first decreases and then +increases. It should be noted that the threshold becomes larger with the higher excited state, i.e., αt = 0.2400 +for the ground state n = 0, αt = 0.2470 for the first state n = 1 and αt = 0.2478 for the second state +n = 2. Whereas as α goes up to the Chern-Simons limit α = 0.25, this non-monotonic behavior of the HSC +will convert to a monotonic increasing function of the temperature, which is contrary to the case of α < αt. +Besides, we find that the normal phase always has a smaller HSC than the superconducting phase except for +the situation of the Chern-Simons limit, namely, the value of the HSC in the normal phase is larger than +that in the superconducting phase for α = 0.25. Under the influence of the Gauss-Bonnet parameter, these +special features of the HSC with respect to the temperature imply that the higher curvature correction makes +a difference to the properties of the spacetime and changes the geometric structure. +Ò +Ó =0.80 +Ô +Õ =1.00 +Ö +× =ØÙÚÛ +n=0 +0.00 +0.05 +0.10 +0.15 +ÜÝ +Þß +à áâã +-0.8 +-0.7 +-0.6 +-0.5 +-0.4 +-0.3 +α +s +ρ +ä +ρ =0.80 +å +ρ =1.00 +æ +ρ = +çèéê +n=1 +0.00 +0.05 +0.10 +0.15 +ë ìíî +ï ðñ ò +-0.8 +-0.7 +-0.6 +-0.5 +-0.4 +-0.3 +α +s +ρ +ó +ρ =0.80 +ô +ρ =1.00 +õ +ρ =ö÷ øù +n=ú +0.00 +0.05 +0.10 +0.15 +û üý þ +ÿ0� � +-0.8 +-0.7 +-0.6 +-0.5 +-0.4 +-0.3 +α +s +ρ +FIG. 7: +The HEE of the scalar operator O+ as a function of the Gauss-Bonnet parameter α at the temperature +T/√ρ = 0.02 with some chosen values of the widths, i.e., l√ρ = 0.80 (blue), l√ρ = 1.00 (red) and l√ρ = 1.20 (green). +The three panels from left to right represent the ground (n = 0), first (n = 1) and second (n = 2) states, respectively. +l +ρ =0.80 +� +ρ =1.00 +� +ρ = +1 �� � +n=0 +0.00 +0.05 +0.10 +0.15 +�� +� +  +-0.9 +-0.8 +-0.7 +-0.6 +-0.5 +-0.4 +-0.3 +α +c +ρ + +ρ =0.80 + +ρ =1.00 + +ρ = + +n=1 +0.00 +0.05 +0.10 +0.15 +  +   +-0.9 +-0.8 +-0.7 +-0.6 +-0.5 +-0.4 +-0.3 +α +c +ρ + +ρ =0.80 + +ρ =1.00 + +ρ = +!" #$ +n=2 +0.00 +0.05 +0.10 +0.15 +% &' ( +)*+ , +-0.9 +-0.8 +-0.7 +-0.6 +-0.5 +-0.4 +-0.3 +α +c +ρ +FIG. 8: +The HSC of the scalar operator O+ as a function of the Gauss-Bonnet parameter α at the temperature +T/√ρ = 0.02 with some chosen values of the widths, i.e., l√ρ = 0.80 (blue), l√ρ = 1.00 (red) and l√ρ = 1.20 (green). +The three panels from left to right represent the ground (n = 0), first (n = 1) and second (n = 2) states, respectively. +On the other hand, for the fixed temperature, from Figs. 5 and 6 we find that the larger Gauss-Bonnet +parameter α leads to the larger values of the HEE and HSC. To further illustrate this, in Figs. 7 and 8, we plot +the HEE and HSC as a function of the Gauss-Bonnet parameter α, respectively, from the ground state to the +second excited state at a fixed temperature T/√ρ = 0.02 with different widths, i.e., l√ρ = 0.80, l√ρ = 1.00 + +16 +and l√ρ = 1.20. We observe clearly that, regardless of the width and the number of nodes, the HEE and HSC +increase with the increase of the Gauss-Bonnet parameter α. Moreover, in each panel, the HEE and HSC +become larger as the width increases. +IV. +CONCLUSION +In this work, we first study the HEE and HSC for the excited states of holographic superconductors with +full backreaction in the 4-dimensional Einstein gravity. We note that the changes of the HEE and HSC both +for the scalar operators O− and O+ are discontinuous at the critical temperature Tc, and Tc in the excited +states is lower than that in the ground state, which indicates that the higher excited state makes the scalar +condensate harder to form. The values of Tc reflected by the HEE and HSC are consistent with the results +obtained from the condensate behavior, which means that both the HEE and HSC can be used as good probes +to the superconducting phase transition in the excited state. However, there are some differences between +the HEE and HSC. We observe that, for the ground state and excited states, the value of the HEE in the +normal phase is larger than that in the superconducting phase and increases as the temperature increases, +which is the opposite to the behavior of the HSC, namely, the normal phase always has a smaller HSC than +the superconducting phase and the HSC decreases as the temperature increases. +Meanwhile, for a given +temperature T in the superconducting phase, the higher excited state leads to a lager value of the HEE but +a smaller value of the HSC. +Next, we extend the investigation to the HEE and HSC for the excited states of backreacting superconductors +in the 4-dimensional Einstein-Gauss-Bonnet gravity. One remarkable feature is that the critical temperature +Tc for the scalar operator ⟨O+⟩ decreases as the higher curvature correction α increases, but slightly increases +as α grows to the Chern-Simons limit, and this non-monotonic bahavior of Tc is more pronounced in the ground +state than in the excited state, which can be supported by the findings obtained from both the HEE and HSC. +The other noteworthy feature is that the effect of α on the relation between the HSC and the temperature +is nontrivial in the ground state and excited states, which is a distinguishing property of the HSC and has +not been found in the HEE. Specifically, the HSC always behaves as a monotonic decreasing function of the +temperature till α reaches some threshold αt. At this critical point αt, the HSC changes non-monotonously +with the temperature, i.e., it first decreases and then increases with the increasing temperature. It is shown +that the value of αt will be closer to the Chern-Simons limit in the higher excited state. More interestingly, +when α approaches to the Chern-Simons limit, the HSC will convert to a monotonic increasing function of + +17 +the temperature. Besides, the value of the HSC in the normal phase is less than that in the superconducting +phase for the case of α ≤ αt, but it is just on the contrary for the Chern-Simons limit. Whereas the HEE +always increases monotonously with the increase of the temperature and its value in the normal phase always +larger than that in the superconducting phase, regardless of α. Lastly, we find that, for the ground state +and excited states, the increase of α makes both the HEE and HSC increase, which is independent of the +strip width. 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Myers, Phys. Rev. D 49, 6587 (1994); arXiv:gr-qc/9312023. + diff --git a/2tAyT4oBgHgl3EQfovik/content/tmp_files/load_file.txt b/2tAyT4oBgHgl3EQfovik/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9c96b41ca205f586aee82b8879bebdcc695bf98c --- /dev/null +++ b/2tAyT4oBgHgl3EQfovik/content/tmp_files/load_file.txt @@ -0,0 +1,1413 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf,len=1412 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00513v1 [hep-th] 2 Jan 2023 Holographic entanglement entropy and complexity for the excited states of holographic superconductors Dong Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Xiongying Qiao1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Qiyuan Pan1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='3∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Chuyu Lai2†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' and Jiliang Jing1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='3‡ 1Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Synergetic Innovation Center for Quantum Effects and Applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' and Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Hunan Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Changsha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Hunan 410081,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' China 2 Center for Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' School of Physics and Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Guangzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Guangzhou 510006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' China and 3Center for Gravitation and Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' College of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Yangzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Yangzhou 225009,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' China Abstract We investigate the holographic entanglement entropy (HEE) and holographic subregion complex- ity (HSC) for the holographic superconductors in the Einstein gravity and Einstein-Gauss-Bonnet gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' For the ground state and excited states, we observe that, in the Einstein gravity, the HSC decreases as the temperature increases and the normal phase has a smaller HSC than the supercon- ducting phase, which is the opposite to the behavior of the HEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' And for a given temperature T in the superconducting phase, the higher excited state leads to a lager value of the HEE but a smaller value of the HSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' However, the Einstein-Gauss-Bonnet gravity has significantly different effect on the HSC, while the HEE always increases monotonously with the increase of the temperature and its value in the normal phase always larger than that in the superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The HEE and HSC provide richer physics in the phase transition and the condensate of the scalar hair for holographic superconductors with excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' PACS numbers: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='Tq, 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='Bw, 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='-z ∗ panqiyuan@hunnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='cn † laichuyu@gzhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='cn ‡ jljing@hunnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='cn 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' INTRODUCTION The anti-de Sitter/conformal field theory (AdS/CFT) correspondence, more generally the gauge/gravity duality [1–3], which relates a weakly coupling gravity theory in a (d + 1)-dimensional spacetime to a strongly coupling field theory on the d-dimensional boundary, has been widely applied to study the strongly correlated systems in the theoretical condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' One of its most remarkable and successful applications is providing a holographically dual description of a high temperature superconducting phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Holo- graphic superconductors can be constructed by coupling an AdS black hole with the charged field and U(1) gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' When the Hawking temperature is decreased to some critical value, the black hole background becomes unstable against perturbations and gets hair by condensing some fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' According to the AdS/CFT duality, this hairy black hole solution can be regarded as a superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The first simple model of the s-wave holographic superconductor was built by Hartnoll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' By considering the Yang-Mills theory/Maxwell complex vector field model or the charged tensor field in the bulk, one can get the p-wave holographic superconductors with a vector order parameter [6, 7] and d-wave holographic superconductors with a tensor order parameter [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Until now, a lot of holographic superconductor models have been con- structed and have attracted considerable interest for their potential applications to the condensed matter physics, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' [10–13] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Most of the works of holographic superconductors focus on the ground state, which is the first state to condense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Since there are many novel and important properties showing up in the excited states for su- perconducting materials in condensed matter systems [14–17], it is interesting and significant to explore the holographic superconductors with excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' As a first step, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' constructed a novel family of solutions of holographic superconductor with excited states in the probe limit where the backreaction of the matter fields on the spacetime metric is neglected [18], and pointed out that the excited states of the holo- graphic superconductors could be related to the metastable states of the mesoscopic superconductors [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Subsequently, they built a fully backreaction holographic model of superconductor with excited states [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Qiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' developed a general analytic technique by including more higher order terms in the expansion of the trial function to investigate the excited states of the holographic dual models in the backgrounds of AdS black hole [22] and AdS soliton [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' investigated the non-equilibrium dynamical transition process between the excited states of holographic superconductors [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Along this line, there have been accumulated 3 interest to study various holographic superconductors with excited states [25–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Within the AdS/CFT duality, the quantities in the boundary field theories are related to certain geometric quantities in the bulk spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Two quantities introduced from the quantum information theory, which play important roles in investigating the quantum gravity and quantum field theory, are the entanglement entropy and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The entanglement entropy is a powerful tool to probe the phase transitions and keep track of the degrees of freedom in a strongly coupled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Holographically it can be computed by Ryu-Takayanagi formula [30, 31], which states that the entanglement entropy of CFTs is associated with the minimal area surface in the gravity side, namely S = Area(γA) 4GN , (1) where GN is the Newtonian constant in the dual gravity theory, and γA is the Ryu-Takayanagi minimal area surface in the bulk, which shares the same boundary ∂A with the subregion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Since this dual description of the entanglement entropy has been checked for several cases, it can be applied to study the properties of holographic superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The initial work was done by Albash and Johnson who evaluated the holographic entanglement entropy (HEE) in the s-wave holographic superconductor [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Subsequently, the HEE in various superconducting phase transition models has also been studied [33–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The entanglement entropy turns out to be a good probe to investigate the critical points and the order of the holographic phase transitions, and provides us new insights into the quantum structure of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' However, the entanglement entropy is not enough to understand the rich geometric structures that exist behind the horizon because it only grows for a very short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Then the holographic dual of the complexity, which essentially describes the minimal number of gates of any quantum circuit to obtain a desired target state from a reference state, has recently been presented by Susskind [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The computation of the complexity in holography is refined into two concrete conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' One is known as “complexity=volume” (CV) conjec- ture [43, 44], which proposes that the holographic complexity is proportional to the volume of the extremal codimension-one bulk hypersurface which meets the asymptotic boundary on the desired time slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The other one is known as “complexity=action” (CA) conjecture [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' It states that the complexity corresponds to the on-shell bulk action in the Wheeler-DeWitt (WDW) patch which is the domain of dependence of some Cauchy surface in the bulk ending on the time slice of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In this work, we will focus on the holo- graphic subregion complexity (HSC) proposed by Alishahiha [47], which is another definition of holographic complexity based on the original CV conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Following Alishahiha’s proposal, we can evaluate the HSC by 4 the codimension-one volume of the time-slice of the bulk geometry enclosed by the extremal codimension-two Ryu-Takayanagi (RT) hypersurface used for the computation of holographic entanglement entropy as C = V olume(γA) 8πLGN , (2) where γA is the RT surface of the corresponding subregion A, and L is the AdS radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Because the complexity measures the difficulty of turning one quantum state into another, it is expected that the holographic complexity should capture the behavior of phase transition of the boundary field theory, and there raises an intensive interest in studying the complexity for different types of holographic superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Many efforts have been made in using the HSC as a probe of phase transition in the s-wave superconductor [48, 49], p-wave superconductor [50], St¨uckelberg superconductor [51] and superconductor with nonlinear elec- trodynamics [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' These works show that the holographic complexity is a good parameter to characterize the superconducting phase transitions, and behaves in the different way with the entanglement entropy which means that the two quantities reflect different information of the holographic superconductor systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In order to further disclose the properties of the holographic superconductors with excited states, here we are aiming to study the HEE and HSC for the excited states of holographic superconductors with backreaction in Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Furthermore, we would like to extend our discussion to the case with the presence of higher curvature corrections to the Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The Einstein-Gauss-Bonnet gravity is one of the natural modifications for Einstein gravity by including Gauss-Bonnet term which arises naturally from the low-energy limit of heterotic string theory [54–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Importantly, the presence of Gauss-Bonnet term does not lead to more than second derivatives of the metric in the corresponding field equations and thus the theory is ghost-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The Gauss-Bonnet theory has earned much attentions in holographic studies in the past decades, and the previous works of the holographic superconductors in the Gauss-Bonnet gravity show that the higher curvature terms have nontrivial contributions to some universal properties in Einstein gravity, for example see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' [58–74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Particularly, it was believed that the Gauss-Bonnet term only plays the role in spacetime with the dimension d ≥ 5 until Glavan and Lin presented a novel 4-dimensional Einstein-Gauss-Bonnet gravity by rescaling the Gauss-Bonnet coupling constant α → α/(d − 4) and taking the limit d → 4, where the Gauss-Bonnet term makes an important contribution to the gravitational dynamics [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Subsequently, the “regularized” versions of the 4-dimensional Einstein-Gauss-Bonnet gravity [76–79] and the consistent theory of d → 4 [80] have also been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' [81], the authors constructed the (2 + 1)-dimensional Gauss-Bonnet superconductors in the probe limit, which shows that the critical temperature first decreases then increases as the Gauss- 5 Bonnet parameter tends towards the Chern-Simons value in a scalar mass dependent fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' This subtle effect of the higher curvature correction on the scalar condensates in the s-wave superconductor in (2 + 1)- dimensions is different from the findings in the higher-dimensional superconductors [58–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In this work, we will also investigate the HEE and HSC for the excited states of the superconductors in the 4-dimensional Gauss-Bonnet gravity away from the probe limit, which can present us some interesting details of excited states of superconductors under the impact of the Gauss-Bonnet curvature correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' This work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In section II, we investigate the entanglement entropy and complexity for the excited states of the holographic superconductors with fully backreaction in the Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In section III, we extend the discussion to the entanglement entropy and complexity of the fully backreacting holographic superconductors in the 4-dimensional Einstein-Gauss-Bonnet gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In section IV, we conclude our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' ENTANGLEMENT ENTROPY AND COMPLEXITY FOR EXCITED STATES OF HOLOGRAPHIC SUPERCONDUCTORS IN EINSTEIN GRAVITY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Holographic model and condensates of the scalar field In a d-dimensional Einstein gravity, we consider a Maxwell field and a charged complex scalar field coupled via the action S = � ddx√−g � 1 2κ2 (R − 2Λ) − 1 4FµνF µν − |∇ψ − iqAψ|2 − m2|ψ|2 � , (3) where κ2 = 8πGd is the gravitational constant and Λ = −(d − 1)(d − 2)/(2L2) is the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' A and ψ respectively represent the gauge field and a scalar field with mass m and charge q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' To include the backreaction, we choose the following ansatz of the metric for the black hole with a planar symmetric horizon ds2 = −f(r)e−χ(r)dt2 + dr2 f(r) + r2hijdxidxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (4) The Hawking temperature of the black hole, which gives the temperature of the holographic superconductor, is expressed as TH = f ′(r+)e−χ(r+)/2 4π , (5) with the radius of the event horizon r+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Considering the ansatz for the matter fields ψ = ψ(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' A = φ(r)dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' we obtain the equations of motion for the matter and metric χ′ + 4κ2r d − 2 � ψ′2 + q2eχφ2ψ2 f 2 � = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (6) 6 f ′ − �(d − 1)r L2 − (d − 3)f r � + 2κ2r d − 2 � m2ψ2 + eχφ′2 2 + f � ψ′2 + q2eχφ2ψ2 f 2 �� = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (7) φ′′ + �d − 2 r + χ′ 2 � φ′ − 2q2ψ2 f φ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (8) ψ′′ + �d − 2 r + f ′ f − χ′ 2 � ψ′ + �q2eχφ2 f 2 − m2 f � ψ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (9) where the prime denotes the derivative with respect to the coordinate r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Just as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' [82], we will set q = 1 and keep κ2 finite when the backreaction is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' For the normal phase, there is no condensate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=', ψ(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Thus, we find that χ is a constant and the analytic solutions to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (7) and (8) lead to the AdS Reissner-Nordstr¨om black holes f(r) = r2 L2 − 1 rd−3 �rd−1 + L2 + (d − 3)κ2ρ2 (d − 2)rd−3 + � + (d − 3)κ2ρ2 (d − 2)r2(d−3) , φ(r) = µ − ρ rd−3 , (10) where µ and ρ are the chemical potential and the charge density in the dual field theory respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' If κ = 0, the metric coefficient f goes back to the case of the Schwarzschild AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In order to get the solutions corresponding to the superconducting phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=', ψ(r) ̸= 0, we have to impose the appropriate boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' At the horizon r = r+, the metric coefficient χ and scalar field ψ are regular, but the metric coefficient f and gauge field φ obey φ(r+) = 0 and f(r+) = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Near the boundary r → ∞, the asymptotic behaviors of the solutions are χ → 0, f ∼ r2 L2 , φ ∼ µ − ρ rd−3 , ψ ∼ ψ− rλ− + ψ+ rλ+ , (11) where the coefficients ψ+ and ψ− are related to the vacuum expectation value of the boundary operator O with the conformal dimension λ± = [(d − 1) ± � (d − 1)2 + 4m2L2]/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' When λ− is larger than the unitarity bound, both the modes are normalizable, and we may impose boundary condition that either ψ− and ψ+ vanishes [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' From the equations of motion (6)-(9), we can get the useful scaling symmetries and the transformation of the relevant quantities r → βr, (t, xi) → 1 β (t, xi), (χ, ψ, L) → (χ, ψ, L), (φ, µ, T ) → β(φ, µ, T ), ρ → βd−2ρ, ψ± → βλ±ψ±, (12) with a real positive number β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Thus, we can choose r+ = 1 and L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' For concreteness, we focus on the 4-dimensional AdS black hole spacetime, and set the backreaction parameter κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 and the mass of the 7 scalar field m2L2 = −2 above the Breitenlohner-Freedman (BF) bound (m2L2 ≥ −9/4 for d = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In the following, we will transform the coordinate as r → z = r+/r for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' n=0 n=1 n=2 n=3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='8 T � () 1 �+ � n=0 n=1 n=2 n=3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='4 T ρ 1 λ- ρ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 1: The condensates of scalar operators O+ (left) and O− (right) with excited states versus temperature for the fixed mass m2L2 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In each panel, the blue, red, green and black lines denote the ground (n = 0), first (n = 1), second (n = 2) and third (n = 3) states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' TABLE I: The critical temperature Tc of scalar operators O+ and O− with excited states for the fixed mass m2L2 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' n 0 1 2 3 4 < O+ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='117710ρ1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='076345ρ1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='058377ρ1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='046861ρ1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='038202ρ1/2 < O− > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='225271ρ1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='092168ρ1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='066290ρ1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='052181ρ1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='042283ρ1/2 In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 1, we exhibit the condensates of scalar operators ⟨O+⟩ (left) and ⟨O−⟩ (right) as a function of temperature from the ground state to the third excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' When the temperature drops below the critical value Tc, the condensates of the scalar field emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The results of the critical temperature Tc for both operators from the ground state to the fourth excited states are presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' It is shown that the critical temperature Tc of excited states is lower than that of the ground state, which means that the higher excited state makes it harder for the scalar hair to form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The results are consistent with the findings in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Fitting the curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 1, we have the condensate behavior of operators with respect to the temperature as ⟨O±⟩ ∼ (1 − T/Tc)1/2 near the critical point, which tells us that for the ground and excited states, the superconducting phase transition of the 4-dimensional holographic model with the backreactions belongs to the second order and the critical exponent of the system takes the mean field value 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' HEE and HSC of the holographic model In this section, we will numerically study the behaviors of the HEE and HSC in the metal/superconductor phase transition with excited states, which will give more physics about the superconducting phase transition of excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 8 Let us consider a subsystem A with a straight strip geometry which is described by −l/2 ≤ x ≤ l/2 and −R/2 ≤ y ≤ R/2 (R → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Here l is defined as the size of region A, and R is a regulator which can be set to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The radial minimal surface γA starts from z = ǫ at x = l/2, and extends into the bulk until it reaches z = z∗, then returns back to the AdS boundary z = ǫ at x = −l/2, where ǫ is a UV cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Therefore, the induced metric on the minimal surface takes the form ds2 induced = r2 + z2 �� 1 + 1 z2f �dz dx �2� dx2 + dy2 � , (13) By using the Ryu-Takayanagi formula given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (1), the entanglement entropy in the strip geometry is S = R 4G4 � l 2 − l 2 r2 + z2 � 1 + 1 z2f �dz dx �2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (14) The minimality condition implies dz dx = 1 z � (z4⋆ − z4)f, (15) in which the constant z∗ satisfies the condition dz dx|z=z∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Setting x(z∗) = 0, we integrate the condition (15) and obtain x(z) = � z∗ z z � (z4⋆ − z4)f dz, (16) with x(ǫ → 0) = l/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' After minimizing the area by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (15), the HEE becomes S = R 2G4 � zǫ ǫ z2 ∗ z3� (z4∗ − z4)f dz = R 2G4 � s + 1 ǫ � , (17) where s is the finite term and 1/ǫ is the divergent term as ǫ → 0 caused by the pure AdS geometry f → z−2 near the UV cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' We can subtract this divergent term from S in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (17), and analyze the physically important finite part s of the HEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Following the proposal given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (2), the HSC in the strip geometry is C = R 4πLG4 � z∗ ǫ x(z)dz z4f = R 4πLG4 � c + F(z∗) ǫ2 � , (18) which also includes a universal term c and a divergent term in the form of F(z∗)/ǫ2 with a function of z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Note that the function F(z∗) has different forms under different situations, we cannot give the general analytical form of the HSC divergence term and subtract it off to find the universal part of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Fortunately, the value of the universal term c is independent of the UV cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' So considering two different values of cutoff ǫ1 and ǫ2, we can numerically compute the value of F(z∗) by F(z∗) = 4πLG4[C(ǫ1) − C(ǫ2)] R(ǫ−2 1 − ǫ−2 2 ) , (19) 9 n=0 n=1 n=2 n=3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='710 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='705 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='700 T � s � n=0 n=1 n=2 n=3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='60 T � s � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='02 0� �� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='716 \x0e \x0f\x10\x11\x12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='710 n=0 n=1 n=2 n=3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='06 \x13\x14 \x15\x16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='12 \x17\x18 \x19\x1a\x1b !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='" #$% &\' ()* T + c , n=0 n=1 n=2 n=3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='78 T c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='789 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='788 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='785 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 2: The HEE and HSC of scalar operators O+ (left) and O− (right) with excited states versus temperature for the fixed width l√ρ = 1, which shows that the higher excited state leads to a lager value of the HEE but a smaller value of the HSC for a given temperature in the superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In each panel, the blue, red, green and black dashed lines denote the ground (n = 0), first (n = 1), second (n = 2) and third (n = 3) states of the superconducting phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The magenta solid line denotes the normal phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' which can help us to pick up the universal term c of the HSC in every situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 2, we plot the HEE (top) and HSC (bottom) as a function of temperature T for the operators O+ and O− with excited states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In each panel, we can see that the HEE and HSC change discontinuously at the critical point where the curves of the normal phase (solid) intersect with those of the superconducting phase (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' It characterizes that the system undergoes a phase transition from a normal phase to a superconducting phase as the temperature decreases below the critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Moreover, both for the operator O+ and operator O−, the discontinuous points of the curves of the HEE and HSC correspond to the critical temperatures from the ground state to the third excited state given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' It is obvious that the critical temperature Tc of the phase transition decreases as the number of nodes n increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' On the other hand, there are some differences between the HEE and HSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Firstly, the HEE increases as the temperature increases, and its value in the normal phase is larger than that in the superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' On the contrary, the HSC decreases with the increasing temperature and always has a smaller value in the normal phase than that in the superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Secondly, it is interesting to find that, for a fixed temperature T , the 10 higher excited state has a larger HEE but a smaller HSC in the superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' ENTANGLEMENT ENTROPY AND COMPLEXITY FOR EXCITED STATES OF HOLOGRAPHIC SUPERCONDUCTORS IN 4-DIMENSIONAL EINSTEIN-GAUSS-BONNET GRAVITY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Holographic model and condensates of the scalar field In this section, we extend the study to the backreacting holographic superconductor in the 4-dimensional Einstein-Gauss-Bonnet gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' We consider the Gauss-Bonnet-AdS black hole solution by using the consistent d → 4 Einstein-Gauss-Bonnet gravity [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In the ADM formalism, we adopt the metric ansatz ds2 = gµνdxµdxν = −N 2dt2 + γij(dxi + N idt)(dxj + N jdt), (20) where N is the lapse function, γij is the spatial metric and N i is the shift vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' We begin with a Maxwell field and a charged complex scalar field coupled via the action S = � dtd3xN√γ � L4D EGB − 1 4FµνF µν − |∇ψ − iqAψ|2 − m2|ψ|2 � , (21) where the Lagrangian density reads L4D EGB = 1 2κ2 � 2R + 6 L2 − M + α 2 � 8R2 − 4RM − M2 − 8 3 � 8RijRij − 4RijMij − MijMij ��� , (22) with the Gauss-Bonnet coupling parameter α and the Ricci tensor of the spatial metric Rij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Here, we have Mij = Rij + Kκ κKij − KiκKκ j , M ≡ Mi i, (23) where Kij ≡ � ˙γij − 2D(iNj) − γijD2λGF � /(2N) with a dot denoting the derivative with respect to the time t, and Di being the covariant derivative compatible with the spatial metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' We simply take the following ansatz for the metric N = � f(r)e−χ(r)/2, N i = 0, γij = diag � 1 f(r), r2, r2 � (24) and consider the matter fields to be real functions of r, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=', ψ = |ψ(r)| and At = φ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' So the equations of motion are χ′ + 2κ2r3 r2 − 2αf � ψ′2 + q2eχφ2ψ2 f 2 � = 0, (25) f ′ − 1 r2 − 2αf �3r3 L2 − rf − αf 2 r � + κ2r3 r2 − 2αf � m2ψ2 + eχφ′2 2 + f � ψ′2 + q2eχφ2ψ2 f 2 �� = 0, (26) 11 φ′′ + �2 r + χ′ 2 � φ′ − 2q2ψ2 f φ = 0, (27) ψ′′ + �2 r + f ′ f − χ′ 2 � ψ′ + �q2eχφ2 f 2 − m2 f � ψ = 0, (28) where the prime denotes differentiation in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' When the Gauss-Bonnet parameter α → 0, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (25)-(28) will reduce to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (6)-(9) with d = 4 for the backreacting holographic superconductors investigated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Here, the Hawking temperature has the same form as in (5), which is interpreted as the temperature of the dual field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' For the normal phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=', ψ(r) = 0, we can get the analytic solutions to the field equations (26) and (27) f(r) = r2 2α � 1 − � 1 − 4α L2 � 1 − r3 + r3 � + 2ακ2ρ2 r+r3 � 1 − r+ r �� , φ(r) = µ − ρ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (29) In the limit α → 0, the solutions reduce to the case of the 4-dimensional AdS Reissner-Nordstr¨om black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' For the superconducting phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=', ψ(r) ̸= 0, the boundary conditions at the horizon and asymptotic AdS boundary have to be imposed to solve Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' (25)-(28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' At the horizon r = r+, we still have the regularity conditions, just as in section II for the Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Near the asymptotic boundary r → ∞, we have χ → 0, f ∼ r2 L2 eff , φ ∼ µ − ρ r , ψ ∼ ψ− rλ− + ψ+ rλ+ , (30) where the effective asymptotic AdS scale is defined by L2 eff = 2α 1 − � 1 − 4α L2 , (31) with the characteristic exponents ∆± = (3 ± � 9 + 4m2L2 eff)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In order to obtain the correct consistent influence due to α in various condensates for all dimensions, we will choose the mass by selecting the value of m2L2 eff, just as pointed out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In the following calculation, we fix the mass by m2L2 eff = −2 and the backreaction parameter κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' And we take the range α ≤ L2/4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=', the so-called Chern-Simons limit, for the Gauss-Bonnet parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' For simplicity, here we concentrate on the scalar operator O+ since the behaviors of the HEE or the HSC both for the scalar operators O+ and O− are the same, just as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 2 for the Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 3, we present the condensates of the scalar operator O+ as a function of the temperature for some chosen values of α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=', α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='24 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='25, in the ground (n = 0), first (n = 1) and second (n = 2) states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' It is observed that the condensate occurs for O+ with different values of α and 12 n=0 /=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0001 1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 2= 3 4 5 6 7= 8 9 : ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' < =>?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 @ AB C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0 DEF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='8 T ρ (< O>) 1 λ+ ρ n=1 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0001 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 α=GH I J α=KL M N PQ RS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0 TU V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='8 T ρ (< W>) 1 λ+ ρ n=X α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0001 α=Y Z[ \\ α=] ^_ ` α=a bc d e fg h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0 i jk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='8 T ρ (< l>) 1 λ+ ρ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 3: The condensates of the scalar operator O+ versus temperature with the fixed mass m2L2 eff = −2 for different Gauss-Bonnet parameters α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=', α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0001 (orange), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 (green), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='24 (red) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='25 (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The three panels from left to right represent the ground (n = 0), first (n = 1) and second (n = 2) states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' n if T < Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' For small condensate, we see that there is a square root behavior ⟨O+⟩ ∼ (1 − T/Tc)1/2, which is shown that, for the ground and excited states, the phase transition of the 4D Gauss-Bonnet holographic superconductors with the backreactions is typical of second order one with the mean field critical exponent 1/2 for all values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' n=0 n=1 n= m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='15 nopq rstu 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='15 vwxy α Tc ρ z{|} ~\x7f\x80 \x81 \x82 \x83\x84 \x86 \x87\x88\x89\x8a FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 4: The critical temperature Tc of the scalar operator O+ as a function of the Gauss-Bonnet parameter α with the fixed mass m2L2 eff = −2 for the ground n = 0 (blue), first n = 1 (red) and second n = 2 (green) states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 4, we give the critical temperature Tc for the condensate of the operator O+ as a function of the Gauss-Bonnet parameter α with the fixed mass m2L2 eff = −2 from the ground state to the second excited state in order to get the effect of the curvature correction on Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' An interesting feature we can find is that the critical temperature Tc decreases as α increases, but slightly increases near the Chern-Simons limit α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Furthermore, this non-monotonic behavior of the critical temperature is more pronounced in the ground state than that in the excited state, which is in good agreement with the results obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' HEE and HSC of the holographic model Now we are ready to study the properties of the HEE and HSC for the backreacting holographic supercon- ductor in the 4-dimensional Einstein-Gauss-Bonnet gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Due to the presence of a Gauss-Bonnet term, we 13 should use a general formula to calculate the HEE [83–86] S = −2π � ddy√−g � ∂L ∂Rµρνσ εµρενσ − � α � ∂2L ∂Rµ1ρ1ν1σ1∂Rµ2ρ2ν2σ2 � α 2Kλ1ρ1σ1Kλ2ρ2σ2 qα + 1 × � (nµ1µ2nν1ν2 − εµ1µ2εν1ν2)nλ1λ2 + (nµ1µ2εν1ν2 + εµ1µ2nν1ν2)ελ1λ2 �� , (32) where nµν and εµν reduce to the metric and Levi-Civita tensor in the two orthogonal directions with all other components vanishing, and qα is treated as “anomaly coefficients”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' This will result in the corrections to expressions (14) and (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Still, we employ the shooting method to carry out our numerical calculation, and use s and c to denote the entanglement entropy and complexity of the universal term, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' n=0 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='115 \x8b\x8c\x8d \x8e\x8f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='706 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='704 \x90 \x91\x92 \x93\x94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='700 T ρ s ρ n=0 α= \x96 \x97 \x98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='410 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='405 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='395 T ρ s ρ n=0 α=\x99\x9a \x9b \x9c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='105 \x9d\x9e\x9f ¡¢£ ¤ ¥¦§ ¨ ©ª« ¬ \xad®¯ ° T ρ s ρ n=1 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='716 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='713 T ρ s ρ n=1 α=± ² ³ ´ µ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='060 ¶ ·¸ ¹º 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='390 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='384 » ¼½ ¾¿ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='380 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='378 T ρ s ρ n=1 α= À Á ÂÃ Ä ÅÆÇ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='345 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='340 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='330 ÈÉÊ ËÌ ÍÎÏ ÐÑ T ρ s ρ n= Ò α=0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7146 T ρ s ρ n= Þ α=ß à áâ ã ä åæ çè é êëì íî 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='384 ï ðñ òó 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='380 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='378 T ρ s ρ n= ô α=õö ÷ ø ùú ûü ý þ ÿ0 �� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='345 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='340 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='330 T ρ s ρ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 5: The HEE of the scalar operator O+ versus temperature from the ground state (n = 0) to the second excited state (n = 2) with a fixed width l√ρ = 1 for different Gauss-Bonnet parameters α, which shows that the HEE always increases monotonously with the increase of the temperature and its value in the normal phase always larger than that in the superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The solid (blue) lines denote the normal phase and the dashed (red) lines are for the superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 5 shows our results from the ground state (n = 0) to the second excited state (n = 2) for the HEE of the scalar operator O+ as a function of the temperature T , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In each panel, the critical temperature Tc of the system can be read from the joint point of the solid line for the normal phase and the dashed line for the superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' For example, in the ground state (n = 0), we have Tc/√ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='117705 for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0001 (top-left panel), Tc/√ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='100418 for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='24 (top-middle panel) and Tc/√ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='105064 for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='25 (top- 14 right panel), which are consistent with those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' It is obvious that, for the ground state, the critical temperature first decreases and then increases as α approaches to the Chern-Simons limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In addition, for the ground state and excited states, we can always find that the value of the HEE in the superconducting phase is less than that in the normal phase when T < Tc, which is independent of the Gauss-Bonnet parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' This behavior of the HEE is due to the fact that the condensate turns on at the critical temperature and the formation of Cooper pairs makes the degrees of freedom decrease in the superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' While we fix the Gauss-Bonnet parameter α, we can see that the value of the HEE becomes larger as the number of nodes n increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' n=0 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='110 � �� �� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='118 ��� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7930 \x0e \x0f\x10 \x11 \x12 \x13\x14 \x15\x16 \x17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7915 T ρ c ρ n=0 α=\x18\x19 \x1a \x1b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='3689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='3688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='3687 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='3686 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='3685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='3684 T ρ c ρ n=0 α= !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' "# 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content="10 $%& ' ()* + ,-." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' / 123 4 567 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='18 T ρ c ρ n=1 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7880 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7870 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7860 T ρ c ρ n=1 α=9 : ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='< = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='060 > ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' @ AB C D EF GH I J KL MN O P QR ST U V WX YZ [ T ρ c ρ n=1 α=\\ ]^ _ ` abc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='06 def gh ijk lm nop qr stu vw xyz {| }~\x7f \x80\x81 T ρ c ρ n=\x82 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='0001 \x83 \x84 \x86\x87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7864 \x88 \x89\x8a \x8b\x8c \x8d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7860 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7858 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7856 T ρ c ρ n=\x8e α=\x8f \x90 \x91\x92 \x93\x94 \x96 \x97\x98 \x99 \x9a\x9b \x9c\x9d \x9e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='040 0.' metadata={'source': 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+page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='045 ´µ¶ ·¸ ¹º» ¼½ ¾¿À Á ÃÄÅ ÆÇ ÈÉÊ ËÌ ÍÎÏ ÐÑ T ρ c ρ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 6: The HSC of the scalar operator O+ versus temperature from the ground state (n = 0) to the second excited state (n = 2) with a fixed width l√ρ = 1 for different Gauss-Bonnet parameters α, which shows that the Gauss-Bonnet parameter has a more subtle effect on the HSC when compared to the HEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The solid (blue) lines denote the normal phase and the dashed (red) lines are for the superconducting phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 6, we plot the HSC of the scalar operator O+ as a function of the temperature T , and find that the curve of the HSC in the normal phase and the one in the superconducting phase intersect at the same critical temperature as that reflected by the HEE in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' It means that the HSC is able to capture the emergence of the phase transitions in the ground state and excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' What is noteworthy is that the Gauss-Bonnet parameter α has an interesting effect on the relation between the HSC and the temperature, which can be seen in the ground state and excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Obviously, there is a threshold αt of the Gauss- 15 Bonnet parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' When α < αt, the value of the HSC decreases as the temperature increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' And the value of the HSC as n increases for the fixed α, which agrees with the finding shown in the bottom-left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' At the threshold α = αt, with the increasing T/√ρ, the value of the HSC first decreases and then increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' It should be noted that the threshold becomes larger with the higher excited state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=', αt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='2400 for the ground state n = 0, αt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='2470 for the first state n = 1 and αt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='2478 for the second state n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Whereas as α goes up to the Chern-Simons limit α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='25, this non-monotonic behavior of the HSC will convert to a monotonic increasing function of the temperature, which is contrary to the case of α < αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Besides, we find that the normal phase always has a smaller HSC than the superconducting phase except for the situation of the Chern-Simons limit, namely, the value of the HSC in the normal phase is larger than that in the superconducting phase for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Under the influence of the Gauss-Bonnet parameter, these special features of the HSC with respect to the temperature imply that the higher curvature correction makes a difference to the properties of the spacetime and changes the geometric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Ò Ó =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='80 Ô Õ =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 Ö × =ØÙÚÛ n=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='15 ÜÝ Þß à áâã 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='3 α s ρ ä ρ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='80 å ρ =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 æ ρ = çèéê n=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='15 ë ìíî ï ðñ ò 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='3 α s ρ ó ρ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='80 ô ρ =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 õ ρ =ö÷ øù n=ú 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='15 û üý þ ÿ0� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='3 α s ρ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 7: The HEE of the scalar operator O+ as a function of the Gauss-Bonnet parameter α at the temperature T/√ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='02 with some chosen values of the widths, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=', l√ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='80 (blue), l√ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 (red) and l√ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='20 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The three panels from left to right represent the ground (n = 0), first (n = 1) and second (n = 2) states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' l ρ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='80 � ρ =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 � ρ = 1 �� � n=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='15 �� � \x0e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='3 α c ρ \x0f ρ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='80 \x10 ρ =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 \x11 ρ = \x12\x13\x14\x15 n=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='15 \x16 \x17\x18\x19 \x1a \x1b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='3 α c ρ ρ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='80 ρ =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 ρ = !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='" #$ n=2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content="15 % &' ( )*+ , 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='3 α c ρ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 8: The HSC of the scalar operator O+ as a function of the Gauss-Bonnet parameter α at the temperature T/√ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='02 with some chosen values of the widths, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=', l√ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='80 (blue), l√ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 (red) and l√ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='20 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The three panels from left to right represent the ground (n = 0), first (n = 1) and second (n = 2) states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' On the other hand, for the fixed temperature, from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 5 and 6 we find that the larger Gauss-Bonnet parameter α leads to the larger values of the HEE and HSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' To further illustrate this, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 7 and 8, we plot the HEE and HSC as a function of the Gauss-Bonnet parameter α, respectively, from the ground state to the second excited state at a fixed temperature T/√ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='02 with different widths, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=', l√ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='80, l√ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='00 16 and l√ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' We observe clearly that, regardless of the width and the number of nodes, the HEE and HSC increase with the increase of the Gauss-Bonnet parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Moreover, in each panel, the HEE and HSC become larger as the width increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' CONCLUSION In this work, we first study the HEE and HSC for the excited states of holographic superconductors with full backreaction in the 4-dimensional Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' We note that the changes of the HEE and HSC both for the scalar operators O− and O+ are discontinuous at the critical temperature Tc, and Tc in the excited states is lower than that in the ground state, which indicates that the higher excited state makes the scalar condensate harder to form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The values of Tc reflected by the HEE and HSC are consistent with the results obtained from the condensate behavior, which means that both the HEE and HSC can be used as good probes to the superconducting phase transition in the excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' However, there are some differences between the HEE and HSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' We observe that, for the ground state and excited states, the value of the HEE in the normal phase is larger than that in the superconducting phase and increases as the temperature increases, which is the opposite to the behavior of the HSC, namely, the normal phase always has a smaller HSC than the superconducting phase and the HSC decreases as the temperature increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Meanwhile, for a given temperature T in the superconducting phase, the higher excited state leads to a lager value of the HEE but a smaller value of the HSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Next, we extend the investigation to the HEE and HSC for the excited states of backreacting superconductors in the 4-dimensional Einstein-Gauss-Bonnet gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' One remarkable feature is that the critical temperature Tc for the scalar operator ⟨O+⟩ decreases as the higher curvature correction α increases, but slightly increases as α grows to the Chern-Simons limit, and this non-monotonic bahavior of Tc is more pronounced in the ground state than in the excited state, which can be supported by the findings obtained from both the HEE and HSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' The other noteworthy feature is that the effect of α on the relation between the HSC and the temperature is nontrivial in the ground state and excited states, which is a distinguishing property of the HSC and has not been found in the HEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Specifically, the HSC always behaves as a monotonic decreasing function of the temperature till α reaches some threshold αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' At this critical point αt, the HSC changes non-monotonously with the temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=', it first decreases and then increases with the increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' It is shown that the value of αt will be closer to the Chern-Simons limit in the higher excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' More interestingly, when α approaches to the Chern-Simons limit, the HSC will convert to a monotonic increasing function of 17 the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Besides, the value of the HSC in the normal phase is less than that in the superconducting phase for the case of α ≤ αt, but it is just on the contrary for the Chern-Simons limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Whereas the HEE always increases monotonously with the increase of the temperature and its value in the normal phase always larger than that in the superconducting phase, regardless of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Lastly, we find that, for the ground state and excited states, the increase of α makes both the HEE and HSC increase, which is independent of the strip width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Thus, we conclude that the HEE and HSC provide richer physics in the phase transition and the condensate of the scalar hair for holographic superconductors with excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' Acknowledgments We would like to acknowledge helpful discussions with Jian-Pin Wu and Guoyang Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' This work was sup- ported by the National Key Research and Development Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfovik/content/2301.00513v1.pdf'} +page_content=' 2020YFC2201400), National Natural Science Foundation of China (Grant Nos.' 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There aren’t many direct ways to build +binary optimal ZCCSs with lengths that are non- power-of- +two (NPT). This letter suggests two direct ways to build binary +optimal ZCCSs with lengths that are NPT. The proposed designs +are based on generalised Boolean functions (GBF). The results +are new and does not exists in the literature. +Index Terms—MC-CDMA, GBF, ZCCS, ZCZ. +I. INTRODUCTION +Golay complementary pair (GCP) postulated by M. J. Golay +[1] is a pair of sequences whose out-of-phase aperiodic auto- +correlation sums are zero. Numerous modern communication +systems like radar [2], OFDM communication systems [3] +etc. make use of GCPs because of their ideal correlation +properties [4]. However, the length of GCPs are very limited +[5]. Complementary sequence set (CSS) is a generalization +of GCP made by Tseng and Liu in [6], wherein each CSS +consists of numerous sequences and has zero aperiodic auto- +correlation sums (AACSs) except at zero time shift. Further the +idea of CSS was extended to a set called mutually orthogonal +complementary sequence sets (MOCSSs). When the set size +and flock size are both equal, we say that MOCSS is a +complete complementary code (CCC). CCCs have been found +application in MC-CDMA systems [7]. One of the biggest +problems with CCC is that the number of users it can support +for MC-CDMA is limited by the number of row sequences in +each complementary matrix, or the number of subcarriers. +Fan et al. in [8] proposed Z-complementary code sets +(ZCCSs) to increase the capacity of MC-CDMA networks +to support more concurrent users. ZCCS refers to a set of +matrices or codes that exhibit zero correlation (ZCZ). ZCCS’s +strong correlation properties allow for the deployment of +interference-free multi-carrier CDMA (MC-CDMA) in quasi- +synchronous channels without the need for power regulation +[9]. For an (M, Z) − ZCCSL +N we have M ≤ N⌊ L +Z ⌋, where +M, N, L, Z refers to set size, sub carrier, length and zero +correlation zone width respectively. When the equality sign +is true, ZCCS is said to be optimal [10]. In the literature, +numerous ZCCSs have been proposed by using direct [11]– +[18] and indirect [19]–[22] method. In [11]–[14] binary ZCCS +of length power-of-two have been presented. In contrast, +towards the challenges of designing binary optimal ZCCS +with non-power-of-two lengths through GBF, up to now, a +little progress has been made. Sarkar et al. in [18] suggested +building q-ary ZCCS with a length of the form 2m + 2 using +GBF. In [22], Sun et al. proposed a way to build q-ary ZCCS +with a length of the form 32m using GBF. In [21], Das et +al. proposed q-ary non-power-of-two lengths ZCCS but the +construction depends on initial choices of sequences. +Motivated by the scarcity of NPT length binary optimal +ZCCS through direct construction, in this letter, we propose +two direct constructions of optimal binary ZCCS. At first, +we propose optimal binary ZCCS of length of this form +(2m−1 + 2m−3)2l where m ≥ 5. Next we propose a optimal +binary ZCCS of length of this form 3(2m−1 + 2m−3) where +m ≥ 5. Both of the two proposed constructions are based +on GBFs. The proposed constructions produce novel class of +ZCCSs with unreported sequence lengths. +The plan for the rest of this paper is shown below. In Section +II, some basic ideas are looked at again. In Sections III and +IV, both ZCCS constructions are put forward. Section V is the +paper’s conclusion. +II. PRELIMINARY +In this section, we introduce some basic ideas and lemmas +that will be applied to the rest of the proposed building. Let +u = (u0, u1, . . . , ul−1) and v = (v0, v1, . . . , vl−1) consist +of two sequences whose elements are complex numbers. +We define the aperiodic cross-correlation function (ACCF) +between u and v as +Ψ(u, v)(τ) = + + + + + +�l−1−τ +k=0 +ukv∗ +k+τ, +0 ≤ τ < l, +�l+τ−1 +k=0 +uk−τv∗ +k, +−l < τ < 0, +0, +otherwise, +(1) +where ∗ denotes the complex conjugate. If u = v, then +the corresponding function is known as the aperiodic auto- +correlation function (AACF) of u, which we will refer to +simply as Ψ(u). +Let Ci = {ui +k +: 0 ≤ k < N} and Cj += {uj +k : +0 ≤ k < N} be collections of N sequences, where ui +k = +(ui +k,0, ui +k,1, . . . , ui +k,l−1) and uj +k = (uj +k,0, uj +k,1, . . . , uj +k,l−1). For +two sets of sequence Ci and Cj the ACCF is defined by +Ψ(Ci, Cj)(τ) = +N−1 +� +k=0 +Ψ(ui +k, uj +k). +Definition 1: Let C = {C0, C1, . . . , CM−1} be collection +of M sequence sets. The code set C is called a (M, Z) − +ZCCSL +N ([4]) if +Ψ(Cµ1, Cµ2)(τ) = + + + + + +Nl, +τ = 0, µ1 = µ2, +0, +0 < |τ| < Z, µ1 = µ2, +0, +|τ| < Z, µ1 ̸= µ2, +(2) + +2 +when M = N and Z = l, we denote the set C by (K, K, l)- +CCC. +A. Generalized Boolean Functions (GBFs) +A Boolean function f with m variables is a mapping from +the set {0, 1}m to the set Zq. The sequence of a Boolean +function f is a Zq-valued sequence of length N = 2m defined +by ψ(f) = (ωf0 +q , . . . , ωf2m−1 +q +) where fr = f(r0, . . . , rm−1) +and (r0, . . . , rm−1) is the binary representation of the in- +teger r. We denote ψj(f) be the truncated sequence after +eliminating first j terms from ψ(f) and ψj(f) be the trun- +cated sequence after eliminating last j terms from ψ(f). Let +C = (f0, . . . , fm−1) be an ordered set of m GBFs. We define +the codes ψj(C), ψj(C) as ψj(C) = (ψj(f0), . . . , ψj(fm−1)) +and ψj(C) = (ψj(f0), . . . , ψj(fm−1)) respectively. +Let us assume Q : {0, 1}m−4 → Zq be a quadratic form in +variables z0, z1, . . . , zm−5, i.e., +Q (z0, z1, . . . , zm−5) = +� +0≤i 1. Let T ∈ EG, A1 and A2 be the components of G − T, and for i ∈ [2], +Gi be the subgraph of G induced by V (Ai) ∪ V (T). +We may assume G1, . . . , Gt′ ⊆ G1 and +Gt′+1, . . . , Gt ⊆ G2 for some 1 ≤ t′ < t. +Then, by Lemma 5 and the induction hypothesis, +m(G, ϕ) = m(G1, ϕ1) + m(G2, ϕ2) = � +i∈[t′] m(Gi, ϕi) + � +i∈[t]\[t′] m(Gi, ϕi) = � +i∈[t] m(Gi, ϕi). +Lemma 7. Let G be a 3-colorable plane triangulation on n vertices and ϕ be the unique proper +3-coloring of G. For any distinct i, j ∈ [3], Gij is connected. Moreover, if n > 3, Gij is 2-connected. +Proof. We prove by induction on n. The triangulations of order at most 6 are listed in Figure 1. +Among these graphs, only the triangle and the octahedron are 3-colorable. It is not hard to verify +that the claims hold for these two graphs. From now on we assume that n > 6. +As G is a 3-colorable triangulation, every vertex of G has an even degree, and hence there exists +v ∈ V (G) with dG(v) = 4. Let v1v2v3v4v1 be the cycle induced by NG(v). We have ϕ(vi) = ϕ(vi+2) +for each i ∈ [2]. Suppose there exists i ∈ [2] such that vi and vi+2 have no common neighbor other +than v, vi+1, vi+3, where v5 := v1. We contract vivvi+2 to obtain G′ and call the new vertex v′. Let +ϕ′ : V (G′) → [3] be such that ϕ′(v′) = ϕ(vi) and ϕ′(u) = ϕ(u) for u ∈ V (G′) \ {v′}. It is clear +3 + +that ϕ′ is the unique proper 3-coloring of the triangulation G′. By the induction hypothesis, G′ +ij is +2-connected for any distinct i, j ∈ [3]. Then, one can easily prove by the construction that Gij is +2-connected for any distinct i, j ∈ [3]. +Suppose for every i ∈ [2], vi and vi+2 have some common neighbor other than v, vi+1, vi+3. +Since G is not the octahedron, it has some separating triangle T. Let A1, A2 be the components of +G−T. We consider the subgraphs Gi of G induced by V (Ai)∪V (T) (i ∈ [2]). Let ϕi be restriction +of ϕ on V (Gi). As |V (Gi)| > 3, it follows from the induction hypothesis that Gi +jk is 2-connected +for any distinct j, k ∈ [3] (i ∈ [2]), from which it immediately follows that Gjk is 2-connected for +any distinct j, k ∈ [3]. +Figure 1: The triangulations of order at most 6. +Let G be a graph with a proper k-coloring ϕ. Denote by cij the number of connected components +of Gij. The number of edges we need to remove from Gij to make ϕ acyclic is |E(Gij)|−|V (Gij)|+cij. +As E(Gij) are edge-disjoint for distinct i, j, and each vertex v of G is contained in k − 1 subgraphs +Gij, we know that +m(G, φ) = +� +1≤i 3, we have m(G, ϕv) ≤ n − 5. +Proof. By Lemma 7, Gij is connected for any distinct i, j ∈ [3]. Hence +m(G, ϕ) = +� +1≤i max{6, k}. +We first consider, for the first statement, that G is not 4-connected, i.e. G has some separating +triangle T. Let A1, A2 be the components of G − T. Let Gi be the subgraphs of G induced by +V (Ai) ∪ V (T) (i ∈ [2]). Denote by ϕi the restriction of ϕ on V (Gi). Write ni := |V (Gi)| and +ki := |ϕi(Gi)|. Note that n1 + n2 = n + 3 and k1 + k2 ≥ k + 3. By the induction hypothesis and +Lemma 4, for each i ∈ [2], there exists E′ +i ⊆ E(Gi) \ E(T) such that |E′ +i| ≤ ni − ki and Gi − E′ +i is +acyclically colored by ϕi. Let E′ := E′ +1 ∪ E′ +2. It is easy to prove that G − E′ is acyclically colored +by ϕ and |E′| = |E′ +1| + |E′ +2| ≤ (n1 − k1) + (n2 − k2) ≤ n − k. +Henceforth, we assume that G has no separating triangle and thus δ(G) = 4, 5. Fix v ∈ V (G) +such that dG(v) = δ(G). Depending on the value of δ(G), we consider two cases. +Case 1: dG(v) = δ(G) = 4. +Let v1v2v3v4v1 be the cycle induced by NG(v). Since n > 6 and G has no separating triangle, +we can assume that v1, v3 have no common neighbor other than v, v2, v4. +If ϕ(v1) ̸= ϕ(v3), we obtain G′ from G by deleting v and adding the edge v1v3. Let ϕ′ be the +restriction of ϕ on V (G′). Denote n′ := |V (G′)| and k′ := |ϕ′(V (G′))|. Note that G′ is 4-connected, +n′ = n − 1 ≥ 6 and k′ = k or k − 1. Moreover, if k′ = k − 1, then v is the only vertex that is colored +by ϕ(v) and hence no 2-colored cycle in G contains v. By the induction hypothesis, there exists +E′′ ⊆ E(G′) such that G′ − E′′ is acyclically colored by ϕ′ and |E′′| = m(G′, ϕ′) ≤ n′ − k′. Define +S := {vv2} if k′ = k, and S := ∅ if k′ = k − 1. Set E′ := (E′′ \ {v1v3}) ∪ S. One can readily show +that G − E′ is acyclically colored by ϕ and |E′| ≤ n − k. If k = k′ = 4, we additionally require +from the induction hypothesis that |E′′| ≤ n′ − 5, which yields in this case that |E′| ≤ n − 5. If +k = 4 and k′ = k − 1, then, suppose ϕ(V (G)) = [4] and ϕ(v) = 4, one can deduce from Lemma 7 +that Gij are connected for all distinct i, j ∈ [3] and hence prove in a similar way as in the proof of +Theorem 8 that m(G, ϕ) = n − 5. +Assume ϕ(v1) = ϕ(v3). First we prove that m(G, ϕ) ≤ n − |ϕ(V (G))|. Let G′ be from G +by contracting v1vv3 to a new vertex v′ and denote the coloring induced from ϕ by ϕ′ so that +ϕ(v′) = ϕ(v1). Denote n′ := |V (G′)| and k′ := |ϕ′(V (G′))|. We have n′ = n − 2 ≥ 5 and k′ = k or +k − 1. By the induction hypothesis and Lemma 4, there exists E′′ ⊆ E(G′) \ {v′v2, v′v4} such that +G′ − E′′ is acyclically colored by ϕ′ and |E′′| = m(G′, ϕ′) ≤ n′ − k′. Note that any path joining +v1, v3 in G−{v, v2, v4} corresponds to a cycle containing v′ in G′ as v1, v3 have no common neighbor +other than v, v2, v4. Define S := {vv2} if k′ = k, and S := ∅ if k′ = k−1. Let E′ := E′′ ∪{v1v2}∪S. +It is clear that |E′| ≤ n − k and G − E′ is acyclically colored by ϕ as v1v2v3v4v1 is the only cycle +that is possibly 2-colored in G − E′′ − v. +It remains to show that if k = 4, then m(G, ϕ) ≤ n−5. If ϕ(v2) ̸= ϕ(v4), we take E′ := E′′ with +|E′| ≤ n′ − 4 = n − 6 and it is easy to show that G − E′ is acyclically colored by ϕ. So we assume +that ϕ(v2) = ϕ(v4). If k′ = 3, then it follows from Theorem 8 that m(G, ϕ) ≤ n − 5. So we assume +k′ = 4; in particular, |E′′| ≤ n′ − 4. If |E′′| = m(G′, ϕ′) ≤ n′ − 5, we take E′ := E′′ ∪ {vv2, v1v2}, so +|E′| = |E′′|+2 ≤ n−5 and G−E′ is acyclically colored by ϕ. This yields that m(G, ϕ) ≤ |E′| ≤ n−5. +Assume m(G′, ϕ′) = |V (G′)| − 4. +As |V (G′)| > 4, by the induction hypothesis, G′ is not +4-connected, and hence contains separating triangles. As G is 4-connected, it follows that each +separating triangle of G′ contains v′ and separates v2 and v4; an example is given in Figure 2. +This implies that TG′ is a path G′ +1 . . . G′ +t (t ≥ 2), with end-vertex G′ +1 containing v2, and the other +end-vertex G′ +t containing v4. +Denote by ϕ′ +i the restriction of ϕ′ on V (G′ +i). By Lemma 6 and Theorem 8, precisely one graph +G′ +i from VG′ has |ϕ′ +i(V (G′ +i))| = 4, m(G′ +i, ϕi) = |V (G′ +i)| − 4 and |ϕ′ +j(V (Gj))| = 3 for all j ∈ [t] \ {i}. +By the induction hypothesis, we know that |V (G′ +i)| ≤ 4 and hence G′ +i is isomorphic to K4. +5 + +Note that G′ +i is not a leaf of TG′, for otherwise, say i = 1, then |ϕ′(V (G′) \ {v2}| = 3. This +implies that ϕ(v2) ̸= ϕ(v4), contradicting the above assumption. +Thus |ϕ′(V (G′ +j))| = 3 and {ϕ′(v′), ϕ′(v2)} ⊂ ϕ′(V (G′ +j)) for j ∈ {1, t}. As G′ +i is an internal +vertex of TG′, we have ϕ′(V (G′ +1)) ̸= ϕ′(V (G′ +t)). Without loss of generality, we may assume that +ϕ′(V (G′ +1)) = [4]\ϕ(v) and ϕ′(V (G′ +t)) = {ϕ(v), ϕ′(v′), ϕ′(v2)}. Let T be the separating triangle of G′ +that is contained in G′ +1. Write V (T) := {v′, u, w} such that ϕ′(u) = ϕ′(v2). Note that ϕ′(w) ̸= ϕ(v). +Let C be the cycle induced by the neighbors of u in G′ +1 (see Figure 2(b) for an example) and eC be +an arbitrary edge of C. By Lemma 4, we may require E′′ ⊆ E(G′) \ ({v′v2, v′v4} ∪ (E(C) \ {eC})) +as ϕ′(v2) = ϕ′(v4) = ϕ′(u) /∈ ϕ′(V (C)), and hence eC ∈ E′′. Let E′ := (E′′ \ {eC}) ∪ {vv2, v1v2}. +We have |E′| = |E′′| + 1 ≤ n − 5. It remains to show that G − E′ is acyclically colored by ϕ. +Again, it is easy to show that G − E′ − eC is acyclically colored by ϕ. Hence, any cycle K which +uses only two colors in G − E′ contains eC and the two colors used in K are ϕ′(w), ϕ′(v′). So K +does not contain v, v2, v4. If {v1, v3} ⊂ V (K), then after contracting the path v1vv3, K becomes +the union of two edge-disjoint cycles in (G′ +ϕ′(v′)ϕ′(w) − E′′) + eC (as v1, v3 have no other common +neighbors than v, v2, v4), a contradiction. If |{v1, v3} ∩ V (K)| ≤ 1, then K corresponds to C. Since +C is a cycle separating v2 and v4 in G′, K is a cycle separating v2 and v4 in G, which is however +impossible since v2vv4 is a path in G not intersecting K. +Case 2: dG(v) = δ(G) = 5. +Let v1v2v3v4v5v1 be the induced cycle on NG(v). +If |ϕ(NG(v))| = 3, we may assume that ϕ(v1) = ϕ(v3) and ϕ(v2) = ϕ(v4). As G is 4-connected +and δ(G) = 5, we may assume that v1, v3 have no common neighbor other than v, v2. Let G′ +be obtained from G by contracting v1vv3 to a new vertex v′. We do not distinguish edges from +E(G′) \ {v′v2} from their corresponding edges in G. Set ϕ′(v′) := ϕ(v1) and ϕ′(u) := ϕ(u) for all +u ∈ V (G′) \ {v′}. Denote n′ := |V (G′)| and k′ := |ϕ′(V (G′))|. We have n′ = n − 2 and k′ = k or +k − 1. By Lemma 4 and the induction hypothesis, there exists E′′ ⊆ E(G′) \ {v′v2} such that ϕ′ is +an acyclic coloring of G′ − E′′ and |E′′| = m(G′, ϕ′) ≤ n′ − k′. Set S := {vv2} if k′ = k and S := ∅ +if k′ = k − 1. Define E′ := E′′ ∪ S. It is easy to show that |E′| ≤ n − k − 1 and ϕ is an acyclic +coloring of G − E′. +If |ϕ(NG(v))| ≥ 4, we may assume that ϕ(vi) = i for each i ∈ [4]. Obtain G′ from G by deleting +v and adding edges v1v3, v1v4. Let ϕ′ be the restriction of ϕ on V (G) \ {v}. Denote n′ := |V (G′)| +and k′ := |ϕ′(V (G′))|. We have n′ = n − 1 and k′ = k or k − 1. By Lemma 4 and the induction +hypothesis, there exists E′′ ⊆ E(G′) \ {v′v3, v′v4} such that ϕ′ is an acyclic coloring of G′ − E′′ and +|E′′| = m(G′, ϕ′) ≤ n′ − k′. Set S := {vv5} if k′ = k and S := ∅ if k′ = k − 1. Define E′ := E′′ ∪ S. +It is easy to show that |E′| ≤ n − k and ϕ is an acyclic coloring of G − E′. We remark that in this +case we have k > 4, thus we do not need to consider the second statement. +The following corollary characterizes plane triangulations G and colorings ϕ that satisfy the +eqaulities m(G, ϕ) = n − 3 and m(G, ϕ) = n − 4, respectively. +Corollary 10. Let G be a plane triangulation on n vertices and ϕ be a coloring of G. Let VG := +{G1, . . . , Gt} and ϕi be the restriction of ϕ on V (Gi) for i ∈ [t]. We have that m(G, ϕ) = n − 3 if +and only if |ϕ(V (G))| = 3; and m(G, ϕ) = n − 4 if and only if there exists i ∈ [t] such that Gi is +isomorphic to K4 and |ϕj(V (Gj))| = 3 for all j ∈ [t] \ {i}. +3 +Acyclic 2CC transerval and upper bounds for mk(G) +Let G be a graph and ϕ a coloring of G. We have shown upper bounds on m(G, ϕ) when G is a +plane triangulation. In this section, we show that we can choose the 2CC transversal E′ so that it +6 + +v +v1 +v3 +v2 +v4 +(a) +v′ +u +w +(b) +Figure 2: (a) A 4-colored plane triangulation G. (b) The plane triangulation G′ obtained from G +by contracting the path v1vv3. The cycle C consists of the thick edges. +induces a forest as well as extend the results to general planar graphs. +Definition 11. Let G be a graph and U ⊆ V (G). An edge set E′ ⊆ E(G) is U-acyclic if the graph +induced by E′ is a forest and contains no path joining two distinct vertices of U. With abuse of +notation, we say an edge set is H-acyclic instead of V (H)-acyclic for any subgraph H of G, and if +H is a graph induced by a single edge e, we write e-acyclic instead of H-acyclic. +Proposition 12. Let G be a plane triangulation and ϕ be a proper coloring of G. For any facial +cycle F of G, there exists an F-acyclic 2CC transversal EF with respect to ϕ. +Proof. We prove by induction on |V (G)|. We shall assume |V (G)| > max{6, |ϕ(V (G))|} as the +small cases can be readily verified. +Suppose G has some separating triangle T. Let A1 and A2 be the components of G − T, and +for i ∈ [2], Gi be the subgraph of G induced by V (Ai) ∪ V (T). Without loss of generality, assume +that F is a facial cycle of G1. By the induction hypothesis, we have an F-acyclic 2CC transversal +E1 +F ⊆ E(G1) of G1 and a T-acyclic 2CC transversal E2 +T ⊆ E(G2) of G2. It is easy to see that the +edge set EF := E1 +F ∪ E2 +T is an F-acyclic 2CC transversal of G. +Henceforth, we assume that G has no separating triangle and thus δ(G) ≥ 4. Fix v ∈ V (G) \ +V (F) such that dG(v) = δ(G) ≤ 5. We consider two cases, depending on dG(v) = 4 or 5. +Case 1: dG(v) = 4. +Let v1v2v3v4v1 be the cycle induced by NG(v). Since |V (G)| > 6 and G has no separating +triangle, we can assume that v1, v3 have no common neighbor other than v, v2, v4. If ϕ(v1) ̸= ϕ(v3), +we obtain G′ from G by deleting v and adding the edge v1v3, and color it with the coloring ϕ′ +induced from ϕ. Clearly, F remains a facial cycle of G′. By the induction hypothesis, there exists +an F-acyclic 2CC transversal E′ +F ⊆ E(G′) of G′. Set EF := (E′ +F \ {v1v3}) ∪ {vv2}. One can readily +check that EF is an F-acyclic 2CC transversal of G. +If ϕ(v1) = ϕ(v3), obtain G′ from G by contracting v1vv3 to a new vertex v′ and denote the +coloring induced from ϕ by ϕ′ so that ϕ(v′) = ϕ(v1). +Let E′ +F ⊆ E(G′) be an F-acyclic 2CC +transversal of G′. Recall that v1, v3 have no common neighbor other than v, v2, v4, and hence any +path joining v1, v3 in G − {v, v2, v4} corresponds to a cycle containing v′ in G′. We construct EF +as follows. +• If E′ +F ∩ {v′v2, v′v4} = ∅, then v1v2v3v4v1 is the only cycle in G − (E′ +F ∪ {vv2}) that possibly +uses only two colors. We claim that there exists j ∈ {1, 3} such that EF := E′ +F ∪ {vv2, vjv2} +7 + +induces a forest not connecting any distinct vertices from V (F). Suppose it does not hold, +then for each j ∈ {1, 3}, the graph induced by E′ +F in G contains some path joining vj and v2, +or contains two disjoint paths each joining one vertex from V (F) and one vertex of vj, v2. In +any case, the graph induced by E′ +F in G′ contains some path joining two vertices from V (F) +or some cycle, a contradiction. As G − EF is acyclically colored by ϕ, EF is the desired edge +set. +• If E′ +F ∩ {v′v2, v′v4} = {v′vi} for some i ∈ {2, 4}, set EF := (E′ +F \ {v′vi}) ∪ {vv2, v1vi, v3vi}. +Similarly to the previous case, it can be shown that G − EF is acyclically colored by ϕ and +the subgraph induced by EF has no cycle and no path joining distinct vertices from V (F). +• If {v′v2, v′v4} ⊆ E′ +F , then there is a unique path P in G′ − E′ +F joining v′ and v2 using only +colors ϕ(v1) and ϕ(v2). Therefore P can be viewed as a path in G − ((E′ +F \ {v′v2, v′v4}) ∪ +E(v1v2v3v4v1)) connecting v2 and vj for some j ∈ {1, 3}. +Since v1, v3 have no common +neighbor other than v, v2, v4 and the neighbor of v′ in P is not v4, the index j is unique. Set +EF := (E′ +F \{v′v2, v′v4})∪{vv2, vjv2, v1v4, v3v4}. Similarly to the previous cases, it is easy to +show that EF is F-acyclic. It is left to show that ϕ is an acyclic coloring of G − EF . Suppose +to the contrary that there is some 2-colored cycle C in G − E′. It is not hard to see that C +contains v4−jv2 but not vj. Then C − v4−jv2 is a path in G′ − E′ +F connecting v′ and v2 yet +different from P, a contradiction. +Case 2: dG(v) = 5. +Let v1v2v3v4v5v1 be the induced cycle on NG(v). +If |ϕ(NG(v))| = 3, we may assume that +ϕ(v1) = ϕ(v3) and ϕ(v2) = ϕ(v4). Suppose v1, v3 have a common neighbor u other than v, v2 and +v2, v4 have a common neighbor u′ other than v, v3. Since G has no separating triangle, u = u′ +and dG(v2) = dG(v3) = 4. +If v2 or v3 is not incident to F, we may revise our choice of v so +that dG(v) = 4. Otherwise, F is the cycle uv2v3u and since dG(v) = 5, there exists some vertex +w ∈ V (G) \ {v, v1, v2, v3, v4, u} such that dG(w) ≤ 5; we may replace v by w. Therefore, without +loss of generality, we may assume that v1, v3 have no common neighbor other than v, v2. +Obtain G′ from G by contracting v1vv3 to a new vertex v′ and denote the coloring induced from +ϕ by ϕ′ so that ϕ(v′) = ϕ(v1). It is clear that F remains a facial cycle of G′. Let E′ +F ⊆ E(G′) be +an F-acyclic 2CC transversal of G′. We construct EF as follows. +• If v′v2 ∈ E′ +F , set EF := (E′ +F \ {v′v2}) ∪ {vv2, v1v2, v2v3}. +• If v′v2 /∈ E′ +F , set EF := E′ +F ∪ {vv2}. +In both cases it is easy to show that EF is an F-acyclic 2CC transversal of G. +If |ϕ(NG(v))| > 3, we may assume that ϕ(vi) = i for each i ∈ [4]. Let G′ be the graph obtained +from G by deleting v and adding edges v1v3, v1v4. Let ϕ′ be the restriction of ϕ on V (G)\{v}. Let +E′ +F be an F-acyclic 2CC transversal of G. One can easily show that EF := (E′ +F \{v1v3, v1v4})∪{vv5} +is an F-acyclic 2CC transversal of G. +We remark that the F-acyclic 2CC transversal EF found in Proposition 12 induces a forest of +at least |V (F)| = 3 components and hence has size at most |V (G)| − 3. In fact, an F-acyclic 2CC +transversal of the optimal size m(G, ϕ) does exist due to the following observation. Note that for +any edge set E′ ⊆ E(G), G − E′ is acyclically colored by a proper k-coloring ϕ of G if and only if +E(G) \ E′ is an independent set of the direct sum of the graphic matroids of Gij (i, j ∈ [k]). This +yields the following corollary. +8 + +Corollary 13. Let G be a plane triangluation, ϕ be a proper coloring of G and F be a facial cycle +of G. There exists an F-acyclic 2CC transversal E′ ⊆ E(G) with |E′| = m(G, ϕ). +Next, we generalize the results to planar graphs. +Theorem 14. Assume G is a planar graph on n vertices and ϕ is a proper coloring of G with +|ϕ(V (G))| = k. Let U ⊆ V (G) that induces a clique of size |U| ≤ 3. There exists a U-acyclic 2CC +transversal EU ⊆ E(G) with |EU| = m(G, ϕ) ≤ n − k. +Proof. We prove by induction on n. It clearly holds when n ≤ k. From now on we consider n > k. +If G has some separator W ⊂ V (G) such that |W| ≤ 3 and W induces a clique, let A1 be a +component of G − W and A2 the union of all other components. Denote by Gi the subgraph of G +induced by V (Ai) ∪ W and by ϕi the restriction of ϕ on V (Gi) (i ∈ [2]). Write ni := |V (Gi)| and +ki := |ϕi(V (Gi)|. We have n1 + n2 = n − |W| and k1 + k2 ≥ k − |W|. Without loss of generality, +we require that U ⊆ V (G1). By the induction hypothesis, there exist a U-acyclic 2CC transversal +E′ +U of G1 with |E′ +U| ≤ n1 − k1 and a W-acyclic 2CC transversal E′ +W of G2 with |E′ +W | ≤ n2 − k2. +It is easy to show that EU := E′ +U ∪ E′ +W is a U-acyclic 2CC transversal with |EU| ≤ n − k. +We assume that G has no separator W ⊂ V (G) such that |W| ≤ 3 and W induces a clique. In +particular, G is 2-connected and every facial boundary of G is a cycle. We add to G as many edges +as possible such that ϕ remains as a proper coloring and G remains as a plane graph. With abuse +of notation, we call the new graph G. It suffices to prove the statement for the new graph G. +If G is a triangulation, we apply Theorem 9 and Corollary 13 to conclude that G has some +U-acyclic 2CC transversal EU with |EU| = m(G, ϕ) ≤ n − k. +If any facial cycle of G has a chord, then the end-vertices of the chord form a separator of G, +contradicting our assumption. +Assume G is not a plane triangulation. As each facial cycle is an induced cycle, and any two +non-adjacent vertices of a face are colored by the same color, there exists a facial cycle v1v2v3v4v1 +in G such that ϕ(v1) = ϕ(v3) and ϕ(v2) = ϕ(v4). If v1, v3 have 3 common neighbors and v2, v4 have +3 common neighbors, then G must be isomorphic to the plane graph obtained from the octahedron +by deleting one vertex since we assume that G has no separating triangle. One can easily verify that +the statement holds for this graph. Thus, without loss of generality, we assume that v1, v3 have no +common neighbor other than v2, v4. Let G′ be obtained from G by identifying v1 and v3 as a new +vertex v′ and ϕ′ be the coloring of G′ induced from ϕ. Denote n′ := |V (G′)| and k′ := |ϕ′(V (G′))|. +We have n′ = n − 1 and k′ = k. Moreover, we can view U as a vertex set of G′ since U contains +at most one of v1, v3. By the induction hypothesis, we have a U-acyclic 2CC transversal E′ +U of G′ +with |E′ +U| = m(G′, ϕ′) ≤ n′ − k′. We construct EU as follows. Since the approach is similar to that +in the proof of Proposition 12, some details will be omitted. +• If E′ +U ∩{v′v2, v′v4} = ∅, then there exists j ∈ {1, 3} such that EU := E′ +U ∪{vjv2} is U-acyclic. +• If E′ +U ∩ {v′v2, v′v4} = {v′vi} for some i ∈ {2, 4}, set EU := (E′ +U \ {v′vi}) ∪ {v1vi, v3vi}. +• If {v′v2, v′v4} ⊆ E′ +U, then there is a unique path P in G′−E′ +U joining v′ and v2 using only colors +ϕ(v1) and ϕ(v2). We can view P as a path in G − ((E′ +F \ {v′v2, v′v4}) ∪ E(v1v2v3v4v1)) con- +necting v2 and vj for some unique j ∈ {1, 3}. Set EU := (E′ +U \{v′v2, v′v4})∪{vjv2, v1v4, v3v4}. +It is not hard to verify that the edge set EU constructed above is a U-acyclic 2CC transversal with +|EU| ≤ n − k. This completes the proof. +Corollary 15. Let G be a planar graph on n vertices. If n ≥ 5, then m4(G) ≤ n − 5. If G is +3-colorable, then m3(G) ≤ n − 3. +9 + +Theorem 16. There are infinitely many 4-connected planar graphs G with m4(G) = |V (G)| − 5, +and infinitely many 3-colorable planar graphs with m3(G) = |V (G)| − 3. +Proof. It follows from Corollary 10 that for any 3-colorable plane triangulation G, m3(G) = +|V (G)| − 3. +Let G be the 4-connected plane triangulation obtained by joining two independent vertices u, v +to every vertex of a cycle C on n−2 vertices with n ≥ 7 odd. It is obvious that G is not 3-colorable. +Let ϕ be any 4-coloring of G. Then, without loss of generality, ϕ(V (C)) = [3] and ϕ(u) = ϕ(v) = 4. +For any i ∈ [3], Gi4 is a connected plane graph with |ϕ−1(i)| faces, and for i, j ∈ [3], Gij is acyclic. +Therefore m(G, ϕ) = � +i∈[3](|ϕ−1(i)| − 1) = n − 5. +4 +Upper bounds for m′ +k(G) +In this section we study the problem of how many edges we need to remove from a planar graph in +order to make it acyclic k-colorable for k = 3, 4. +Theorem 17. Let G be a planar graph on n vertices. +We have m3(G) ≤ (13n − 42)/10 and +m4(G) ≤ (3n − 12)/5. +Proof. We first prove that m4(G) ≤ (3n − 12)/5. As every plane graph is a spanning subgraph +of some plane triangulation, we may assume that G is a plane triangulation on n vertices. Let +ϕ : V (G) → [5] be an acyclic 5-coloring of G. Without loss of generality, assume that +� +v∈ϕ−1(5) +(dG(v) − 3) ≤ 1 +5 +� +v∈V (G) +(dG(v) − 3) = 3n − 12 +5 +. +Let v be any vertex in ϕ−1(5). Since the neighbors of v span some cycle and ϕ is acyclic, there exist +v1, v2, v3 ∈ NG(v) whose colors are pairwise distinct. Define Ev to be the set of edges incident to v +other than vv1, vv2 and vv3, and set ϕ′(v) to be the color from [4] other than ϕ(v1), ϕ(v2), ϕ(v3). To +complete the construction, we set E′ := � +v∈ϕ−1(5) Ev and set ϕ′(u) := ϕ(u) for all u ∈ � +i∈[4] ϕ−1(i). +It is readily to verify that ϕ′ is a proper 4-coloring of G′ := G − E′ and |E′| = � +v∈ϕ−1(5)(dG(v) − +3) ≤ +3n−12 +5 +. +Suppose ϕ′ is not an acyclic coloring of G′, then there is a cycle C contained in +ϕ′−1(i)∪ϕ′−1(j) for some distinct i, j ∈ [4]. Note that C cannot contain any v ∈ ϕ−1(5) since v has +precisely three neighbors of three different colors in G′. Therefore C is contained in G′[(ϕ′−1(i) ∪ +ϕ′−1(j)) \ ϕ−1(5)] = G[ϕ−1(i) ∪ ϕ−1(j)], a contradiction. +This approach can be repeated to show that m3(G) ≤ (13n − 42)/10. More precisely, we may +assume that +� +v∈ϕ′−1(4) +(dG′(v) − 2) ≤ 1 +4 +� +v∈V (G′) +(dG′(v) − 2) = 4n − 12 − 2|E′| +4 +. +It is not hard to see that for any v ∈ V (G′), |ϕ′(NG′(v))| ≥ 2. Let v ∈ ϕ′−1(4) and v1, v2 ∈ NG′(v) +be of different colors. Define E′ +v to be the set of edges incident to v other than vv1 and vv2, and +set ϕ′′(v) to be the color from [3] other than ϕ(v1), ϕ(v2). Set E′′ := E′ ∪ � +v∈ϕ′−1(4) E′ +v and set +ϕ′′(u) := ϕ′(u) for all u ∈ � +i∈[3] ϕ′−1(i). Again, it is readily to verify that ϕ′′ is a proper 3-coloring +of G′′ := G − E′′ and +|E′′| = |E′| + +� +v∈ϕ′−1(4) +(dG′(v) − 2) ≤ 13n − 42 +10 +. +Similarly as before, one can show that ϕ′′ is an acyclic 3-coloring of G′′ and hence the result +follows. +10 + +We remark that there exist infinitely many planar graphs G on n vertices so that G − E′ is +not acyclically 4-colorable for any E′ ⊆ E(G) with |E′| < (n − 2)/4. Let H be a 2-face-colorable +triangulation and T be a family of |E(H)|/3 edge-disjoint facial triangles of H. Let G be obtained +from H by replacing each triangle from T by an octahedron. Therefore E(G) is partitioned into +|E(H)|/3 octahedra, and n = |V (H)|+|E(H)| = 4|V (H)|−6. As the octahedron is not acyclically +4-colorable, any E′ ⊆ E(G) satisfying that G − E′ is acyclically 4-colorable has size at least +|E(H)|/3 = n−2 +4 . +Acknowledgments +The research of On-Hei Solomon Lo was supported by a Postdoctoral Fellowship of Japan Society +for the Promotion of Science and by Natural Sciences and Engineering Research Council of Canada. +The research of Ben Seamone was supported by Natural Sciences and Engineering Research Council +of Canada. The research of Xuding Zhu was supported by National Natural Science Foundation of +China grant NSFC 11971438 and U20A2068. +References +[1] O. V. Borodin. On acyclic colorings of planar graphs. Discrete Math., 25(3):211–236, 1979. +[2] O. V. Borodin. Colorings of plane graphs: A survey. Discrete Math., 313(4):517–539, 2013. +[3] B. Grünbaum. Acyclic colorings of planar graphs. Israel J. Math., 14:390–408, 1973. +[4] D. Mondal, R. I. Nishat, Md. S. Rahman, and S. Whitesides. Acyclic coloring with few division +vertices. J. Discrete Algorithms, 23:42–53, 2013. +11 + diff --git a/4tFJT4oBgHgl3EQfkCwt/content/tmp_files/load_file.txt b/4tFJT4oBgHgl3EQfkCwt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4a79555e306e06e6a86ac4233e91a9a83bbe3e3c --- /dev/null +++ b/4tFJT4oBgHgl3EQfkCwt/content/tmp_files/load_file.txt @@ -0,0 +1,484 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf,len=483 +page_content='Defective acyclic colorings of planar graphs On-Hei Solomon Lo∗ Ben Seamone†‡ Xuding Zhu§ Abstract This paper studies two variants of defective acyclic coloring of planar graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' For a graph G and a coloring ϕ of G, a 2CC transversal is a subset E′ of E(G) that intersects every 2-colored cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let k be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We denote by mk(G) the minimum integer m such that G has a proper k-coloring which has a 2CC transerval of size m, and by m′ k(G) the minimum size of a subset E′ of E(G) such that G − E′ is acyclic k-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We prove that for any n-vertex 3-colorable planar graph G, m3(G) ≤ n−3 and for any planar graph G, m4(G) ≤ n−5 provided that n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We show that these upper bounds are sharp: there are infinitely many planar graphs attaining these upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Moreover, the minimum 2CC transversal E′ can be chosen in such a way that E′ induces a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We also prove that for any planar graph G, m′ 3(G) ≤ (13n − 42)/10 and m′ 4(G) ≤ (3n − 12)/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' 1 Introduction An acyclic k-coloring of a graph G is a proper k-coloring of G with no 2-colored cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Confirming a conjecture of Grünbaum [3], Borodin [1] proved that every planar graph has an acyclic 5-coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' This celebrated result is best possible as there are planar graphs that are not acyclic 4-colorable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' the octahedron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Acyclic coloring has been studied extensively for several decades and applied to solve other problems on graph coloring and partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We refer to [2] for a comprehensive survey on this subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' This paper studies defective acyclic k-coloring of planar graphs mainly for k = 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' In other words, we study k-colorings of planar graphs for which the condition of being an acyclic coloring is not completely satisfied, however, we want to limit the violation of the acyclicity rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We consider two variants of defective acyclic coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Given a graph G and a proper coloring ϕ of G, a 2-colored cycle transversal (2CC transversal) with respect to ϕ is a subset E′ of E(G) that intersects all 2-colored cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' In other words, G − E′ contains no 2-colored cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let G be a graph and k be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We define two parameters mk(G) and m′ k(G) as follows: mk(G) := minE′⊆E(G){|E′| : E′ is a 2CC transversal with respect to a proper k-coloring}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' m′ k(G) := minE′⊆E(G){|E′| : G − E′ has an acyclic k-coloring}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' ∗Faculty of Environment and Information Sciences, Yokohama National University, Yokohama 240-8501, Japan †Mathematics Department, Dawson College, Montreal, QC, Canada ‡Département d’informatique et de recherche opérationnelle, Université de Montréal, Montreal, QC, Canada §Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content='11577v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content='CO] 27 Jan 2023 Note that mk(G) = m′ k(G) = 0 if and only if G is acyclic k-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' If G has no proper k- coloring, then mk(G) is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' In this case, we let mk(G) := ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' It follows from the definition that for any graph G and integer k, mk(G) ≥ m′ k(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We are interested in the case that G is a planar graph and k = 3, 4 as Borodin’s theorem asserts that m5(G) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' To obtain an upper bound for mk(G), we need to construct a proper k-coloring ϕ of G and find a 2CC transerval E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' One immediate difficulty is that, for k = 4, the existence of a proper 4-coloring of a planar graph follows from the Four Color Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' For k = 3, it is NP-complete to decide whether a planar graph G is 3-colorable, and hence there is no easy way to construct a proper 3-coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Fortunately, it turns out that tight upper bounds for m4(G) and m3(G) for the whole family of planar graphs and the whole family of 3-colorable planar graphs do not depend on a particular proper coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' For any proper coloring ϕ of a graph G, define m(G, ϕ) := min E′⊆E(G){|E′| : E′ is a 2CC transerval with respect to ϕ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We prove in Section 3 that for any planar graph G on n vertices and any proper coloring ϕ of G, m(G, ϕ) ≤ n − |ϕ(V (G))|, where |ϕ(V (G))| denotes the number of colors used in ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' To this end, we study the case when G is a plane triangulation in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Moreover, we show that if n ≥ 5, then there is a 4-coloring ϕ of G with m(G, ϕ) ≤ n − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We apply these results to prove that for every planar graph G, m4(G) ≤ n − 5 provided that n ≥ 5, and m3(G) ≤ n − 3 provided that G is 3-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' These two bounds are tight as there are infinitely many 3-colorable planar graphs G with m3(G) = n − 3 and infinitely many planar graphs G with m4(G) = n − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Besides, we show in Section 3 that for any proper coloring ϕ of a planar graph G, we can find a 2CC transerval E′ with |E′| = m(G, ϕ) that induces a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' In Section 4 we study the parameter m′ k(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We show that m′ 3(G) ≤ (13n − 42)/10 and m′ 4(G) ≤ (3n − 12)/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We shall mention an application of our results on acyclic colorings of subdivisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' For a graph G and a positive integer k, define m′′ k(G) to be the minimum size of an edge set E′ ⊆ E(G) such that the graph obtained from G by subdividing each edge in E′ by one vertex is acyclically k- colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' It is easy to observe that mk(G) ≥ m′′ k(G) ≥ m′ k(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' It was shown in [4] that for any n-vertex planar graph G, m′′ 4(G) ≤ n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Our upper bound for m4(G) immediately improves it to m′′ 4(G) ≤ n − 5 for n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' All graphs considered in this paper are finite and simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We denote by V (G) and E(G) the vertex set and the edge set of G, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' For v ∈ V (G), denote by NG(v) the set of vertices adjacent to v and by dG(v) the degree of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' For a positive integer k, denote [k] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' A k-coloring ϕ of G is a function which assigns a color ϕ(v) ∈ [k] to each vertex v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We say a coloring ϕ is proper if ϕ(u) ̸= ϕ(v) for any uv ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' In fact, we always consider proper colorings unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Given a k-coloring ϕ of G, we define the color classes by ϕ−1(i) := {v ∈ V (G) : ϕ(v) = i} for any i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' For any distinct i, j ∈ [k], define Gij to be the subgraph of G induced by ϕ−1(i) ∪ ϕ−1(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' 2 Upper bounds for m(G, ϕ) In this section we prove upper bounds on the parameter m(G, ϕ) for planar graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We first present several lemmas for plane triangulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let G be a plane triangulation on at least 4 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Denote by EG the set of sepa- rating triangles of G, and by VG the set of maximal connected subgraphs of G without separating triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' The graph TG is defined to be the graph on VG with edge set EG such that G1, G2 ∈ VG are joined by T ∈ EG if and only if both G1 and G2 contain T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' 2 It is easy to see that VG is a family of 4-connected plane triangulations and TG is a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let VG := {G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' , Gt} and EG := {T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' , Tt−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' The graph G can be retrieved from the vertex- disjoint union of G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' , Gt by identifying the copies of triangle T in Gi, Gj for each T = GiGj ∈ EG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Hence � i∈[t] |V (Gi)| = |V (G)| + 3(t − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let G be a graph and ϕ be a proper coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' If A is an edge set of G such that A ∩ E(Gij) is an acyclic edge set for any distinct i, j ∈ [k], then there exists E′ ⊆ E(G) \\ A satisfying that |E′| = m(G, ϕ) and ϕ is an acyclic coloring of G − E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let E′ ⊆ E(G) be such that |E′| = m(G, ϕ), ϕ is an acyclic coloring of G − E′ and, subject to this, |E′ ∩ A| is minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Suppose there exists uv ∈ E′ ∩ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' There is precisely one cycle C in Gϕ(u)ϕ(v) − (E′ − uv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' As A ∩ E(Gϕ(u)ϕ(v)) is acyclic, there exists e′ ∈ E(C) \\ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Then Gϕ(u)ϕ(v)−(E′−uv+e′) is acyclic, |E′−uv+e′| = |E′| = m(G, ϕ) and |(E′−uv+e′)∩A| < |E′∩A|, contradicting our choice of E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Hence E′ ⊆ E(G) \\ A as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let G be a plane graph, T be a separating triangle of G and ϕ be a proper coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let A1 and A2 be the components of G − T, and for i ∈ [2], Gi be the subgraph of G induced by V (Ai) ∪ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Then m(G, ϕ) = m(G1, ϕ1) + m(G2, ϕ2), where ϕi denotes the restriction of ϕ on V (Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Without loss of generality, we let V (T) = {v1, v2, v3} with ϕ(vi) = i for i ∈ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' By Lemma 4, there exists E′ ⊆ E(G) \\ E(T) such that |E′| = m(G, ϕ) and ϕ is an acyclic coloring of G − E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' As Gi − (E′ ∩ E(Gi)) is acyclically colored by ϕi (i ∈ [2]), we have m(G, ϕ) = |E′| = |E′ ∩ E(G1)| + |E′ ∩ E(G2)| ≥ m(G1, ϕ1) + m(G2, ϕ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Similarly, by Lemma 4, let E′ i ⊆ E(Gi) \\ E(T) be such that |E′ i| = m(Gi, ϕi) and Gi − E′ i is acyclically colored by ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let E′ := E′ 1 ∪ E′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Observe that if there is a cycle C which is colored by only two colors in G − E′, then C must contain two vertices of T, say v1, v2, and C + v1v2 contains some cycle in G1 −E′ 1 or G2 −E′ 2 which uses only two colors as well, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Hence G−E′ is acyclically colored and m(G, ϕ) ≤ |E′| = |E′ 1| + |E′ 2| = m(G1, ϕ1) + m(G2, ϕ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let G be a plane triangulaion on at least 4 vertices and ϕ be a proper coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let VG := {G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' , Gt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We have m(G, ϕ) = � i∈[t] m(Gi, ϕi), where ϕi denotes the restriction of ϕ on V (Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We prove by induction on |VG|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' It trivially holds when |VG| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Suppose |VG| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let T ∈ EG, A1 and A2 be the components of G − T, and for i ∈ [2], Gi be the subgraph of G induced by V (Ai) ∪ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We may assume G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' , Gt′ ⊆ G1 and Gt′+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' , Gt ⊆ G2 for some 1 ≤ t′ < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Then, by Lemma 5 and the induction hypothesis, m(G, ϕ) = m(G1, ϕ1) + m(G2, ϕ2) = � i∈[t′] m(Gi, ϕi) + � i∈[t]\\[t′] m(Gi, ϕi) = � i∈[t] m(Gi, ϕi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let G be a 3-colorable plane triangulation on n vertices and ϕ be the unique proper 3-coloring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' For any distinct i, j ∈ [3], Gij is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Moreover, if n > 3, Gij is 2-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We prove by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' The triangulations of order at most 6 are listed in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Among these graphs, only the triangle and the octahedron are 3-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' It is not hard to verify that the claims hold for these two graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' From now on we assume that n > 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' As G is a 3-colorable triangulation, every vertex of G has an even degree, and hence there exists v ∈ V (G) with dG(v) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let v1v2v3v4v1 be the cycle induced by NG(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We have ϕ(vi) = ϕ(vi+2) for each i ∈ [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Suppose there exists i ∈ [2] such that vi and vi+2 have no common neighbor other than v, vi+1, vi+3, where v5 := v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We contract vivvi+2 to obtain G′ and call the new vertex v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let ϕ′ : V (G′) → [3] be such that ϕ′(v′) = ϕ(vi) and ϕ′(u) = ϕ(u) for u ∈ V (G′) \\ {v′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' It is clear 3 that ϕ′ is the unique proper 3-coloring of the triangulation G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' By the induction hypothesis, G′ ij is 2-connected for any distinct i, j ∈ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Then, one can easily prove by the construction that Gij is 2-connected for any distinct i, j ∈ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Suppose for every i ∈ [2], vi and vi+2 have some common neighbor other than v, vi+1, vi+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Since G is not the octahedron, it has some separating triangle T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let A1, A2 be the components of G−T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' We consider the subgraphs Gi of G induced by V (Ai)∪V (T) (i ∈ [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let ϕi be restriction of ϕ on V (Gi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' As |V (Gi)| > 3, it follows from the induction hypothesis that Gi jk is 2-connected for any distinct j, k ∈ [3] (i ∈ [2]), from which it immediately follows that Gjk is 2-connected for any distinct j, k ∈ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Figure 1: The triangulations of order at most 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Let G be a graph with a proper k-coloring ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' Denote by cij the number of connected components of Gij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' The number of edges we need to remove from Gij to make ϕ acyclic is |E(Gij)|−|V (Gij)|+cij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tFJT4oBgHgl3EQfkCwt/content/2301.11577v1.pdf'} +page_content=' As E(Gij) are edge-disjoint for distinct i, j, and each vertex v of G is contained in k − 1 subgraphs Gij, we know that m(G, φ) = � 1≤i 102 points, the Allan variance still shows 1/τA de- +pendency, as indicated by Eq. (5), whereas the MSE saturates +at 0. In this region, Cm(r) is almost 1, i.e., almost all of the +embedded vectors are neighbors of each other. This saturation +effect is one of the differences between the MSE and Allan +variance. +Figures 1(b) and (d) show the MSE and Allan variance of +1/ f noise, respectively. Theoretically, both are predicted to be +independent of τA, whereas the simulation results show some +τA dependency. This may be a result of the difficulty in prop- +erly generating 1/ f noise on scales from τA = 1 to 103, how- +ever it is noteworthy that the MSE, −Cm(r) and Allan variance +exhibit similar curves as a function of τA. +0 +1 +2 +3 +MSE +10 0 +10 1 +10 2 +10 3 +10 −3 +10 −2 +10 −1 +10 0 +Allan Var (arb. units) +10 −4 +10 −2 +10 0 +10 0 +10 1 +10 2 +10 3 +WGN +1/f +(a) +(b) +(c) +(d) +FIG. 1: Comparison of the MSE and Allan variance for WGN +and 1/ f noise. (a,b) The MSE and Cm(r) of WGN and 1/ f +noise, respectively. The black, red, and blue curves represent +the MSE, C3(r) and C2(r), respectively. Each curve is the av- +erage of independent results from 30 trials, and the filled area +around the curve represents the first and third quartile. The +variance of the computed results is so slight that it is diffi- +cult to observe the painted areas. (c,d) The Allan variance of +WGN and 1/ f noise, respectively. The Allan variance plots +and Cm(r) plots are logarithmic on both axes to observe the +1/τA dependency of WGN clearly, while the y-axis is linear +for the MSE plots, because the MSE is defined with the loga- +rithm of Cm(r). +B. +Physiological signal +Costa et al.1 introduced the MSE to demonstrate that physi- +ological signals observed in young, healthy individuals exhibit +versatile signal levels across a variety of time scales, whereas +unhealthy subjects do not. In this section, we compare the +MSE and Allan variance of the RR interval19 series, which are +sequences of intervals between adjacent R waves in electro- +cardiograms, for healthy subjects and those with congestive +heart failure (CHF) or atrial fibrillation (AF), following Costa +et al. Each dataset was obtained from PhysioNet.11–14 +The lengths of the time series varied, but the entirety of +each time series was used in the calculations, because adjust- +ing to the shorter ones would reduce the accuracy of the esti- +mation. To compare the Allan variance, the standard deviation +of each time series was normalized to 1. This normalization +is consistent with the MSE calculation, because r was fixed at +0.15×SD(s). The MSE and Allan variance for each dataset is +shown in Fig. 2. The Allan variance, MSE and −Cm(r) show +trends that are similar except for the location of the peaks. + +Information-theoretical analysis of statistical measures for multiscale dynamics +4 +Figures 2(a) and (e) respectively show the MSE and Allan +variance overlaid for all three cases. Figure 2(a) shows that +the RR interval for healthy subjects is the most complex on +long time scales, which is consistent with the results of the +previous study.1 As shown in Fig. 2(e), the ordering of the +different classes of subjects for the Allan variance and MSE +show a similar trend. However, the exact scale factor τA at +which each curve intersects differs from that of the MSE. It +seems plausible that the analysis of RR interval time series in +general could be performed using Allan variance instead of +the MSE. +C. +Laser chaos +The MSE and Allan variance of the noise time series exhibit +monotonic properties. However, the MSE and Allan variance +of physiological signals show more complex properties, but +the underlying dynamics are not completely known, which +makes further analysis difficult. +As an example of a time series that includes multiple time +scales and for which the underlying dynamics are known, +we discuss a phenomenon called LFF23 exhibited by a semi- +conductor laser with optical feedback. The Lang–Kobayashi +equations,10 a set of model equations, was used to generate the +time series. An example of a time series is shown in Fig. 3, +where fast chaotic oscillations on the order of GHz coexist +with irregular intensity dropouts and recovery on the order of +MHz. The results are presented in Fig. 4. As the simulation +step and time scale of the dynamics are important, τA was con- +verted to time. +Figure 4 shows that the MSE, Cm(r) and Allan variance all +capture the dynamics of fast oscillations on the order of GHz, +corresponding to the fast peak in the MSE and Allan variance, +and dropouts on the order of MHz, corresponding to the slow +peak. +IV. +INFORMATION THEORETICAL CONNECTIONS +BETWEEN THE MSE AND ALLAN VARIANCE +A. +Cm(r) and Allan variance +In Sec. III, we observe that −Cm(r) exhibits behavior sim- +ilar to that of the Allan variance. In this section, we discuss +the underlying mechanisms. +1. +Cm(r) decomposition +First, we consider decomposing Cm(r) by the difference in +the indices of the embedding vectors. Let M = ⌊N/τA⌋ be +the length of the coarse-grained time series, and Dm(i, j) be +dcheb(vm(i),vm( j)). Considering the expectation and symme- +try of i and j, and assuming s(t) or its first-order difference +s(t)−s(t −1) is strongly stationary, +⟨Cm(r)⟩ = +∑M−m +i=1 ∑M−m +j=1 +j̸=i +⟨u(r −Dm(i, j))⟩ +(M −m−1)(M −m) +(16) += +∑M−m +i=1 ∑M−m +j=1 +j̸=i +Pr(Dm(i, j) < r) +(M −m−1)(M −m) +(17) += ∑M−m +i=1 ∑M−m +j=i+1 2Pr(Dm(i, j) < r) +(M −m−1)(M −m) +(18) += ∑M−m−1 +k=1 +(M −m−k)Pr(Dm(1,1+k) < r) +�M−m +2 +� +. (19) +First we clarify the meaning of expectation. In the following, +we regard the time series s(t), t ∈ {1,2,··· ,N} under study +as a sequence of random variables following a specific prob- +ability distribution function (PDF). In this sense, Cm(r) and +u(r−Dm(i, j)) are also random variables, and thus each of the +possible values of Cm(r) and u(r −Dm(i, j)) has some proba- +bility of occurrence. The expectation is the weighted average +of the likelihood of every potential value. +Here we explain the transformation of the equations from +(16) to (19) line by line. +The only change made from +Eq. (16) to (17) is that the expectation term ⟨u(r −Dm(i, j))⟩ +is changed to the probability term Pr(Dm(i, j) < r). We as- +sume that a random variable sequence s(t), t ∈ {1,2,··· ,N} +follows a PDF, and calculate u(r −Dm(i, j)). As previously +mentioned, u(r −Dm(i, j)) is also a random variable that is +either 0 or 1 with the following probability: +u(r −Dm(i, j)) = +� +0 +(1−Pr(Dm(i, j) < r)) +1 +(Pr(Dm(i, j) < r)) +. +(20) +Consequently, +⟨u(r −Dm(i, j))⟩ = Pr(Dm(i, j) < r) +(21) +holds. +From Eq. (17) to (18), the factor 2 is added to the nu- +merator, and the range of summation over j is restricted to +j > i. Here, we use the fact that Dm(i, j) = Dm(j,i). From +Eq. (18) to (19), we must assume that Pr(Dm(i, j) < r) de- +pends only on the difference in the indices k = j−i. For exam- +ple, Pr(Dm(i+1, j +1) < r) is the same as Pr(Dm(i, j) < r). +This assumption is satisfied when the original signal s(t) or +its first-order difference s(t) − s(t − 1) is strongly stationary. +This is a reasonable assumption when studying a time series +from a dynamical system. The factor (M −m−k) in Eq. (19) +comes from the fact that, for each k, the number of pairs (i, j) +satisfying k = j −i is (M −m−k). For example, when k = 1, +the pairs are (1,2),(2,3),··· ,(M −m−1,M −m), as the total +number of embedded vectors is M −m. +The probability in Eq. (19) can be decomposed further +by considering conditional probabilities. Using the proposi- + +Information-theoretical analysis of statistical measures for multiscale dynamics +5 +0.03 +0.04 +0.05 +0.06 +0.07 +0 +0.5 +1 +1.5 +2 +MSE +healthy +CHF +AF +10 −2 +10 −1 +10 0 +Allan Var (arb. units) +20 +5 +10 +15 +20 +5 +10 +15 +20 +5 +10 +15 +20 +5 +10 +15 +Healthy +subjects +CHF +subjects +AF +subjects +(a) +(b) +(e) +(f) +(c) +(g) +(d) +(h) +200 +400 +600 +800 1000 +Beat Number +0.4 +0.5 +0.6 +Interval (sec) +200 +400 +600 +800 1000 +Beat Number +0.6 +0.8 +1 +Interval (sec) +200 +400 +600 +800 1000 +Beat Number +0 +2 +4 +Interval (sec) +FIG. 2: Comparison of the MSE and Allan variance for the RR interval time series. (a, e) Overlaid figure of the MSE and Allan +variance of the RR interval time series from healthy subjects and subjects with CHF or AF, respectively. (b,c,d) The MSE and +Cm(r) of the RR interval time series for healthy subject, CHF, and AF, respectively. The black, red, and blue curves represent +the MSE, C3(r) and C2(r), respectively. Each curve is the average of independent results from 147 subjects for healthy, 29 for +CHF, and 84 for AF, and the filled area around the MSE curve represents the first and third quartile. The first and third quartiles +of Cm(r) are omitted for legibility. (f,g,h) Allan variance of the corresponding time series. The filled area around the curve +represents the first and third quartile. Each dataset was obtained from PhysioNet.11–14 +0 +500 +1000 +Time (ns) +0 +5 +10 +15 +0 +500 +1000 +Time (ns) +1.4 +1.6 +1.8 +2 +2.2 +(a) +(b) +FIG. 3: Example of an LFF time series. The intensity is nor- +malized to that without optical feedback. (a) Original time +series. (b) Time series obtained by applying an ideal low-pass +filter with a cutoff frequency of 100 MHz to the time series in +(a). Sudden dropouts with gradual recovery are observed. +10 −2 10 −1 10 0 10 1 10 2 10 3 +0 +0.5 +1 +1.5 +2 +2.5 +MSE +10 −3 +10 −2 +10 −1 +10 0 +10 −2 10 −1 10 0 10 1 10 2 10 3 +10 −2 +10 −1 +10 0 +Allan Var (arb. units) +(a) +(b) +FIG. 4: Comparison of the MSE and Allan variance for an +LFF time series. (a) The MSE and Cm(r). The black, red, and +blue curves represent the MSE, C3(r) and C2(r), respectively. +(b) Allan variance. Note that the MSE, −Cm(r), and the Allan +variance exhibits similar τA dependencies. +tion (14), +Pr(Dm(1,1+k) < r) += Pr(dcheb(v(1),v(1+k)) < r) += Pr(|s(1)−s(1+k)| < r ∧|s(2)−s(2+k)| < r +∧···∧|s(m)−s(m+k)| < r) += Pr(D1(1,1+k) < r ∧D1(2,2+k) < r +∧···∧D1(m,m+k) < r) +(22) +holds. The probability of a product event, such as Eq. (22), +can be represented as a product of conditional probabilities. +For example, the probability that propositions A, B and C hold +simultaneously is +Pr(A∧B∧C) = Pr(B∧C | A)Pr(A) += Pr(C | A∧B)Pr(B | A)Pr(A). +(23) +In the same way, +Pr(Dm(1,1+k) < r) +=Pr(D1(1,1+k) < r) +m +∏ +i=2 +Pr +� +D1(i,i+k) < r | Di−1(1,1+k) < r +� +(24) +holds. +In the following, we show that the above conditional prob- +ability behaves oppositely to the Allan variance regardless of +k, while the unconditional probability does not. This point is +the key to understanding why −Cm(r), the MSE and Allan +variance show similar τA dependency. + +Information-theoretical analysis of statistical measures for multiscale dynamics +6 +Figure 6(a) shows a logarithmic-scale color map of +Pr(D1(1,1 + k) < r), which is the first term on the right- +hand side of Eq. +(24), whereas Fig. 6(b) shows that of +Pr(D1(2,2 + k) < r | D1(1,1 + k) < r), corresponding to the +conditional probability in the second term on the right-hand +side of Eq. (24) for i = 2. +Each probability was calculated for the LFF time series. +For k = 1, corresponding to the lowest row in Fig. 6(a) and +the solid curve in Fig. 6(c), the unconditioned probability +Pr(D1(1,2) < r) behaves opposite to the Allan variance (see +Fig. 4(b)). However, for larger k, the behavior differs from +that of the k = 1 case, as shown in the dashed plot in Fig. 6(c). +This can be understood as follows. +First, the following +equation holds: +D1(1,1+k) +=|¯s(τA)(k +1)− ¯s(τA)(1)| +=|¯s(τA)(k +1)− ¯s(τA)(k)+ ¯s(τA)(k)−···+ ¯s(τA)(2)− ¯s(τA)(1)| += +����� +k+1 +∑ +j=2 +∆¯s(τA)( j) +�����. +(25) +For k = 1, whether D1(1,2) = |∆¯s(τA)(2)| is smaller than r is +closely related to the Allan variance. If the Allan variance for +a certain τA is smaller than that of another τA, the ∆¯s(τA)(l) dis- +tribution is biased toward the center, because the mean value +of ∆¯s(τA)(l) is zero by definition. Consequently, the probabil- +ity D1(1,2) = |∆¯s(τA)(2)| < r is high for that τA. For larger +k, the ∑k+1 +j=2 ∆¯s(τA)( j) distribution is affected by the time cor- +relation of ¯s(τA)(l) that the time series under study inherently +has, and the Allan variance Var(∆¯s(τA)(l)) cannot predict the +∑k+1 +j=2 ∆¯s(τA)( j) distribution well, so Pr(D1(1,1+k) < r) does +not behave in a manner that is strongly correlated with the +Allan variance. +Conversely, the conditional probability Pr(D1(2,2 + k) < +r | D1(1,1 + k) < r) depicted in Fig. 6 (b) shows a similar +trend to that of the Allan variance regardless of k, which we +can also observe in Fig. 6(d). That is, Pr(D1(2,2 + k) < r | +D1(1,1+k) < r) is the main connection that explains the sim- +ilarity between the MSE and Allan variance. To examine this +further, we discuss the variance of the corresponding quan- +tity, rather than considering the probabilities in the following +sections. +2. +Neighborhood-Likelihood-to-Variance-Relationship +(NLVR) +In this section, we introduce an assumption called the +Neighborhood-Likelihood-to-Variance-Relationship (NLVR) +to connect the probability discussion to the variance dis- +cussion. +First, define ∆Pr(τA) as the difference between +Pr(D1(i,i+k) < r | Di−1(1,1+k) < r) for τA and τA−1. In the +same way, ∆Var(τA) is the difference between Var(¯s(τA)(i + +k) − ¯s(τA)(i) | Di−1(1,1 + k) < r) for τA and τA − 1. Please +note that the conditional probability Pr(D1(i,i + k) < r | +Di−1(1,1+k) < r) is the same as that in Eq. (24), and the ab- +solute value of ¯s(τA)(i+k)− ¯s(τA)(i), referred to in ∆Var(τA), +is identical to D1(i,i+k) in the definition of ∆Pr(τA). +Then, to connect the probability with the variance, we as- +sume the Neighborhood-Likelihood-to-Variance-Relationship +(NLVR), as follows: +∆Pr(τA)∆Var(τA) < 0. +(26) +The inequality (26) states that the signs of ∆Pr(τA) and +∆Var(τA) are always opposite. +We will explain the NLVR in more detail. Both Pr(D1(i,i+ +k) < r | Di−1(1,1 + k) < r) and Var(¯s(τA)(i + k) − ¯s(τA)(i) | +Di−1(1,1+k) < r) are statistics of ¯s(τA)(i+k)− ¯s(τA)(i) under +the same condition of Di−1(1,1+k) < r. Considering the PDF +of ¯s(τA)(i+k)− ¯s(τA)(i), Pr(D1(i,i+k) < r | Di−1(1,1+k) < r) +represents the area under the PDF in the range [−r, r]. In ad- +dition, when there is no condition, the mean of ¯s(τA)(i + k) − +¯s(τA)(i) is zero by definition. Here we also assume that the +mean value of ¯s(τA)(i+k)− ¯s(τA)(i) is close to zero under the +condition Di−1(1,1+k) < r. When ∆Pr(τA) > 0, the distribu- +tion of ¯s(τA)(i+k)− ¯s(τA)(i) is more centrally biased. Accord- +ingly, when ∆Pr(τA) > 0, ∆Var(τA) is likely to be negative. +Figure 5 schematically illustrates the concept of the NLVR. +The black curves in Figs. 5 (a) and (b) represent the PDF of +¯s(τA)(i+k)− ¯s(τA)(i) under the condition Di−1(1+k,1) < r for +τA −1 and τA, respectively. The red-filled areas and black bars +show Pr(D1(i,i+k) < r | Di−1(1,1+k) < r) and Var(¯s(τA)(i+ +k) − ¯s(τA)(i) | Di−1(1,1 + k) < r), respectively. ∆Pr(τA) and +∆Var(τA) are the difference in the red-filled areas and black +bars between Figs. 5(b) and (a), respectively. Please note that +the plots shown here, based on Gaussian distributions, are for +explanatory purposes only and are not obtained from a time +series. The validity of the NLVR is discussed in Sec. IV C. +(a) +(b) +FIG. 5: A visual representation of the NLVR. (a, b) ¯s(τA)(i + +k)− ¯s(τA)(i) distribution under the condition Di−1(1+k,1) < r +for τA −1 and τA, respectively. The red-filled area and the the +black bars represent Pr(D1(i,i + k) < r | Di−1(1,1 + k) < r) +and Var(¯s(τA)(i + k) − ¯s(τA)(i) | Di−1(1,1 + k) < r), respec- +tively. +3. +Variance decomposition +Here +we +further +discuss +the +conditional +variance +Var(¯s(τA)(i + k) − ¯s(τA)(i) | Di−1(1,1 + k) < r), instead of the + +Information-theoretical analysis of statistical measures for multiscale dynamics +7 +conditional probability Pr(D1(i,i+k) < r | Di−1(1,1+k) < r) +following the NLVR assumption, as mentioned in Sec. IV A 1. +By decomposing ¯s(τA)(i+k)− ¯s(τA)(i) as +¯s(τA)(i+k)− ¯s(τA)(i) += ¯s(τA)(i+k)− ¯s(τA)(i+k −1)+ ¯s(τA)(i+k −1) +− ¯s(τA)(i)+ ¯s(τA)(i−1)− ¯s(τA)(i−1) +=∆¯s(τA)(i+k)−∆¯s(τA)(i)+(¯s(τA)(i+k −1)− ¯s(τA)(i−1)), +(27) +the +conditional +variance, +Var(¯s(τA)(i + k) − ¯s(τA)(i) | +Di−1(1,1+k) < r), can be decomposed as follows: +Var(¯s(τA)(i+k)− ¯s(τA)(i) | Di−1(1,1+k) < r) +=Var(∆¯s(τA)(i+k) | ···) ++Var(∆¯s(τA)(i) | ···) +−2Cov(∆¯s(τA)(i+k),∆¯s(τA)(i) | ···) ++Var(¯s(τA)(i+k −1)− ¯s(τA)(i−1) | ···) ++2Cov(∆¯s(τA)(i+k),(¯s(τA)(i+k −1)− ¯s(τA)(i−1)) | ···) +−2Cov(∆¯s(τA)(i),(¯s(τA)(i+k −1)− ¯s(τA)(i−1)) | ···). +(28) +Here, the condition Di−1(1,1 + k) < r is abbreviated by the +symbol ··· on the right-hand side of Eq. (28). +[Overview of the variance decomposition] +There are six terms on the right-hand side of Eq. (28). +The logarithmic scale color map of each term is shown in +Figs. 7(a)–(f) to determine which term has the greatest in- +fluence on the conditional variance. Figure 7(g) represents +the summation of each term, which is the original conditional +variance. Meanwhile, Fig. 7(h) show cross-sectional profiles +when k = 50 regarding the first term (Fig. 7(a)), the third term +(Fig. 7(c)) and the total conditional variance (Fig. 7(g)). +As discussed shortly below, the first through third terms in +Eq. (28) are dominant, as shown in Figs. 7(a)–(c), whereas +they show a similar τA dependency with the Allan variance +regardless of k, as indicated in Fig. 7(h). +Please note that we plotted the absolute values of the +covariance terms, because the covariance can be negative. +However, the τA dependency of Cov(∆¯s(τA)(i + k),∆¯s(τA)(i) | +Di−1(1,1+k) < r), shown in Figs. 7(c) and (h), is still similar +to that of the Allan variance. Conversely, the remaining panels +(Figs. 7(d)–(f)) exhibit small values, regardless of k and τA. +Therefore, +we observe that the conditional variance +Var(¯s(τA)(i + k) − ¯s(τA)(i) | Di−1(1,1 + k) < r), which is +shown in Figs. 7(g) and (h), and the conditional probability +Pr(|¯s(τA)(i + k) − ¯s(τA)(i)| < r | Di−1(1,1 + k) < r), shown in +Fig. 6(b) for the case of i = 2, exhibit the opposite τA depen- +dency to the Allan variance from the NLVR. +[The first and the second terms] +The first and second terms are the conditional variances of +∆¯s(τA)(i+k) and ∆¯s(τA)(i), namely the conditional Allan vari- +ances, respectively. Although the condition somewhat affects +the ∆¯s(τA)(i + k) and ∆¯s(τA)(i) distribution, the size relation- +ship of the variance of ∆¯s(τA)(i + k) and ∆¯s(τA)(i) concerning +τA is almost the same as the Allan variance. +[The third term] +However, the third term is strongly influenced by the con- +dition. In the absence of the condition, if k is sufficiently +large, there would be little correlation between ∆¯s(τA)(i + k) +and ∆¯s(τA)(i); thus, the covariances are expected to be small. +Fig. 8(a) shows the unconditional covariance of ∆¯s(τA)(i + k) +and ∆¯s(τA)(i), where the color scale is the same as in Fig. 7. +Comparing to the conditional covariance plotted in Fig. 7(c), +this covariance without the condition is smaller in most places. +By contrast, when the condition is satisfied, i − 1 consec- +utive points up to ¯s(τA)(i + k − 1) and ¯s(τA)(i − 1) are close +values. In that case, ∆¯s(τA)(i+k) and ∆¯s(τA)(i), i.e., the differ- +ence between these points and their consecutively following +point, often have the same sign, and the distribution of their +magnitudes follows the Allan variance. +Figure 8(b) shows an example of a pair of ∆¯s(τA)(i + k) +and ∆¯s(τA)(i) satisfying the condition for i = 3. Because we +can regard v2(7) and v2(17), denoted by the blue dots, as +identical under the tolerance r, the red arrows, representing +∆¯s(τA)(9) and ∆¯s(τA)(19) are likely to be similar. By the def- +inition of covariance, if ∆¯s(τA)(i + k) and ∆¯s(τA)(i) are simi- +lar, Cov(∆¯s(τA)(i+k),∆¯s(τA)(i)) is close to Var(∆¯s(τA)(i+k)), +namely the Allan variance. From these considerations, the +conditional covariance behaves similarly to the Allan vari- +ance. +Please note that the form of the condition in the Fig. 8(b), +namely D2(7,17) < r is slightly different from Di−1(1,1 + +k) < r. However, as we assume stationary, it is allowed to +shift the indices. +For example, in this case, we examined +∆¯s(τA)(19) and ∆¯s(τA)(9) under the condition of D2(7,17) < r, +and this corresponds to the shifting of indices by six. To be +more specific, we investigated ∆¯s(τA)(19) = ∆¯s(τA)(3+10+6) +and ∆¯s(τA)(3 + 6), under the condition of D2(7,17) < r = +D2(1+6,1+10+6), where i = 3 and k = 10. +[The fourth to the sixth terms] +The fourth to sixth terms do not contribute significantly to +the conditional variance, as mentioned earlier. This is due to +the fact that ¯s(τA)(i+k−1)− ¯s(τA)(i−1), which is involved in +the variance or covariance in the fourth to sixth terms, is small +when the condition Di−1(1,1+k) < r is satisfied. Once again, +the condition implies a sort of similarity on the time scale of +i−1 points, and when these points are similar, the variance of +these terms containing the difference will be small. +[Summary of the decomposition] +Adding the above six terms together, we see that the con- +ditional variance eventually behaves similarly to the Allan +variance, as shown in Figs. 7(g) and (h). Because the con- +ditional variances and conditional probabilities are assumed +by the NLVR to act in opposite directions for increasing scale +factors, the conditional probabilities and Cm(r), represented +by the weighted average of the product of probability without +conditions Pr(D1(1,1+k) < r) and the conditional probabili- +ties Pr(D1(i,i+k) < r | Di−1(1,1+k) < r), in turn, behave in + +Information-theoretical analysis of statistical measures for multiscale dynamics +8 +the opposite direction for increasing scale factors for the Allan +variance. +Although we observe the conditional variances and covari- +ances from an LFF time series as an example, the discus- +sion of behavior shown by the variances and covariances is +valid for other time series to some extent. In the discussion, +we assumed strong stationarity of the signal s(t) or its first- +order difference to estimate probabilities and variances from +the obtained time series. This is because Pr(Dm(i, j) < r) in +Eq. (17) cannot be estimated from a single time series. Con- +versely, the NLVR assumption between probability and vari- +ance can also be applied to Pr(|¯s(τA)(i) − ¯s(τA)( j)| < r) and +Var(¯s(τA)(i) − ¯s(τA)( j)) and could be valid to a certain extent. +Thus, similar behavior exhibited by the MSE and Allan vari- +ance should hold for a more general class of signals. +20 +40 +60 +80 +100 +10−1 +100 +10 −2 10 −1 10 0 +10 1 +10 2 +10 3 +10 −1 +10 0 +Probability +10 −2 10 −1 10 0 +10 1 +10 2 +10 3 +(a) +(b) +(c) +(d) +FIG. 6: (a,b) Logarithmic scale color maps of +(a) Pr(D1(1,1+k) < r) and +(b) Pr(D1(2,2+k) < r | D1(1,1+k) < r) +for the LFF time series up to k = 100, respectively. (c,d) One- +dimensional plots of the two-dimensional data displayed in (a) +and (b) along the two arrows (k = 1 and k = 100). Solid and +dashed plots correspond to k = 1 and k = 100, respectively. +B. +The MSE and Allan variance +In Sec. IV A, the connection between Cm(r) and the Allan +variance was discussed based on the connection between the +conditional probability and variance. We can also apply this +discussion to the connection between the MSE and Allan vari- +ance. +To simplify the discussion, we assume a sufficiently long +(strongly stationary) time series, so that we can consider +Cm(r) to be the same as the expectation ⟨Cm(r)⟩. +This is +equivalent to assuming ergodicity. +Under this condition, +MSE +=−log Cm+1(r) +Cm(r) +=−log ∑M−m−1 +k=1 +(M −m−k)Pr(Dm+1(1,1+k) < r) +∑M−m−1 +k=1 +(M −m−k)Pr(Dm(1,1+k) < r) . (29) +Pr(Dm+1(1,1 + k) < r), in the numerator of Eq. (29) can be +decomposed similarly, as shown in Sec. IV A, as follows: +Pr(Dm+1(1,1+k) < r) +=Pr(D1(m+1,m+1+k) < r | Dm(1,1+k) < r) +Pr(Dm(1,1+k) < r). +(30) +Let ak = (M − m − k)Pr(Dm(1,1 + k) < r) and bk = +Pr(D1(m + 1,m + 1 + k) < r | Dm(1,1 + k) < r). Consider- +ing the equality (M − m − k)Pr(Dm+1(1,1 + k) < r) = akbk, +Eq. (29) can be rewritten as follows: +MSE = −log Cm+1(r) +Cm(r) += −log ∑M−m−1 +k=1 +akbk +∑M−m−1 +k=1 +ak +(31) +Equation (31) states that the MSE is the negative logarithm +of the weighted average of bk. Because bk is the conditional +probability, the behavior of the MSE can be explained by the +conditional probability as well as Cm(r). +More precisely, we must care about the expectation. Cm(r) +can be represented with probability when considering the ex- +pectation ⟨Cm(r)⟩. The expectation of the MSE is +⟨MSE⟩ = +� +−log Cm+1(r) +Cm(r) +� +(32) +≥ −log +�Cm+1(r) +Cm(r) +� +. +(33) +From Eq. (32) to (33), we use the fact that the negative log- +arithm is convex, along with Jensen’s inequality. Eq. (33) in +general is not the same as +−log ⟨Cm+1(r)⟩ +⟨Cm(r)⟩ += −log ∑M−m−1 +k=1 +akbk +∑M−m−1 +k=1 +ak +. +(34) +However, it is noteworthy that Costa et al.3 showed that the +MSE values calculated theoretically, using probability, agree +well for the WGN and 1/ f noise cases. +C. +Connection between the probability and variance +In Secs. IV A and IV B, we discussed the connection be- +tween Cm(r) or the MSE and Allan variance based on the as- +sumption of NLVR. In this section, we discuss the extent to +which NLVR is valid, and what happens if it is violated. + +Information-theoretical analysis of statistical measures for multiscale dynamics +9 +20 +40 +60 +80 +100 +10-2 10-1 100 101 102 103 +20 +40 +60 +80 +100 +10-2 10-1 100 101 102 103 +10-3 +10-2 +10-1 +100 +10-2 10-1 100 101 102 103 10-2 10-1 100 101 102 10310-6 +10-4 +10-2 +100 +102 +Variance +from (g) +from (a) +from (c) +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) + +FIG. 7: Element-wise logarithmic-scale color maps of the conditional variance Var(¯s(τA)(i + k) − ¯s(τA)(i) | Di−1(1,1 + k) < r) +given by Eq. (28). There are six terms of an LFF time series up to k = 100: +(a) Var(∆¯s(τA)(i+k) | Di−1(1,1+k) < r), +(b) Var(∆¯s(τA)(i) | Di−1(1,1+k) < r), +(c) 2|Cov(∆¯s(τA)(i+k),∆¯s(τA)(i) | Di−1(1,1+k) < r)|, +(d) Var(¯s(τA)(i+k −1)− ¯s(τA)(i−1) | Di−1(1,1+k) < r), +(e) 2|Cov(∆¯s(τA)(i+k),(¯s(τA)(i+k −1)− ¯s(τA)(i−1)) | Di−1(1,1+k) < r)| and +(f) 2|Cov(∆¯s(τA)(i),(¯s(τA)(i+k −1)− ¯s(τA)(i−1)) | Di−1(1,1+k) < r)|. +(g) The term before the above decomposition, which is Var(¯s(τA)(i+k)− ¯s(τA)(i) | Di−1(1,1+k) < r). +(h) Cross-sectional profiles when k = 50 from (a) (red curve), (c) (blue curve), and (g) (black curve) indicated by the arrows. +Please note that for parts (c), (e), and (f) we plot the absolute value of the covariance, because covariance can be negative, and +our color scale is logarithmic. However, 2Cov(∆¯s(τA)(i+k),∆¯s(τA)(i) | Di−1(1,1+k) < r), shown in (c), is mostly positive. In +(g), although the points with negative values are neglected, almost the entire curve is depicted. +[Considering independent identical distribution (i.i.d.)] +For simplicity, we consider the case of independent identi- +cal distributions (i.i.d.). In this case, the conditional probabil- +ity distribution of (¯s(τA)(i + k) − ¯s(τA)(i) | Di−1(1,1 + k) < r) +is identical to the probability distribution of (¯s(τA)(i + k) − +¯s(τA)(i)), which is independent of the condition (Di−1(1,1 + +k) < r). +The distribution does not depend on k (see Ap- +pendix A). Therefore, it is sufficient to consider the distri- +bution of ¯s(τA)(i + 1) − ¯s(τA)(i). +In addition, the variances +of ¯s(τA)(i) and ¯s(τA)(i + 1) − ¯s(τA)(i) are Var(s(1))/τA and +2Var(s(1))/τA, respectively. +Therefore, the Allan variance for an i.i.d. system is in- +versely proportional to τA. +For the WGN case, owing to +the reproductive property, the PDF of ¯s(τA)(i) and ¯s(τA)(i + +1) − ¯s(τA)(i) are also Gaussian, with variances Var(s(1))/τA +and 2Var(s(1))/τA, respectively. +In addition, because the +mean value of ¯s(τA)(i + 1) − ¯s(τA)(i) is zero, the probability +|¯s(τA)(i+1)− ¯s(τA)(i)| < r can be represented as follows: +Pr(|¯s(τA)(i+1)− ¯s(τA)(i)| < r) += +� +1 +2π2σ/τA +� r +−r exp +� +− +x2 +2·(2σ/τA)2 +� +dx, +(35) +where σ2 = Var(s(1)), and increases monotonically with in- + +Information-theoretical analysis of statistical measures for multiscale dynamics +10 +10 −1 +10 0 +10 1 +10 2 +10 3 +20 +40 +60 +80 +100 +0 +1 +2 +3 +4 +5 +6 +5 +10 +15 +20 +(a) +(b) +FIG. 8: +(a) The color map of the unconditional version +of the third term of the conditional variance (Eq. (28): +2|Cov(∆¯s(τA)(i+k),∆¯s(τA)(i)|. The color scale is the same as +in Fig. 7. (b) ∆¯s(τA)(i + k) and ∆¯s(τA)(i) under the condition +of D2(7,7 + 10) < r. Here, i = 3 and k = 10. The blue dots +represent the two consecutive points which we can regard as +v2(7) and v2(17) satisfying the condition D2(7,7 + 10) < r. +The red dots denote ¯s(τA)(9) and ¯s(τA)(19) and the red arrows +show ∆¯s(τA)(9) and ∆¯s(τA)(19). +creasing τA. Therefore, NLVR holds for all τA. +The variance of ¯s(τA)(i + 1) − ¯s(τA)(i) continuously de- +creases when τA increases in the i.i.d. case. That is, ∆Var(τA) +is negative for all τA. The cases in which NLVR does not +hold are those in which the area under the PDF of ¯s(τA)(i + +1)− ¯s(τA)(i) in the range [−r, r] decreases with increasing τA, +which means ∆Pr(τA) is also negative. This situation seems +unlikely to occur when the i.i.d. process is unimodal. That +is to say, the reduction of the variance means the shrinking of +the distribution toward the center, which means an increase in +the probabilities around the center. +[When NLVR is invalid] +However, situations that violate NLVR are likely to occur +when, for example, the original system exhibits a multimodal +distribution. It should be noted that ¯s(τA)(i) of a multimodal +distribution can have more peaks than the distribution of s(t), +owing to the effect of averaging. Thus, ¯s(τA)(i + 1) − ¯s(τA)(i) +may also have many peaks. +The number of peaks increases as τA increases; however, +beyond a certain point, the number of peaks eventually de- +creases because the peaks fuse with each other. According +to the central limit theorem, the distribution of ¯s(τA)(i) and +¯s(τA)(i + 1) − ¯s(τA)(i) finally assumes shapes close to a Gaus- +sian distribution in the i.i.d. case. As the number of peaks +increases, the area of the PDF of ¯s(τA)(i + 1) − ¯s(τA)(i) near +the center is distributed to each peak, thus, ∆Pr(τA) is neg- +ative until the effects of this distribution and peaks merging +become antagonistic. +[Violation of NLVR by bimodal distributions] +Figure 9 presents an example of this discussion. Figure 9(a) +shows a histogram of the original bimodal i.i.d. system com- +posed of two Gaussians centered around ±1. Figures 9(c), +(e), and (g) show the histograms of the coarse-grained time +series for τA = 1, 2, 5, 100, respectively. Recall that coarse- +graining means averaging over neighboring points. Because +the process is i.i.d., two consecutive points have a 50% prob- +ability of coming from opposite sides of the origin. Thus, for +τA = 2 shown in Fig. 9(b), a third peak appears around the +origin. More precisely, the leftmost peak shown in Fig. 9(c) +corresponds to the case in which two consecutive s(t) points +are negative, i.e., from the left peak in Fig. 9(a), which has a +probability of 50% × 50% = 25%. Similarly, the rightmost +peak in Fig. 9(c) corresponds to two consecutive positive s(t) +points, and the center peak corresponds to either positive and +negative, or negative and positive points. The number of peaks +increases up to a certain τA, until they start to merge with each +other. For a sufficiently large τA, the distribution approaches +a Gaussian distribution, as shown in Fig. 9(g). +Figures 9(b), (d), (f), and (h) show the histograms of the dif- +ference between adjacent coarse-grained points ¯s(τA)(i+1)− +¯s(τA)(i) for τA = 1, 2, 5, 100, respectively. The red filled area +represents the range [−r, r] and the black bar denotes the stan- +dard deviation of the distribution. In fact, the distributions can +be regarded as convolutions of the corresponding distribution +of ¯s(τA)(i). Similar to ¯s(τA)(i), the number of peaks increases +until a certain τA and then decreases. In such cases, ∆Pr(τA) +is negative until a certain τA, which can be observed from the +reduction of the red area in Fig. 9(b)–(f), whereas ∆Var(τA) is +always negative. Thus, the NLVR is violated. +0 +0.02 +0.04 +Probability +0 +0.02 +0.04 +Probability +−1 +0 +1 +0 +0.02 +0.04 +Probability +−2 +−1 +0 +1 +2 +0 +0.02 +0.04 +Probability +0 +0.02 +0.04 +Probability +0 +0.02 +0.04 +Probability +0 +0.02 +0.04 +Probability +0 +0.02 +0.04 +Probability +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +FIG. 9: Example of histograms of a bimodal i.i.d. +sys- +tem. (a,c,e,g) Histograms of the coarse-grained time series +for τA = 1, 2, 5, 100, respectively. +(b,d,f,h) Histograms +of the difference between adjacent coarse-grained points for +τA = 1, 2, 5, 100, respectively. Red areas represent the range +[−r, r]. Black bars denote the standard deviation. +[Disagreement of MSE and Allan variance] +Plots of Pr(D1(i,i + 1) = |¯s(τA)(i + 1) − ¯s(τA)(i)| < r), the +MSE, Cm(r) and Allan variance of this bimodal i.i.d. sys- +tem are presented in Fig. 10. +The dashed black line in +Fig. 10(a) shows Pr(|¯s(τA)(i+1)− ¯s(τA)(i)| < r). In contrast to +the continuously decreasing Allan variance, Pr(|¯s(τA)(i+1)− + +Information-theoretical analysis of statistical measures for multiscale dynamics +11 +¯s(τA)(i)| < r) decreases until τA = 8 and then increases. Cm(r) +(red and blue curves) shows the same trend, and the MSE +(black curve) inherits the inverted trend. Actually ⟨Cm(r)⟩ = +Pr(|¯s(τA)(i+1)− ¯s(τA)(i)| < r)m and MSE ≃ −logPr(|¯s(τA)(i+ +1) − ¯s(τA)(i)| < r) (see Appendix A). In contrast, the Allan +variance curve plotted in Fig. 10(b) shows 1/τA dependency +(note that the plot is logarithmically scaled), as discussed +above. Because the example under study here was an i.i.d. +process, the Allan variance result may be considered the more +reliable one, as it does not detect any specific time scales of +relevance, whereas the MSE indicates a higher level of com- +plexity for τA = 8 than at other scales. +10 0 +10 1 +10 2 +10 3 +0 +0.5 +1 +1.5 +MSE +10 −3 +10 −2 +10 −1 +10 0 +Probability +10 0 +10 1 +10 2 +10 3 +10 −3 +10 −2 +10 −1 +10 0 +Allan Var (arb. units) +(a) +(b) +FIG. 10: Comparison of the MSE and Allan variance for the +bimodal system shown in Fig. 9(a). (a) The MSE, Cm(r) and +Pr(|¯s(τA)(i + 1) − ¯s(τA)(i)| < r) of a time series obtained from +the system. +The black, red, and blue curves represent the +MSE, C3(r) and C2(r), respectively. The dashed black curve +represents Pr(D1(i,i + 1) = |¯s(τA)(i + 1) − ¯s(τA)(i)| < r). (b) +The Allan variance of the corresponding time series. Herein, +the MSE and the Allan variance clearly exhibit different τA +dependencies. +From these discussions we can see that the NLVR is not al- +ways valid. For example, the condition may not hold when the +original time series s(t) contains a multimodal distribution. If +the NLVR is violated, meaning that ∆Pr(τA)∆Var(τA) > 0 for +a large number of k, Cm(r), expressed as a weighted average +of the probability, is expected to change in the same direc- +tion as the Allan variance. As a result, the Allan variance and +MSE change with the opposite sign. In i.i.d. cases, ∆Pr(τA) +and ∆Var(τA) are independent of k, so if NLVR does not hold +for a specific τA and k, it is violated for all k. +Therefore, in the range of τA where NLVR holds, the MSE +and the Allan variance show similar τA dependence; in the +range where the NLVR is violated, they behave oppositely +to τA. If the time series are time-correlated, as in dynamical +systems, the NLVR may be violated by even more complex +mechanisms. +D. +Empirical validation of the NLVR +Finally, we examine the extent to which the NLVR is sat- +isfied based on empirical data. Specifically, the change in +conditional probability (∆Pr(τA)) and conditional variance +(∆Var(τA)) are calculated for the time series studied in Sec- +tions III and IV C up to k = 100. +First, Fig. 11(a) shows a histogram of the intensity ob- +served in the LFF time series. The intensity is normalized +to that without optical feedback, i.e., the intensity in the sin- +gle mode.24 The histogram shown in Fig. 11(a) has a strong +peak near the origin and a smaller peak near the normalized +intensity of 3. That is, the probability distribution exhibits a +somewhat multimodal distribution, which may cause a viola- +tion of the NLVR. +Figures 11(b), (c) and (d) show the scatter plot of +(∆Pr(τA),∆Var(τA)) for LFF, WGN, and bimodal distribu- +tion, respectively, as explained below. If the NLVR holds, +a positive ∆Pr(τA) means a negative ∆Var(τA) and a negative +∆Pr(τA) means a positive ∆Var(τA). That is to say, the points +in the scatter diagram should be in the second or fourth quad- +rants. +From Fig. 11(b), almost all the sampling points are con- +centrated in the second and fourth quadrants, i.e., in regions +where the signs of ∆Pr(τA) and ∆Var(τA) are opposite. That is +to say, even though the probability distribution contains multi- +modality, this LFF system mostly does not violate the NLVR. +From these observations, we speculate that the peaks of the +distribution would need to be more separated to lead to a dis- +agreement of the Allan variance and MSE. +Similarly, Fig. 11(c) shows ∆Pr(τA) and ∆Var(τA) for the +WGN time series. As discussed above, ∆Pr(τA) should al- +ways be positive, whereas ∆Var(τA) is always negative, i.e., +every point should be in the second quadrant. However, the +distribution calculated from the time series is not precisely the +Gaussian distribution, so some points are in the third quadrant. +Finally, Fig. 11(d) shows the same scatter plot for the bi- +modal case. In contrast to the LFF and WGN cases, many +points are in the third quadrant in violation of the NLVR. This +was expected, as we constructed this case specifically as a +counter-example. +LFF +WGN +Bimodal +(a) +(b) +(c) +(d) +FIG. 11: ∆Pr(τA) and ∆Var(τA) for m = 2 and i = 2, up to +k = 100. (a) Histogram of the LFF time series. (b, c, d) Scatter +plot of ∆Pr(τA) and ∆Var(τA) for (b) the LFF time series, (c) +WGN time series, and (d) the bimodal time series. When the +NLVR holds, the points in the scatter diagram should be in the +second or fourth quadrants. + +0 +-0.5 +0 +5 +10 +15 +-5 +I / Io +0.05 +0.05 +△Pr(TA) +△Pr(TA) +0 +0 +-0.05 +-0.05 +.1 +0 +1 +△Var(TA)0 +5 +△Var(TA) +0 +1 +△Var(TA)0.01 +0.5 +Probability +△Pr(TA) +0.005Information-theoretical analysis of statistical measures for multiscale dynamics +12 +E. +Computation of the MSE and Allan variance +As introduced in Sec. II B, the definition of MSE involves +probabilities, whereas that of the Allan variance is based on +the variability of the data under study. Section IV C reveals +the common underlying mechanism from an information- +theoretic viewpoint despite the seemingly different definitions +of the statistical measures of multiscale dynamics. In this sec- +tion, we discuss the difference from the viewpoint of compu- +tation between the MSE and Allan variance. +For the MSE, calculating Cm(r) is a computationally de- +manding task. Computing Eq. (9) for all i requires O(N2) +calculations, as all pairs of embedded vectors are compared. +Some implementations also require O(N2) memory to store +all the calculated Chebyshev distances. Thus, the MSE eval- +uates the details of the probability distribution at a high com- +putational cost. +In contrast, the Allan variance calculation requires only +O(N) computations and memory. The Allan variance does +not consider the probability distribution; it depends only on +the differences ∆¯s(τA)(l) of successive coarse-grained points +¯s(τA)(l). As discussed in Sec. IV C, the MSE for the i.i.d. pro- +cess depends on the distribution, whereas the Allan variance +does not. +Despite being computationally cheaper, it is not obvious +that the Allan variance has any disadvantages when compared +to the MSE for extracting multiscale features. The Allan vari- +ance is a statistical measure that is similar but not identical to +the MSE, and that quantifies slightly different aspects of the +time series. +In the literature, the range of applications of the MSE is +versatile, such as bearing fault detection25 and sleep level +qualification.26 However, the MSE suffers from severe com- +putational difficulties as discussed above. +Concerning the +similar properties of the MSE and Allan variance, as well +as the computationally lightweight nature of Allan variance, +the extension of the Allan variance to real-time applications, +such as bearing fault detection or prediction of epilepsy from +electroencephalography (EEG), would be an interesting future +topic. +V. +CONCLUSION +In this study, we examined the similarities shown by the +multiscale statistics the MSE and Allan variance, and dis- +cussed the underlying mechanisms through an information- +theoretic analysis. It is noteworthy that although the apparent +definitions of the MSE and Allan variance are significantly +different, they show a similar behavior for a wide range of +time-series data. +We experimentally confirmed the similar +properties of the MSE and Allan variance observed in LFF +in chaotic lasers and physiological heartbeat data, as well as +white Gaussian and 1/f noise. The connection can be un- +derstood by decomposing the conditional probabilities in the +MSE and extracting the dominant contributions. We derived +a condition which must be satisfied for the MSE and Allan +variance to exhibit similar tendencies via a discussion of con- +ditional probabilities. Then, we artificially constructed a ran- +dom sequence that violates the condition, leading to incon- +sistent MSE and Allan variance behavior. We also quantita- +tively demonstrated that the aforementioned LFF and heart- +beat, which are physically plausible systems, mostly satisfy +the condition. +Finally, we discussed future research topics. +Using Al- +lan variance instead of the MSE may lead to more computa- +tionally lightweight applications that are suitable for real-time +tasks. In addition, there is a possibility of integrating further +developments that have been devised for the MSE9 and Allan +variance.27 +Furthermore, more research on the theoretical foundations +of coarse-graining and the MSE is needed. The MSE research +to date has focused mainly on its application as a statistical +tool, and there has been little research on its theoretical foun- +dations. The relationship between the dynamics of a coarse- +grained time series and those of the original time series is +unclear. Coarse-graining can be regarded as a combination +of a moving-average filter and downsampling; however, ac- +cording to a previous study,28 a linear filter applied to the +original time series during Takens’ embedding preserves its +topological properties. From this theorem, it may be possi- +ble to discuss the theoretical basis of coarse-graining from +the viewpoint of dynamical invariants, including entropies. +Meanwhile, the theoretical foundations of the MSE should be +more complicated, as the MSE shares the tolerance r for all +scales. As pointed out by Humeau-Heurtier,9 more and more +embedded vectors may be regarded as neighbors of each other +as the scale factor τA increases, owing to the reduction of the +variance of the coarse-grained time series. Notably, Costa et +al.3, the original proposers of the MSE, pointed out that the +variance changes induced by coarse-graining are related to the +temporal structures of the original time series. We may need a +framework that allows us to connect the dynamical invariants +at each time scale. +ACKNOWLEDGMENTS +This work was supported in part by the CREST Project +(JPMJCR17N2) funded by the Japan Science and Tech- +nology Agency and Grants-in-Aid for Scientific Research +(JP20H00233), +and Transformative Research Areas (A) +(JP22H05197) funded by the Japan Society for the Promotion +of Science (JSPS). AR is supported by JSPS as an Interna- +tional Research Fellow. +Appendix A: Independent identical distributions case +In this section, we discuss the properties of conditional +probability Pr(D1(i,i+k) < r | Di−1(1,1+k) < r) and condi- +tional variance Var(¯s(τA)(i+k)− ¯s(τA)(i) | Di−1(1,1+k) < r) +for i.i.d. systems. +Let s(t) be the i.i.d. time series under study. We can regard +s(t) as a sequence of random variables. Obviously, the coarse- + +Information-theoretical analysis of statistical measures for multiscale dynamics +13 +grained time series defined as +¯s(τA)(l) = 1 +τA +lτA +∑ +t=(l−1)τA+1 +s(t) +(A1) +is also an i.i.d. sequence of random variables. To simplify +the symbols, let the random variable Yl be ¯s(τA)(l). We now +define the PDF of Yl = ¯s(τA)(l) as g(τA)(yl). Please note that +the function itself is independent of l. +The distribution of ¯s(τA)(i+k)− ¯s(τA)(i) under the condition +Di−1(1,1+k) < r is then computed. For visibility, we define +the random variable Zk as follows: +Zk = ¯s(τA)(i+k)− ¯s(τA)(i). +(A2) +Let h(τA) +k +(zk) be the PDF of Zk under the condition of +Di−1(1,1 + k) < r. h(τA) +k +(zk) can be represented by g(τA)(yl) +as follows: +h(τA) +k +(zk) = d +dzk +Pr(Zk < zk | Di−1(1,1+k) < r) += d +dzk +Pr(Zk < zk ∧Di−1(1,1+k) < r) +Pr(Di−1(1,1+k) < r) += d +dzk +� +T (τA) +k,1 +∧T (τA) +k,2 ∏i+k +j=1 g(τA)(yj)dyj +� +T (τA) +k,2 ∏i+k +j=1 g(τA)(yj)dy j +, +(A3) +where +T (τA) +k,1 += {(y1,y2,··· ,yi+k) | yi+k −yi < zk}, +(A4) +T (τA) +k,2 += {(y1,y2,··· ,yi+k) | Di−1(1,1+k) < r}. +(A5) +Here we used the fact that the joint PDF of i.i.d. random vari- +ables is equal to the product of PDFs of each random variable. +Because the conditions in Eqs. (A4) and (A5) refer to differ- +ent variables, the integral in the numerator of Eq. (A3) can be +separated into the integrals of variables yi and yi+k, and the +other terms, as follows: +� +T (τA) +k,1 +∧T (τA) +k,2 +i+k +∏ +j=1 +g(τA)(yj)dy j += +� +˜T (τA) +k,1 +g(τA)(yi)g(τA)(yi+k)dyidyi+k +� +˜T (τA) +k,2 +i+k−1 +∏ +j=1 +j̸=i +g(τA)(yj)dy j, +(A6) +where +˜T (τA) +k,1 += {(yi,yi+k) | yi+k −yi < zk}, +(A7) +˜T (τA) +k,2 += {(y1,··· ,yi−1,yi+1,··· ,yi+k−1) | Di−1(1,1+k) < r}. +(A8) +Here, different from Eqs. (A4) and (A5), there is no variable +overlap between Eqs. (A7) and (A8). Similarly, the denomi- +nator can also be decomposed as follows: +� +T (τA) +k,2 +i+k +∏ +j=1 +g(τA)(yj)dyj += +� +R2 g(τA)(yi)g(τA)(yi+k)dyidyi+k +� +˜T (τA) +k,2 +i+k−1 +∏ +j=1 +j̸=i +g(τA)(yj)dyj += +� +˜T (τA) +k,2 +i+k−1 +∏ +j=1 +j̸=i +g(τA)(yj)dyj. +(A9) +Here we used the fact that +� +R2 g(τA)(yi)g(τA)(yi+k)dyidyi+k = 1. +(A10) +As a result, Eq. (A3) can be reduced to +h(τA) +k +(zk) = d +dzk +� +˜T (τA) +k,1 +g(τA)(yi)g(τA)(yi+k)dyidyi+k. +(A11) +Equation (A11) is the PDF of Zk without any conditions. In +conclusion, the condition Di−1(1,1+k) < r does not matter in +i.i.d. cases. In addition, the above calculation does not depend +on k and i except for k = 1, as i + k − 1 = i. Here yi appears +in both T (τA) +k,1 +and T (τA) +k,2 , so we cannot divide the integral in the +same manner. However, the same conclusion can be derived +for k = 1. Here we introduce a variable transformation, as +follows: +uj = +� +y j+1 −yj +( j = 1,2,··· ,i) +yi+1 +( j = i+1). +(A12) +Using uj, the integral of the numerator in Eq. (A3) can be +written as follows: +� ∞ +−∞ dui+1 +� zk +−∞ dui +� r +−r +i+1 +∏ +j=1 +g(τA) +� +2ui+1 − +� +i+1 +∑ +l=j +ul +�� +i−1 +∏ +j=1 +duj. +(A13) +Similarly, the integral in the denominator in Eq. (A3) is +� ∞ +−∞ dui+1 +� ∞ +−∞ dui +� r +−r +i+1 +∏ +j=1 +g(τA) +� +2ui+1 − +� +i+1 +∑ +l=j +ul +�� +i−1 +∏ +j=1 +duj. +(A14) +Because the only difference between Eqs. (A13) and (A14) is +the range of the integral for ui, we can cancel the integrals for +u1,u2,··· ,ui−1, and the remaining integrals for the numerator +and denominator are +� ∞ +−∞ dui+1 +� zk +−∞ dui g(τA)(ui+1)g(τA)(ui+1 −ui) +(A15) +and +� ∞ +−∞ dui+1 +� ∞ +−∞ dui g(τA)(ui+1)g(τA)(ui+1 −ui) = 1, +(A16) + +Information-theoretical analysis of statistical measures for multiscale dynamics +14 +respectively. Consequently, the resulting PDF is +h(τA) +1 +(z1) = d +dz1 +� ∞ +−∞ dui+1 +� z1 +−∞ dui g(τA)(ui+1)g(τA)(ui+1 −ui) += d +dz1 +� +˜T (τA) +1,1 +g(τA)(yi)g(τA)(yi+1)dyidyi+1. +(A17) +This is equivalent to the PDF of Z1 without any conditions. +Summarizing the results thus far, h(τA) +k +is the same as the PDF +of Zk without any conditions for all k. In addition, the calcu- +lation does not depend on i. Consequently, it is sufficient to +discuss the distribution of ¯s(τA)(2)− ¯s(τA)(1) in i.i.d. cases. +From the above results, Pr(Dm(1,1 + k) < r) can be ex- +pressed as follows: +Pr(Dm(1,1+k) < r) +=Pr(D1(1,1+k) < r) +m +∏ +i=2 +Pr +� +D1(i,i+k) < r | Di−1(1,1+k) < r +� +(A18) +=Pr(D1(1,1+k) < r) +m +∏ +i=2 +Pr +� +D1(i,i+k) < r +� +(A19) +=Pr(D1(1,1+k) < r)m +(A20) +=Pr(D1(1,2) < r)m. +(A21) +Here, we obtain Eq. (A18) in the same manner as Eq. (24). +From Eq. (A18) to (A19), we ignored the condition term, as +discussed above. From Eq. (A19) to(A20) and (A21), we used +the fact that the distribution of ¯s(τA)(i+k)− ¯s(τA)(i) is indepen- +dent of i and k. Consequently, ⟨Cm(r)⟩ = Pr(D1(1,2) < r)m, +and MSE ≃ −logPr(D1(1,2) < r) holds. +It is noteworthy that the variances of ¯s(τA)(1) and ¯s(τA)(2)− +¯s(τA)(1) are 1/τA and 2/τA respectively, with regard to the +variance of s(1), as +Var(¯s(τA)(1)) = Var +� +1 +τA +τA +∑ +t=1 +s(t) +� += 1 +τ2 +A +τA +∑ +t=1 +Var(s(t)) += 1 +τA +Var(s(1)), +(A22) +and +Var(¯s(τA)(2)− ¯s(τA)(1)) += Var(¯s(τA)(2))+Var(¯s(τA)(1)) += 2 +τA +Var(s(1)) +(A23) +hold because s(t) and ¯s(τA)(l) are i.i.d. +1M. Costa, A. L. Goldberger, and C.-K. Peng, “Multiscale entropy analysis +of complex physiologic time series,” Phys. Rev. Lett. 89, 068102 (2002). +2A. Catarino, O. Churches, S. Baron-Cohen, A. Andrade, +and H. Ring, +“Atypical eeg complexity in autism spectrum conditions: A multiscale en- +tropy analysis,” Clinical Neurophysiology 122, 2375–2383 (2011). +3M. Costa, A. L. Goldberger, and C.-K. Peng, “Multiscale entropy analysis +of biological signals,” Phys. Rev. E 71, 021906 (2005). +4E. Martina, E. 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Wang, “Automatic +stage scoring of single-channel sleep eeg by using multiscale entropy and +autoregressive models,” IEEE Transactions on Instrumentation and Mea- +surement 61, 1649–1657 (2012). +27V. Kroupa, Frequency stability (Wiley-IEEE PRESS, 2012). +28T. Sauer, J. A. Yorke, and M. Casdagli, “Embedology,” Journal of Statisti- +cal Physics 65, 579–616 (1991). + diff --git a/6dAzT4oBgHgl3EQf-P4i/content/tmp_files/load_file.txt b/6dAzT4oBgHgl3EQf-P4i/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a8148a4aa08c0ae0ef084688d31ec5c14000bcd8 --- /dev/null +++ b/6dAzT4oBgHgl3EQf-P4i/content/tmp_files/load_file.txt @@ -0,0 +1,926 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf,len=925 +page_content='Information-theoretical analysis of statistical measures for multiscale dynamics Information-theoretical analysis of statistical measures for multiscale dynamics Naoki Asuke,1 Tomoki Yamagami,1 Takatomo Mihana,1 André Röhm,1 Ryoichi Horisaki,1 and Makoto Naruse1 Department of Information Physics and Computing, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan (*Electronic mail: jintengzhizi196@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='com) (Dated: 6 January 2023) Multiscale entropy (MSE) has been widely used to examine nonlinear systems involving multiple time scales, such as biological and economic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Conversely, Allan variance has been used to evaluate the stability of oscillators, such as clocks and lasers, ranging from short to long time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Although these two statistical measures were developed independently for different purposes in different fields in the literature, their interest is to examine multiscale temporal structures of physical phenomena under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We show that, from an information-theoretical perspective, they share some foundations and exhibit similar tendencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We experimentally confirmed that similar properties of the MSE and Allan variance can be observed in low-frequency fluctuations (LFF) in chaotic lasers and physiological heartbeat data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Furthermore, we calculated the condition under which this consistency between the MSE and Allan variance exists, which is related to certain conditional probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Heuristically, physical systems in nature including the aforemen- tioned LFF and heartbeat data mostly satisfy this condition, and hence the MSE and Allan variance demonstrate similar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' As a counterexample, an artificially constructed random sequence is demonstrated, for which the MSE and Allan variance exhibit different trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' INTRODUCTION Multiscale entropy (MSE)1 has been used widely to eval- uate nonlinear systems that involve multiple time scales in biology,2,3 economics,4 transportation,5 and other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Meanwhile, Allan variance6 has been used to evaluate the sta- bility of oscillators,6–8 such as atomic clocks or lasers, over many time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Although these two statistical measures were developed independently in different domains, the MSE and its variants9 and Allan variance have a similar objective: to characterize dynamical systems containing multiple time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' However, the relevance and differences between the MSE and Allan variance have yet to be examined in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In this study, we discuss the similarities and differences between the MSE and Allan variance from an information-theoretical perspective and from the viewpoint of computational cost, and reveal the mechanisms behind them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We experimentally validated the similar tendencies of the MSE and Allan vari- ance observed in low-frequency fluctuations (LFF) in chaotic lasers that are numerically simulated by the Lang–Kobayashi equations,10 and physiological heartbeat data available in the public domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='11–14 Furthermore, we present the underlying mechanism behind the consistency between the MSE and Al- lan variance based on conditional probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We artificially constructed a random sequence that violated this condition, leading to a case exhibiting inconsistent results between the MSE and Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' As discussed in detail below, the LFF and heartbeat con- tain both slow and fast dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' they represent typical phys- ical phenomena with multiple time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Furthermore, re- cent studies on physics-based computing and communica- tions, such as reservoir computing,15 laser-chaos-based secure communication,16,17 and laser networks for solving reinforce- ment learning problems,18 work across multiple time scales;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' hence, understanding fundamental attributes in multiple time scales through statistical measures, such as the MSE and Al- lan variance, is essential for furthering our comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We review the Allan variance and MSE in Sections II A and II B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Section III examines the MSE and Allan vari- ance for several time series, namely noise, RR interval,19 and laser chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' A similar tendency of the MSE and Allan vari- ance is shown for each time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Section IV discusses the mechanism behind the similarity between the two measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Additionally, the cause of the differences in behavior is dis- cussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Section V concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' THEORY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Allan variance Coarse-graining of a time series s(t) (t ∈ {1,2,··· ,N}) with a scale factor τA refers to the operation that obtains an- other time series ¯s(τA)(l) (l ∈ {1,2,··· ,⌊N/τA⌋}) as follows: ¯s(τA)(l) = 1 τA lτA ∑ t=(l−1)τA+1 s(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (1) Here, ⌊·⌋ denotes floor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The time series ¯s(τA)(l) is called a coarse-grained time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' This operation obtains a new point by taking the average of every τA point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Using coarse-graining, the Allan variance6 of a time series s(t) with scale factor τA is defined as σ2 s (τA) = 1 2⌊N/τA⌋ ⌊N/τA⌋ ∑ l=2 � ∆¯s(τA)(l) �2 , (2) where ∆¯s(τA)(l) = ¯s(τA)(l)− ¯s(τA)(l −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (3) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='01930v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='CD] 5 Jan 2023 Information-theoretical analysis of statistical measures for multiscale dynamics 2 The Allan variance considers the variance of the difference between successive points in the coarse-grained time series under study, with respect to the time scale given by the coarse- graining time τA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Meanwhile, it is known that the Allan variance can be ex- pressed using the power spectral density S(f) of the signal s(t) as follows, under the assumption that s(t) is stationary and ergodic20: lim N→∞σ2 s (τA) = 2 � ∞ 0 S( f)sin4(π fτA) (π fτA)2 d f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (4) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (4), for example, the Allan variance of white noise is as follows: 2 � ∞ 0 h0 sin4(π fτA) (π fτA)2 d f = h0 2τA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (5) Similarly, for 1/ f noise, the Allan variance is 2 � ∞ 0 h−1 f sin4(π fτA) (π fτA)2 d f = 2h−1 ln2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (6) Here h0 and h−1 represent the intensities of the white and 1/ f components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We can then estimate h0 and h−1 from the Allan variance with various τA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In this way, the Allan variance has been used to evaluate the variability characteris- tics of time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Multiscale entropy (MSE) Before discussing multiscale entropy,1 it is necessary to introduce original sample entropy (SaEn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='21 The theoretical background and examples of SaEn were described in detail by Richman and Moorman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='22 We will use their definition of SaEn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In short, SaEn quantifies the regularity of a time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' SaEn has several parameters: the embedding dimension m, tolerance r and length of the time series N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The SaEn of a time series s(t) is defined with the correlation integral Cm(r) as follows: SaEn(m,r,N) = −log Cm+1(r) Cm(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (7) Several steps must be taken to define and compute Cm(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' First, the embedded vector series vm(t) is constructed as fol- lows: vm(t) = [s(t),s(t +1),··· ,s(t +m−1)]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (8) Please note that the length of the vector series defined as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (8) is N − m + 1 because the m-th component of vm(N − m+1) is s(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Second, for each embedded vector, Cm i (r) and Cm+1 i (r) are defined as follows: Cm i (r) = 1 N −m−1 N−m ∑ j=1 j̸=i u(r −dcheb(vm(i),vm( j))), (9) Cm+1 i (r) = 1 N −m−1 N−m ∑ j=1 j̸=i u(r −dcheb(vm+1(i),vm+1( j))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (10) Here, u denotes the unit step function, defined as u(x) = � 0 (x < 0) 1 (x ≥ 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (11) dcheb is the Chebyshev distance and dcheb(vm(i),vm( j)) is de- fined as follows: dcheb(vm(i),vm( j)) = max k∈{0,1,···,m−1}|s(i+k)−s( j +k)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (12) Please note that the maximum j is not N − m + 1 but N − m in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (9) because we also calculate Cm+1 i (r) to compute SaEn and the total number of embedded vectors in the m + 1 dimension is N −m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Intuitively, the definition of Cm i (r) is the probability of a randomly chosen vm(j) (j ̸= i) satisfying dcheb(vm(i),vm( j)) < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (13) It is noteworthy that, by the definition of the Chebyshev dis- tance, the following proposition holds: dcheb(vm(i),vm( j)) < r iff ∀k ∈ {0,1,··· ,m−1}, |s(i+k)−s( j +k)| < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (14) This proposition plays an important role in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' From the definition, Cm i (r) can be regarded as the prob- ability that a randomly chosen vm(j) is a neighbor of the embedded vector vm(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In other words, it is the likelihood of patterns [s( j),s(j +1),···s( j +m−1)]T which are consid- ered repetitions of the pattern [s(i),s(i+1),··· ,s(i+m−1)]T under the tolerance r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' With Cm i (r), the correlation integral Cm(r) is defined as Cm(r) = 1 N −m N−m ∑ i=1 Cm i (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (15) Intuitively, Cm(r) is the average of Cm i (r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' therefore, we can regard Cm(r) as the probability that randomly chosen vectors vm(i) and vm( j) (i ̸= j) are close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' A large Cm(r) indicates that time series s(t) contains many repeating structures of length m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' SaEn then quantifies whether, when m consecutive points in the time series are considered repeated, the (m+1)-th point is also considered repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In other words, it quantifies whether the (m+1)-th point is predictable by looking at the m preceding points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' SaEn can be applied to real-world data without assuming any model gov- erning the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Information-theoretical analysis of statistical measures for multiscale dynamics 3 The MSE is defined as the SaEn of a coarse-grained time series and can be plotted as a function of τA, where the same r is used for every τA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The MSE was proposed to quantify com- plexity at various time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 1/f noise has a long-time cor- relation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' therefore, it is considered more complex than white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' However, SaEn assigns the maximum value to white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In contrast, the MSE of white noise monotonically de- creases with increasing τA, whereas the MSE of 1/ f noise is constant,1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=', 1/ f noise has more complexity than white noise over a longer time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' As introduced in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' II A and II B, the definitions of the MSE and Allan variance are apparently completely different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In this study, we demonstrate that the MSE and Allan vari- ance exhibit similar τA dependency, and reveal the underlying mechanism from an information-theoretical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Note that for 1/ f noise, it has already been shown theoretically that both the Allan variance and MSE are constants, independent of τA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' COMPARISON OF THE MSE AND ALLAN VARIANCE To compare the behavior of the MSE and the correlation in- tegral Cm(r) to that of the Allan variance, they were calculated for the three types of signals described later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' m was fixed to 2, and r was fixed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='15×SD(s),1 where SD(s) denotes the standard deviation of the signal s(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Noise Thirty temporal waveforms of white Gaussian noise (WGN) and 1/f noise containing 107 points each were gener- ated numerically, and the MSE and Allan variance were calcu- lated up to τA = 103 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We fixed the length of the coarse- grained time series to 104 points for the MSE calculation ow- ing to limitations of our computing environment, whereas the entire time series was used for the Allan variance calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The Allan variance and MSE or −Cm(r) show similar trends with regard to τA, with appropriate scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Please note that the scales of the y-axes in Cm(r) plots and the Allan vari- ance plots are identical (logarithmic), while the y-axes in the MSE plots are linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Because the MSE is defined by the log- arithm of Cm(r), it is natural to plot it linearly when Cm(r) is plotted logarithmically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Figures 1(a) and (c) show the MSE and Allan variance of WGN, respectively, where both decrease monotonically as a function of τA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' This trend agrees well up to τA = 102 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' For τA > 102 points, the Allan variance still shows 1/τA de- pendency, as indicated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (5), whereas the MSE saturates at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In this region, Cm(r) is almost 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=', almost all of the embedded vectors are neighbors of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' This saturation effect is one of the differences between the MSE and Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Figures 1(b) and (d) show the MSE and Allan variance of 1/ f noise, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Theoretically, both are predicted to be independent of τA, whereas the simulation results show some τA dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' This may be a result of the difficulty in prop- erly generating 1/ f noise on scales from τA = 1 to 103, how- ever it is noteworthy that the MSE, −Cm(r) and Allan variance exhibit similar curves as a function of τA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 0 1 2 3 MSE 10 0 10 1 10 2 10 3 10 −3 10 −2 10 −1 10 0 Allan Var (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' units) 10 −4 10 −2 10 0 10 0 10 1 10 2 10 3 WGN 1/f (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 1: Comparison of the MSE and Allan variance for WGN and 1/ f noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (a,b) The MSE and Cm(r) of WGN and 1/ f noise, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The black, red, and blue curves represent the MSE, C3(r) and C2(r), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Each curve is the av- erage of independent results from 30 trials, and the filled area around the curve represents the first and third quartile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The variance of the computed results is so slight that it is diffi- cult to observe the painted areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (c,d) The Allan variance of WGN and 1/ f noise, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The Allan variance plots and Cm(r) plots are logarithmic on both axes to observe the 1/τA dependency of WGN clearly, while the y-axis is linear for the MSE plots, because the MSE is defined with the loga- rithm of Cm(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Physiological signal Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='1 introduced the MSE to demonstrate that physi- ological signals observed in young, healthy individuals exhibit versatile signal levels across a variety of time scales, whereas unhealthy subjects do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In this section, we compare the MSE and Allan variance of the RR interval19 series, which are sequences of intervals between adjacent R waves in electro- cardiograms, for healthy subjects and those with congestive heart failure (CHF) or atrial fibrillation (AF), following Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Each dataset was obtained from PhysioNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='11–14 The lengths of the time series varied, but the entirety of each time series was used in the calculations, because adjust- ing to the shorter ones would reduce the accuracy of the esti- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' To compare the Allan variance, the standard deviation of each time series was normalized to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' This normalization is consistent with the MSE calculation, because r was fixed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='15×SD(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The MSE and Allan variance for each dataset is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The Allan variance, MSE and −Cm(r) show trends that are similar except for the location of the peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Information-theoretical analysis of statistical measures for multiscale dynamics 4 Figures 2(a) and (e) respectively show the MSE and Allan variance overlaid for all three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Figure 2(a) shows that the RR interval for healthy subjects is the most complex on long time scales, which is consistent with the results of the previous study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='1 As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 2(e), the ordering of the different classes of subjects for the Allan variance and MSE show a similar trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' However, the exact scale factor τA at which each curve intersects differs from that of the MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' It seems plausible that the analysis of RR interval time series in general could be performed using Allan variance instead of the MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Laser chaos The MSE and Allan variance of the noise time series exhibit monotonic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' However, the MSE and Allan variance of physiological signals show more complex properties, but the underlying dynamics are not completely known, which makes further analysis difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' As an example of a time series that includes multiple time scales and for which the underlying dynamics are known, we discuss a phenomenon called LFF23 exhibited by a semi- conductor laser with optical feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The Lang–Kobayashi equations,10 a set of model equations, was used to generate the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' An example of a time series is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 3, where fast chaotic oscillations on the order of GHz coexist with irregular intensity dropouts and recovery on the order of MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' As the simulation step and time scale of the dynamics are important, τA was con- verted to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Figure 4 shows that the MSE, Cm(r) and Allan variance all capture the dynamics of fast oscillations on the order of GHz, corresponding to the fast peak in the MSE and Allan variance, and dropouts on the order of MHz, corresponding to the slow peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' INFORMATION THEORETICAL CONNECTIONS BETWEEN THE MSE AND ALLAN VARIANCE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Cm(r) and Allan variance In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' III, we observe that −Cm(r) exhibits behavior sim- ilar to that of the Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In this section, we discuss the underlying mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Cm(r) decomposition First, we consider decomposing Cm(r) by the difference in the indices of the embedding vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Let M = ⌊N/τA⌋ be the length of the coarse-grained time series, and Dm(i, j) be dcheb(vm(i),vm( j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Considering the expectation and symme- try of i and j, and assuming s(t) or its first-order difference s(t)−s(t −1) is strongly stationary, ⟨Cm(r)⟩ = ∑M−m i=1 ∑M−m j=1 j̸=i ⟨u(r −Dm(i, j))⟩ (M −m−1)(M −m) (16) = ∑M−m i=1 ∑M−m j=1 j̸=i Pr(Dm(i, j) < r) (M −m−1)(M −m) (17) = ∑M−m i=1 ∑M−m j=i+1 2Pr(Dm(i, j) < r) (M −m−1)(M −m) (18) = ∑M−m−1 k=1 (M −m−k)Pr(Dm(1,1+k) < r) �M−m 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (19) First we clarify the meaning of expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In the following, we regard the time series s(t), t ∈ {1,2,··· ,N} under study as a sequence of random variables following a specific prob- ability distribution function (PDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In this sense, Cm(r) and u(r−Dm(i, j)) are also random variables, and thus each of the possible values of Cm(r) and u(r −Dm(i, j)) has some proba- bility of occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The expectation is the weighted average of the likelihood of every potential value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Here we explain the transformation of the equations from (16) to (19) line by line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The only change made from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (16) to (17) is that the expectation term ⟨u(r −Dm(i, j))⟩ is changed to the probability term Pr(Dm(i, j) < r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We as- sume that a random variable sequence s(t), t ∈ {1,2,··· ,N} follows a PDF, and calculate u(r −Dm(i, j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' As previously mentioned, u(r −Dm(i, j)) is also a random variable that is either 0 or 1 with the following probability: u(r −Dm(i, j)) = � 0 (1−Pr(Dm(i, j) < r)) 1 (Pr(Dm(i, j) < r)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (20) Consequently, ⟨u(r −Dm(i, j))⟩ = Pr(Dm(i, j) < r) (21) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (17) to (18), the factor 2 is added to the nu- merator, and the range of summation over j is restricted to j > i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Here, we use the fact that Dm(i, j) = Dm(j,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (18) to (19), we must assume that Pr(Dm(i, j) < r) de- pends only on the difference in the indices k = j−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' For exam- ple, Pr(Dm(i+1, j +1) < r) is the same as Pr(Dm(i, j) < r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' This assumption is satisfied when the original signal s(t) or its first-order difference s(t) − s(t − 1) is strongly stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' This is a reasonable assumption when studying a time series from a dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The factor (M −m−k) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (19) comes from the fact that, for each k, the number of pairs (i, j) satisfying k = j −i is (M −m−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' For example, when k = 1, the pairs are (1,2),(2,3),··· ,(M −m−1,M −m), as the total number of embedded vectors is M −m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The probability in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (19) can be decomposed further by considering conditional probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Using the proposi- Information-theoretical analysis of statistical measures for multiscale dynamics 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='07 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='5 2 MSE healthy CHF AF 10 −2 10 −1 10 0 Allan Var (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' units) 20 5 10 15 20 5 10 15 20 5 10 15 20 5 10 15 Healthy subjects CHF subjects AF subjects (a) (b) (e) (f) (c) (g) (d) (h) 200 400 600 800 1000 Beat Number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='6 Interval (sec) 200 400 600 800 1000 Beat Number 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='8 1 Interval (sec) 200 400 600 800 1000 Beat Number 0 2 4 Interval (sec) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 2: Comparison of the MSE and Allan variance for the RR interval time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (a, e) Overlaid figure of the MSE and Allan variance of the RR interval time series from healthy subjects and subjects with CHF or AF, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (b,c,d) The MSE and Cm(r) of the RR interval time series for healthy subject, CHF, and AF, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The black, red, and blue curves represent the MSE, C3(r) and C2(r), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Each curve is the average of independent results from 147 subjects for healthy, 29 for CHF, and 84 for AF, and the filled area around the MSE curve represents the first and third quartile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The first and third quartiles of Cm(r) are omitted for legibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (f,g,h) Allan variance of the corresponding time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The filled area around the curve represents the first and third quartile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Each dataset was obtained from PhysioNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='11–14 0 500 1000 Time (ns) 0 5 10 15 0 500 1000 Time (ns) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='2 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 3: Example of an LFF time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The intensity is nor- malized to that without optical feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (a) Original time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (b) Time series obtained by applying an ideal low-pass filter with a cutoff frequency of 100 MHz to the time series in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Sudden dropouts with gradual recovery are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 10 −2 10 −1 10 0 10 1 10 2 10 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='5 MSE 10 −3 10 −2 10 −1 10 0 10 −2 10 −1 10 0 10 1 10 2 10 3 10 −2 10 −1 10 0 Allan Var (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' units) (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 4: Comparison of the MSE and Allan variance for an LFF time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (a) The MSE and Cm(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The black, red, and blue curves represent the MSE, C3(r) and C2(r), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (b) Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Note that the MSE, −Cm(r), and the Allan variance exhibits similar τA dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' tion (14), Pr(Dm(1,1+k) < r) = Pr(dcheb(v(1),v(1+k)) < r) = Pr(|s(1)−s(1+k)| < r ∧|s(2)−s(2+k)| < r ∧···∧|s(m)−s(m+k)| < r) = Pr(D1(1,1+k) < r ∧D1(2,2+k) < r ∧···∧D1(m,m+k) < r) (22) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The probability of a product event, such as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (22), can be represented as a product of conditional probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' For example, the probability that propositions A, B and C hold simultaneously is Pr(A∧B∧C) = Pr(B∧C | A)Pr(A) = Pr(C | A∧B)Pr(B | A)Pr(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (23) In the same way, Pr(Dm(1,1+k) < r) =Pr(D1(1,1+k) < r) m ∏ i=2 Pr � D1(i,i+k) < r | Di−1(1,1+k) < r � (24) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In the following, we show that the above conditional prob- ability behaves oppositely to the Allan variance regardless of k, while the unconditional probability does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' This point is the key to understanding why −Cm(r), the MSE and Allan variance show similar τA dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Information-theoretical analysis of statistical measures for multiscale dynamics 6 Figure 6(a) shows a logarithmic-scale color map of Pr(D1(1,1 + k) < r), which is the first term on the right- hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (24), whereas Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 6(b) shows that of Pr(D1(2,2 + k) < r | D1(1,1 + k) < r), corresponding to the conditional probability in the second term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (24) for i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Each probability was calculated for the LFF time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' For k = 1, corresponding to the lowest row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 6(a) and the solid curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 6(c), the unconditioned probability Pr(D1(1,2) < r) behaves opposite to the Allan variance (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' However, for larger k, the behavior differs from that of the k = 1 case, as shown in the dashed plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' This can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' First, the following equation holds: D1(1,1+k) =|¯s(τA)(k +1)− ¯s(τA)(1)| =|¯s(τA)(k +1)− ¯s(τA)(k)+ ¯s(τA)(k)−···+ ¯s(τA)(2)− ¯s(τA)(1)| = ����� k+1 ∑ j=2 ∆¯s(τA)( j) �����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (25) For k = 1, whether D1(1,2) = |∆¯s(τA)(2)| is smaller than r is closely related to the Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' If the Allan variance for a certain τA is smaller than that of another τA, the ∆¯s(τA)(l) dis- tribution is biased toward the center, because the mean value of ∆¯s(τA)(l) is zero by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Consequently, the probabil- ity D1(1,2) = |∆¯s(τA)(2)| < r is high for that τA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' For larger k, the ∑k+1 j=2 ∆¯s(τA)( j) distribution is affected by the time cor- relation of ¯s(τA)(l) that the time series under study inherently has, and the Allan variance Var(∆¯s(τA)(l)) cannot predict the ∑k+1 j=2 ∆¯s(τA)( j) distribution well, so Pr(D1(1,1+k) < r) does not behave in a manner that is strongly correlated with the Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Conversely, the conditional probability Pr(D1(2,2 + k) < r | D1(1,1 + k) < r) depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 6 (b) shows a similar trend to that of the Allan variance regardless of k, which we can also observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 6(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' That is, Pr(D1(2,2 + k) < r | D1(1,1+k) < r) is the main connection that explains the sim- ilarity between the MSE and Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' To examine this further, we discuss the variance of the corresponding quan- tity, rather than considering the probabilities in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Neighborhood-Likelihood-to-Variance-Relationship (NLVR) In this section, we introduce an assumption called the Neighborhood-Likelihood-to-Variance-Relationship (NLVR) to connect the probability discussion to the variance dis- cussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' First, define ∆Pr(τA) as the difference between Pr(D1(i,i+k) < r | Di−1(1,1+k) < r) for τA and τA−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In the same way, ∆Var(τA) is the difference between Var(¯s(τA)(i + k) − ¯s(τA)(i) | Di−1(1,1 + k) < r) for τA and τA − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Please note that the conditional probability Pr(D1(i,i + k) < r | Di−1(1,1+k) < r) is the same as that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (24), and the ab- solute value of ¯s(τA)(i+k)− ¯s(τA)(i), referred to in ∆Var(τA), is identical to D1(i,i+k) in the definition of ∆Pr(τA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Then, to connect the probability with the variance, we as- sume the Neighborhood-Likelihood-to-Variance-Relationship (NLVR), as follows: ∆Pr(τA)∆Var(τA) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (26) The inequality (26) states that the signs of ∆Pr(τA) and ∆Var(τA) are always opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We will explain the NLVR in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Both Pr(D1(i,i+ k) < r | Di−1(1,1 + k) < r) and Var(¯s(τA)(i + k) − ¯s(τA)(i) | Di−1(1,1+k) < r) are statistics of ¯s(τA)(i+k)− ¯s(τA)(i) under the same condition of Di−1(1,1+k) < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Considering the PDF of ¯s(τA)(i+k)− ¯s(τA)(i), Pr(D1(i,i+k) < r | Di−1(1,1+k) < r) represents the area under the PDF in the range [−r, r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In ad- dition, when there is no condition, the mean of ¯s(τA)(i + k) − ¯s(τA)(i) is zero by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Here we also assume that the mean value of ¯s(τA)(i+k)− ¯s(τA)(i) is close to zero under the condition Di−1(1,1+k) < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' When ∆Pr(τA) > 0, the distribu- tion of ¯s(τA)(i+k)− ¯s(τA)(i) is more centrally biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Accord- ingly, when ∆Pr(τA) > 0, ∆Var(τA) is likely to be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Figure 5 schematically illustrates the concept of the NLVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The black curves in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 5 (a) and (b) represent the PDF of ¯s(τA)(i+k)− ¯s(τA)(i) under the condition Di−1(1+k,1) < r for τA −1 and τA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The red-filled areas and black bars show Pr(D1(i,i+k) < r | Di−1(1,1+k) < r) and Var(¯s(τA)(i+ k) − ¯s(τA)(i) | Di−1(1,1 + k) < r), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' ∆Pr(τA) and ∆Var(τA) are the difference in the red-filled areas and black bars between Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 5(b) and (a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Please note that the plots shown here, based on Gaussian distributions, are for explanatory purposes only and are not obtained from a time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The validity of the NLVR is discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' IV C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 5: A visual representation of the NLVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (a, b) ¯s(τA)(i + k)− ¯s(τA)(i) distribution under the condition Di−1(1+k,1) < r for τA −1 and τA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The red-filled area and the the black bars represent Pr(D1(i,i + k) < r | Di−1(1,1 + k) < r) and Var(¯s(τA)(i + k) − ¯s(τA)(i) | Di−1(1,1 + k) < r), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Variance decomposition Here we further discuss the conditional variance Var(¯s(τA)(i + k) − ¯s(τA)(i) | Di−1(1,1 + k) < r), instead of the Information-theoretical analysis of statistical measures for multiscale dynamics 7 conditional probability Pr(D1(i,i+k) < r | Di−1(1,1+k) < r) following the NLVR assumption, as mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' IV A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' By decomposing ¯s(τA)(i+k)− ¯s(τA)(i) as ¯s(τA)(i+k)− ¯s(τA)(i) = ¯s(τA)(i+k)− ¯s(τA)(i+k −1)+ ¯s(τA)(i+k −1) − ¯s(τA)(i)+ ¯s(τA)(i−1)− ¯s(τA)(i−1) =∆¯s(τA)(i+k)−∆¯s(τA)(i)+(¯s(τA)(i+k −1)− ¯s(τA)(i−1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (27) the conditional variance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Var(¯s(τA)(i + k) − ¯s(τA)(i) | Di−1(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='1+k) < r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' can be decomposed as follows: Var(¯s(τA)(i+k)− ¯s(τA)(i) | Di−1(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='1+k) < r) =Var(∆¯s(τA)(i+k) | ···) +Var(∆¯s(τA)(i) | ···) −2Cov(∆¯s(τA)(i+k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='∆¯s(τA)(i) | ···) +Var(¯s(τA)(i+k −1)− ¯s(τA)(i−1) | ···) +2Cov(∆¯s(τA)(i+k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='(¯s(τA)(i+k −1)− ¯s(τA)(i−1)) | ···) −2Cov(∆¯s(τA)(i),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='(¯s(τA)(i+k −1)− ¯s(τA)(i−1)) | ···).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (28) Here, the condition Di−1(1,1 + k) < r is abbreviated by the symbol ··· on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' [Overview of the variance decomposition] There are six terms on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The logarithmic scale color map of each term is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7(a)–(f) to determine which term has the greatest in- fluence on the conditional variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Figure 7(g) represents the summation of each term, which is the original conditional variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Meanwhile, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7(h) show cross-sectional profiles when k = 50 regarding the first term (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7(a)), the third term (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7(c)) and the total conditional variance (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' As discussed shortly below, the first through third terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (28) are dominant, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7(a)–(c), whereas they show a similar τA dependency with the Allan variance regardless of k, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Please note that we plotted the absolute values of the covariance terms, because the covariance can be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' However, the τA dependency of Cov(∆¯s(τA)(i + k),∆¯s(τA)(i) | Di−1(1,1+k) < r), shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7(c) and (h), is still similar to that of the Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Conversely, the remaining panels (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7(d)–(f)) exhibit small values, regardless of k and τA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Therefore, we observe that the conditional variance Var(¯s(τA)(i + k) − ¯s(τA)(i) | Di−1(1,1 + k) < r), which is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7(g) and (h), and the conditional probability Pr(|¯s(τA)(i + k) − ¯s(τA)(i)| < r | Di−1(1,1 + k) < r), shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 6(b) for the case of i = 2, exhibit the opposite τA depen- dency to the Allan variance from the NLVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' [The first and the second terms] The first and second terms are the conditional variances of ∆¯s(τA)(i+k) and ∆¯s(τA)(i), namely the conditional Allan vari- ances, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Although the condition somewhat affects the ∆¯s(τA)(i + k) and ∆¯s(τA)(i) distribution, the size relation- ship of the variance of ∆¯s(τA)(i + k) and ∆¯s(τA)(i) concerning τA is almost the same as the Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' [The third term] However, the third term is strongly influenced by the con- dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In the absence of the condition, if k is sufficiently large, there would be little correlation between ∆¯s(τA)(i + k) and ∆¯s(τA)(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' thus, the covariances are expected to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 8(a) shows the unconditional covariance of ∆¯s(τA)(i + k) and ∆¯s(τA)(i), where the color scale is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Comparing to the conditional covariance plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7(c), this covariance without the condition is smaller in most places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' By contrast, when the condition is satisfied, i − 1 consec- utive points up to ¯s(τA)(i + k − 1) and ¯s(τA)(i − 1) are close values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In that case, ∆¯s(τA)(i+k) and ∆¯s(τA)(i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=', the differ- ence between these points and their consecutively following point, often have the same sign, and the distribution of their magnitudes follows the Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Figure 8(b) shows an example of a pair of ∆¯s(τA)(i + k) and ∆¯s(τA)(i) satisfying the condition for i = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Because we can regard v2(7) and v2(17), denoted by the blue dots, as identical under the tolerance r, the red arrows, representing ∆¯s(τA)(9) and ∆¯s(τA)(19) are likely to be similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' By the def- inition of covariance, if ∆¯s(τA)(i + k) and ∆¯s(τA)(i) are simi- lar, Cov(∆¯s(τA)(i+k),∆¯s(τA)(i)) is close to Var(∆¯s(τA)(i+k)), namely the Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' From these considerations, the conditional covariance behaves similarly to the Allan vari- ance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Please note that the form of the condition in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 8(b), namely D2(7,17) < r is slightly different from Di−1(1,1 + k) < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' However, as we assume stationary, it is allowed to shift the indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' For example, in this case, we examined ∆¯s(τA)(19) and ∆¯s(τA)(9) under the condition of D2(7,17) < r, and this corresponds to the shifting of indices by six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' To be more specific, we investigated ∆¯s(τA)(19) = ∆¯s(τA)(3+10+6) and ∆¯s(τA)(3 + 6), under the condition of D2(7,17) < r = D2(1+6,1+10+6), where i = 3 and k = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' [The fourth to the sixth terms] The fourth to sixth terms do not contribute significantly to the conditional variance, as mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' This is due to the fact that ¯s(τA)(i+k−1)− ¯s(τA)(i−1), which is involved in the variance or covariance in the fourth to sixth terms, is small when the condition Di−1(1,1+k) < r is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Once again, the condition implies a sort of similarity on the time scale of i−1 points, and when these points are similar, the variance of these terms containing the difference will be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' [Summary of the decomposition] Adding the above six terms together, we see that the con- ditional variance eventually behaves similarly to the Allan variance, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7(g) and (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Because the con- ditional variances and conditional probabilities are assumed by the NLVR to act in opposite directions for increasing scale factors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' the conditional probabilities and Cm(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' represented by the weighted average of the product of probability without conditions Pr(D1(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='1+k) < r) and the conditional probabili- ties Pr(D1(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i+k) < r | Di−1(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='1+k) < r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' in turn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' behave in Information-theoretical analysis of statistical measures for multiscale dynamics 8 the opposite direction for increasing scale factors for the Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Although we observe the conditional variances and covari- ances from an LFF time series as an example, the discus- sion of behavior shown by the variances and covariances is valid for other time series to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In the discussion, we assumed strong stationarity of the signal s(t) or its first- order difference to estimate probabilities and variances from the obtained time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' This is because Pr(Dm(i, j) < r) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (17) cannot be estimated from a single time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Con- versely, the NLVR assumption between probability and vari- ance can also be applied to Pr(|¯s(τA)(i) − ¯s(τA)( j)| < r) and Var(¯s(τA)(i) − ¯s(τA)( j)) and could be valid to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Thus, similar behavior exhibited by the MSE and Allan vari- ance should hold for a more general class of signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 20 40 60 80 100 10−1 100 10 −2 10 −1 10 0 10 1 10 2 10 3 10 −1 10 0 Probability 10 −2 10 −1 10 0 10 1 10 2 10 3 (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 6: (a,b) Logarithmic scale color maps of (a) Pr(D1(1,1+k) < r) and (b) Pr(D1(2,2+k) < r | D1(1,1+k) < r) for the LFF time series up to k = 100, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (c,d) One- dimensional plots of the two-dimensional data displayed in (a) and (b) along the two arrows (k = 1 and k = 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Solid and dashed plots correspond to k = 1 and k = 100, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The MSE and Allan variance In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' IV A, the connection between Cm(r) and the Allan variance was discussed based on the connection between the conditional probability and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We can also apply this discussion to the connection between the MSE and Allan vari- ance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' To simplify the discussion, we assume a sufficiently long (strongly stationary) time series, so that we can consider Cm(r) to be the same as the expectation ⟨Cm(r)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' This is equivalent to assuming ergodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Under this condition, MSE =−log Cm+1(r) Cm(r) =−log ∑M−m−1 k=1 (M −m−k)Pr(Dm+1(1,1+k) < r) ∑M−m−1 k=1 (M −m−k)Pr(Dm(1,1+k) < r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (29) Pr(Dm+1(1,1 + k) < r), in the numerator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (29) can be decomposed similarly, as shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' IV A, as follows: Pr(Dm+1(1,1+k) < r) =Pr(D1(m+1,m+1+k) < r | Dm(1,1+k) < r) Pr(Dm(1,1+k) < r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (30) Let ak = (M − m − k)Pr(Dm(1,1 + k) < r) and bk = Pr(D1(m + 1,m + 1 + k) < r | Dm(1,1 + k) < r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Consider- ing the equality (M − m − k)Pr(Dm+1(1,1 + k) < r) = akbk, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (29) can be rewritten as follows: MSE = −log Cm+1(r) Cm(r) = −log ∑M−m−1 k=1 akbk ∑M−m−1 k=1 ak (31) Equation (31) states that the MSE is the negative logarithm of the weighted average of bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Because bk is the conditional probability, the behavior of the MSE can be explained by the conditional probability as well as Cm(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' More precisely, we must care about the expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Cm(r) can be represented with probability when considering the ex- pectation ⟨Cm(r)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The expectation of the MSE is ⟨MSE⟩ = � −log Cm+1(r) Cm(r) � (32) ≥ −log �Cm+1(r) Cm(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (33) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (32) to (33), we use the fact that the negative log- arithm is convex, along with Jensen’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (33) in general is not the same as −log ⟨Cm+1(r)⟩ ⟨Cm(r)⟩ = −log ∑M−m−1 k=1 akbk ∑M−m−1 k=1 ak .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (34) However, it is noteworthy that Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='3 showed that the MSE values calculated theoretically, using probability, agree well for the WGN and 1/ f noise cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Connection between the probability and variance In Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' IV A and IV B, we discussed the connection be- tween Cm(r) or the MSE and Allan variance based on the as- sumption of NLVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In this section, we discuss the extent to which NLVR is valid, and what happens if it is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Information-theoretical analysis of statistical measures for multiscale dynamics 9 20 40 60 80 100 10-2 10-1 100 101 102 103 20 40 60 80 100 10-2 10-1 100 101 102 103 10-3 10-2 10-1 100 10-2 10-1 100 101 102 103 10-2 10-1 100 101 102 10310-6 10-4 10-2 100 102 Variance from (g) from (a) from (c) (a) (b) (c) (d) (e) (f) (g) (h) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7: Element-wise logarithmic-scale color maps of the conditional variance Var(¯s(τA)(i + k) − ¯s(τA)(i) | Di−1(1,1 + k) < r) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' There are six terms of an LFF time series up to k = 100: (a) Var(∆¯s(τA)(i+k) | Di−1(1,1+k) < r), (b) Var(∆¯s(τA)(i) | Di−1(1,1+k) < r), (c) 2|Cov(∆¯s(τA)(i+k),∆¯s(τA)(i) | Di−1(1,1+k) < r)|, (d) Var(¯s(τA)(i+k −1)− ¯s(τA)(i−1) | Di−1(1,1+k) < r), (e) 2|Cov(∆¯s(τA)(i+k),(¯s(τA)(i+k −1)− ¯s(τA)(i−1)) | Di−1(1,1+k) < r)| and (f) 2|Cov(∆¯s(τA)(i),(¯s(τA)(i+k −1)− ¯s(τA)(i−1)) | Di−1(1,1+k) < r)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (g) The term before the above decomposition, which is Var(¯s(τA)(i+k)− ¯s(τA)(i) | Di−1(1,1+k) < r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (h) Cross-sectional profiles when k = 50 from (a) (red curve), (c) (blue curve), and (g) (black curve) indicated by the arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Please note that for parts (c), (e), and (f) we plot the absolute value of the covariance, because covariance can be negative, and our color scale is logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' However, 2Cov(∆¯s(τA)(i+k),∆¯s(τA)(i) | Di−1(1,1+k) < r), shown in (c), is mostly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In (g), although the points with negative values are neglected, almost the entire curve is depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' [Considering independent identical distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=')] For simplicity, we consider the case of independent identi- cal distributions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In this case, the conditional probabil- ity distribution of (¯s(τA)(i + k) − ¯s(τA)(i) | Di−1(1,1 + k) < r) is identical to the probability distribution of (¯s(τA)(i + k) − ¯s(τA)(i)), which is independent of the condition (Di−1(1,1 + k) < r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The distribution does not depend on k (see Ap- pendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Therefore, it is sufficient to consider the distri- bution of ¯s(τA)(i + 1) − ¯s(τA)(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In addition, the variances of ¯s(τA)(i) and ¯s(τA)(i + 1) − ¯s(τA)(i) are Var(s(1))/τA and 2Var(s(1))/τA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Therefore, the Allan variance for an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' system is in- versely proportional to τA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' For the WGN case, owing to the reproductive property, the PDF of ¯s(τA)(i) and ¯s(τA)(i + 1) − ¯s(τA)(i) are also Gaussian, with variances Var(s(1))/τA and 2Var(s(1))/τA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' because the mean value of ¯s(τA)(i + 1) − ¯s(τA)(i) is zero,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' the probability |¯s(τA)(i+1)− ¯s(τA)(i)| < r can be represented as follows: Pr(|¯s(τA)(i+1)− ¯s(τA)(i)| < r) = � 1 2π2σ/τA � r −r exp � − x2 2·(2σ/τA)2 � dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (35) where σ2 = Var(s(1)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' and increases monotonically with in- Information-theoretical analysis of statistical measures for multiscale dynamics 10 10 −1 10 0 10 1 10 2 10 3 20 40 60 80 100 0 1 2 3 4 5 6 5 10 15 20 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 8: (a) The color map of the unconditional version of the third term of the conditional variance (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (28): 2|Cov(∆¯s(τA)(i+k),∆¯s(τA)(i)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The color scale is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (b) ∆¯s(τA)(i + k) and ∆¯s(τA)(i) under the condition of D2(7,7 + 10) < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Here, i = 3 and k = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The blue dots represent the two consecutive points which we can regard as v2(7) and v2(17) satisfying the condition D2(7,7 + 10) < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The red dots denote ¯s(τA)(9) and ¯s(τA)(19) and the red arrows show ∆¯s(τA)(9) and ∆¯s(τA)(19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' creasing τA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Therefore, NLVR holds for all τA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The variance of ¯s(τA)(i + 1) − ¯s(τA)(i) continuously de- creases when τA increases in the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' That is, ∆Var(τA) is negative for all τA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The cases in which NLVR does not hold are those in which the area under the PDF of ¯s(τA)(i + 1)− ¯s(τA)(i) in the range [−r, r] decreases with increasing τA, which means ∆Pr(τA) is also negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' This situation seems unlikely to occur when the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' process is unimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' That is to say, the reduction of the variance means the shrinking of the distribution toward the center, which means an increase in the probabilities around the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' [When NLVR is invalid] However, situations that violate NLVR are likely to occur when, for example, the original system exhibits a multimodal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' It should be noted that ¯s(τA)(i) of a multimodal distribution can have more peaks than the distribution of s(t), owing to the effect of averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Thus, ¯s(τA)(i + 1) − ¯s(τA)(i) may also have many peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The number of peaks increases as τA increases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' however, beyond a certain point, the number of peaks eventually de- creases because the peaks fuse with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' According to the central limit theorem, the distribution of ¯s(τA)(i) and ¯s(τA)(i + 1) − ¯s(τA)(i) finally assumes shapes close to a Gaus- sian distribution in the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' As the number of peaks increases, the area of the PDF of ¯s(τA)(i + 1) − ¯s(τA)(i) near the center is distributed to each peak, thus, ∆Pr(τA) is neg- ative until the effects of this distribution and peaks merging become antagonistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' [Violation of NLVR by bimodal distributions] Figure 9 presents an example of this discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Figure 9(a) shows a histogram of the original bimodal i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' system com- posed of two Gaussians centered around ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Figures 9(c), (e), and (g) show the histograms of the coarse-grained time series for τA = 1, 2, 5, 100, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Recall that coarse- graining means averaging over neighboring points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Because the process is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=', two consecutive points have a 50% prob- ability of coming from opposite sides of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Thus, for τA = 2 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 9(b), a third peak appears around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' More precisely, the leftmost peak shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 9(c) corresponds to the case in which two consecutive s(t) points are negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=', from the left peak in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 9(a), which has a probability of 50% × 50% = 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Similarly, the rightmost peak in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 9(c) corresponds to two consecutive positive s(t) points, and the center peak corresponds to either positive and negative, or negative and positive points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The number of peaks increases up to a certain τA, until they start to merge with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' For a sufficiently large τA, the distribution approaches a Gaussian distribution, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 9(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Figures 9(b), (d), (f), and (h) show the histograms of the dif- ference between adjacent coarse-grained points ¯s(τA)(i+1)− ¯s(τA)(i) for τA = 1, 2, 5, 100, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The red filled area represents the range [−r, r] and the black bar denotes the stan- dard deviation of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In fact, the distributions can be regarded as convolutions of the corresponding distribution of ¯s(τA)(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Similar to ¯s(τA)(i), the number of peaks increases until a certain τA and then decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In such cases, ∆Pr(τA) is negative until a certain τA, which can be observed from the reduction of the red area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 9(b)–(f), whereas ∆Var(τA) is always negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Thus, the NLVR is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='04 Probability 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='04 Probability −1 0 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='04 Probability −2 −1 0 1 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='04 Probability 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='04 Probability 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='04 Probability 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='04 Probability 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='04 Probability (a) (b) (c) (d) (e) (f) (g) (h) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 9: Example of histograms of a bimodal i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (a,c,e,g) Histograms of the coarse-grained time series for τA = 1, 2, 5, 100, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (b,d,f,h) Histograms of the difference between adjacent coarse-grained points for τA = 1, 2, 5, 100, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Red areas represent the range [−r, r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Black bars denote the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' [Disagreement of MSE and Allan variance] Plots of Pr(D1(i,i + 1) = |¯s(τA)(i + 1) − ¯s(τA)(i)| < r), the MSE, Cm(r) and Allan variance of this bimodal i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' sys- tem are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The dashed black line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 10(a) shows Pr(|¯s(τA)(i+1)− ¯s(τA)(i)| < r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In contrast to the continuously decreasing Allan variance, Pr(|¯s(τA)(i+1)− Information-theoretical analysis of statistical measures for multiscale dynamics 11 ¯s(τA)(i)| < r) decreases until τA = 8 and then increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Cm(r) (red and blue curves) shows the same trend, and the MSE (black curve) inherits the inverted trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Actually ⟨Cm(r)⟩ = Pr(|¯s(τA)(i+1)− ¯s(τA)(i)| < r)m and MSE ≃ −logPr(|¯s(τA)(i+ 1) − ¯s(τA)(i)| < r) (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In contrast, the Allan variance curve plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 10(b) shows 1/τA dependency (note that the plot is logarithmically scaled), as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Because the example under study here was an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' process, the Allan variance result may be considered the more reliable one, as it does not detect any specific time scales of relevance, whereas the MSE indicates a higher level of com- plexity for τA = 8 than at other scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 10 0 10 1 10 2 10 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='5 MSE 10 −3 10 −2 10 −1 10 0 Probability 10 0 10 1 10 2 10 3 10 −3 10 −2 10 −1 10 0 Allan Var (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' units) (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 10: Comparison of the MSE and Allan variance for the bimodal system shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 9(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (a) The MSE, Cm(r) and Pr(|¯s(τA)(i + 1) − ¯s(τA)(i)| < r) of a time series obtained from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The black, red, and blue curves represent the MSE, C3(r) and C2(r), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The dashed black curve represents Pr(D1(i,i + 1) = |¯s(τA)(i + 1) − ¯s(τA)(i)| < r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (b) The Allan variance of the corresponding time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Herein, the MSE and the Allan variance clearly exhibit different τA dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' From these discussions we can see that the NLVR is not al- ways valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' For example, the condition may not hold when the original time series s(t) contains a multimodal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' If the NLVR is violated, meaning that ∆Pr(τA)∆Var(τA) > 0 for a large number of k, Cm(r), expressed as a weighted average of the probability, is expected to change in the same direc- tion as the Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' As a result, the Allan variance and MSE change with the opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' cases, ∆Pr(τA) and ∆Var(τA) are independent of k, so if NLVR does not hold for a specific τA and k, it is violated for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Therefore, in the range of τA where NLVR holds, the MSE and the Allan variance show similar τA dependence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' in the range where the NLVR is violated, they behave oppositely to τA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' If the time series are time-correlated, as in dynamical systems, the NLVR may be violated by even more complex mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Empirical validation of the NLVR Finally, we examine the extent to which the NLVR is sat- isfied based on empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Specifically, the change in conditional probability (∆Pr(τA)) and conditional variance (∆Var(τA)) are calculated for the time series studied in Sec- tions III and IV C up to k = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' First, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 11(a) shows a histogram of the intensity ob- served in the LFF time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The intensity is normalized to that without optical feedback, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=', the intensity in the sin- gle mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='24 The histogram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 11(a) has a strong peak near the origin and a smaller peak near the normalized intensity of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' That is, the probability distribution exhibits a somewhat multimodal distribution, which may cause a viola- tion of the NLVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Figures 11(b), (c) and (d) show the scatter plot of (∆Pr(τA),∆Var(τA)) for LFF, WGN, and bimodal distribu- tion, respectively, as explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' If the NLVR holds, a positive ∆Pr(τA) means a negative ∆Var(τA) and a negative ∆Pr(τA) means a positive ∆Var(τA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' That is to say, the points in the scatter diagram should be in the second or fourth quad- rants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 11(b), almost all the sampling points are con- centrated in the second and fourth quadrants, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=', in regions where the signs of ∆Pr(τA) and ∆Var(τA) are opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' That is to say, even though the probability distribution contains multi- modality, this LFF system mostly does not violate the NLVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' From these observations, we speculate that the peaks of the distribution would need to be more separated to lead to a dis- agreement of the Allan variance and MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Similarly, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 11(c) shows ∆Pr(τA) and ∆Var(τA) for the WGN time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' As discussed above, ∆Pr(τA) should al- ways be positive, whereas ∆Var(τA) is always negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=', every point should be in the second quadrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' However, the distribution calculated from the time series is not precisely the Gaussian distribution, so some points are in the third quadrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 11(d) shows the same scatter plot for the bi- modal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In contrast to the LFF and WGN cases, many points are in the third quadrant in violation of the NLVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' This was expected, as we constructed this case specifically as a counter-example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' LFF WGN Bimodal (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 11: ∆Pr(τA) and ∆Var(τA) for m = 2 and i = 2, up to k = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (a) Histogram of the LFF time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (b, c, d) Scatter plot of ∆Pr(τA) and ∆Var(τA) for (b) the LFF time series, (c) WGN time series, and (d) the bimodal time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' When the NLVR holds, the points in the scatter diagram should be in the second or fourth quadrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='5 0 5 10 15 5 I / Io 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='05 △Pr(TA) △Pr(TA) 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='1 0 1 △Var(TA)0 5 △Var(TA) 0 1 △Var(TA)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='5 Probability △Pr(TA) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='005Information-theoretical analysis of statistical measures for multiscale dynamics 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Computation of the MSE and Allan variance As introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' II B, the definition of MSE involves probabilities, whereas that of the Allan variance is based on the variability of the data under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Section IV C reveals the common underlying mechanism from an information- theoretic viewpoint despite the seemingly different definitions of the statistical measures of multiscale dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In this sec- tion, we discuss the difference from the viewpoint of compu- tation between the MSE and Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' For the MSE, calculating Cm(r) is a computationally de- manding task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Computing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (9) for all i requires O(N2) calculations, as all pairs of embedded vectors are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Some implementations also require O(N2) memory to store all the calculated Chebyshev distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Thus, the MSE eval- uates the details of the probability distribution at a high com- putational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In contrast, the Allan variance calculation requires only O(N) computations and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The Allan variance does not consider the probability distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' it depends only on the differences ∆¯s(τA)(l) of successive coarse-grained points ¯s(τA)(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' IV C, the MSE for the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' pro- cess depends on the distribution, whereas the Allan variance does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Despite being computationally cheaper, it is not obvious that the Allan variance has any disadvantages when compared to the MSE for extracting multiscale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The Allan vari- ance is a statistical measure that is similar but not identical to the MSE, and that quantifies slightly different aspects of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In the literature, the range of applications of the MSE is versatile, such as bearing fault detection25 and sleep level qualification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='26 However, the MSE suffers from severe com- putational difficulties as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Concerning the similar properties of the MSE and Allan variance, as well as the computationally lightweight nature of Allan variance, the extension of the Allan variance to real-time applications, such as bearing fault detection or prediction of epilepsy from electroencephalography (EEG), would be an interesting future topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' CONCLUSION In this study, we examined the similarities shown by the multiscale statistics the MSE and Allan variance, and dis- cussed the underlying mechanisms through an information- theoretic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' It is noteworthy that although the apparent definitions of the MSE and Allan variance are significantly different, they show a similar behavior for a wide range of time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We experimentally confirmed the similar properties of the MSE and Allan variance observed in LFF in chaotic lasers and physiological heartbeat data, as well as white Gaussian and 1/f noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The connection can be un- derstood by decomposing the conditional probabilities in the MSE and extracting the dominant contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We derived a condition which must be satisfied for the MSE and Allan variance to exhibit similar tendencies via a discussion of con- ditional probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Then, we artificially constructed a ran- dom sequence that violates the condition, leading to incon- sistent MSE and Allan variance behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We also quantita- tively demonstrated that the aforementioned LFF and heart- beat, which are physically plausible systems, mostly satisfy the condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Finally, we discussed future research topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Using Al- lan variance instead of the MSE may lead to more computa- tionally lightweight applications that are suitable for real-time tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In addition, there is a possibility of integrating further developments that have been devised for the MSE9 and Allan variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='27 Furthermore, more research on the theoretical foundations of coarse-graining and the MSE is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The MSE research to date has focused mainly on its application as a statistical tool, and there has been little research on its theoretical foun- dations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The relationship between the dynamics of a coarse- grained time series and those of the original time series is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Coarse-graining can be regarded as a combination of a moving-average filter and downsampling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' however, ac- cording to a previous study,28 a linear filter applied to the original time series during Takens’ embedding preserves its topological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' From this theorem, it may be possi- ble to discuss the theoretical basis of coarse-graining from the viewpoint of dynamical invariants, including entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Meanwhile, the theoretical foundations of the MSE should be more complicated, as the MSE shares the tolerance r for all scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' As pointed out by Humeau-Heurtier,9 more and more embedded vectors may be regarded as neighbors of each other as the scale factor τA increases, owing to the reduction of the variance of the coarse-grained time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Notably, Costa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='3, the original proposers of the MSE, pointed out that the variance changes induced by coarse-graining are related to the temporal structures of the original time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We may need a framework that allows us to connect the dynamical invariants at each time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported in part by the CREST Project (JPMJCR17N2) funded by the Japan Science and Tech- nology Agency and Grants-in-Aid for Scientific Research (JP20H00233), and Transformative Research Areas (A) (JP22H05197) funded by the Japan Society for the Promotion of Science (JSPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' AR is supported by JSPS as an Interna- tional Research Fellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Appendix A: Independent identical distributions case In this section, we discuss the properties of conditional probability Pr(D1(i,i+k) < r | Di−1(1,1+k) < r) and condi- tional variance Var(¯s(τA)(i+k)− ¯s(τA)(i) | Di−1(1,1+k) < r) for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Let s(t) be the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' time series under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We can regard s(t) as a sequence of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Obviously, the coarse- Information-theoretical analysis of statistical measures for multiscale dynamics 13 grained time series defined as ¯s(τA)(l) = 1 τA lτA ∑ t=(l−1)τA+1 s(t) (A1) is also an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' sequence of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' To simplify the symbols, let the random variable Yl be ¯s(τA)(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' We now define the PDF of Yl = ¯s(τA)(l) as g(τA)(yl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Please note that the function itself is independent of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' The distribution of ¯s(τA)(i+k)− ¯s(τA)(i) under the condition Di−1(1,1+k) < r is then computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' For visibility, we define the random variable Zk as follows: Zk = ¯s(τA)(i+k)− ¯s(τA)(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A2) Let h(τA) k (zk) be the PDF of Zk under the condition of Di−1(1,1 + k) < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' h(τA) k (zk) can be represented by g(τA)(yl) as follows: h(τA) k (zk) = d dzk Pr(Zk < zk | Di−1(1,1+k) < r) = d dzk Pr(Zk < zk ∧Di−1(1,1+k) < r) Pr(Di−1(1,1+k) < r) = d dzk � T (τA) k,1 ∧T (τA) k,2 ∏i+k j=1 g(τA)(yj)dyj � T (τA) k,2 ∏i+k j=1 g(τA)(yj)dy j , (A3) where T (τA) k,1 = {(y1,y2,··· ,yi+k) | yi+k −yi < zk}, (A4) T (τA) k,2 = {(y1,y2,··· ,yi+k) | Di−1(1,1+k) < r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A5) Here we used the fact that the joint PDF of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' random vari- ables is equal to the product of PDFs of each random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Because the conditions in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A4) and (A5) refer to differ- ent variables, the integral in the numerator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A3) can be separated into the integrals of variables yi and yi+k, and the other terms, as follows: � T (τA) k,1 ∧T (τA) k,2 i+k ∏ j=1 g(τA)(yj)dy j = � ˜T (τA) k,1 g(τA)(yi)g(τA)(yi+k)dyidyi+k � ˜T (τA) k,2 i+k−1 ∏ j=1 j̸=i g(τA)(yj)dy j, (A6) where ˜T (τA) k,1 = {(yi,yi+k) | yi+k −yi < zk}, (A7) ˜T (τA) k,2 = {(y1,··· ,yi−1,yi+1,··· ,yi+k−1) | Di−1(1,1+k) < r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A8) Here, different from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A4) and (A5), there is no variable overlap between Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A7) and (A8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Similarly, the denomi- nator can also be decomposed as follows: � T (τA) k,2 i+k ∏ j=1 g(τA)(yj)dyj = � R2 g(τA)(yi)g(τA)(yi+k)dyidyi+k � ˜T (τA) k,2 i+k−1 ∏ j=1 j̸=i g(τA)(yj)dyj = � ˜T (τA) k,2 i+k−1 ∏ j=1 j̸=i g(τA)(yj)dyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A9) Here we used the fact that � R2 g(τA)(yi)g(τA)(yi+k)dyidyi+k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A10) As a result, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A3) can be reduced to h(τA) k (zk) = d dzk � ˜T (τA) k,1 g(τA)(yi)g(τA)(yi+k)dyidyi+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A11) Equation (A11) is the PDF of Zk without any conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In conclusion, the condition Di−1(1,1+k) < r does not matter in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In addition, the above calculation does not depend on k and i except for k = 1, as i + k − 1 = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Here yi appears in both T (τA) k,1 and T (τA) k,2 , so we cannot divide the integral in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' However, the same conclusion can be derived for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Here we introduce a variable transformation, as follows: uj = � y j+1 −yj ( j = 1,2,··· ,i) yi+1 ( j = i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A12) Using uj, the integral of the numerator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A3) can be written as follows: � ∞ −∞ dui+1 � zk −∞ dui � r −r i+1 ∏ j=1 g(τA) � 2ui+1 − � i+1 ∑ l=j ul �� i−1 ∏ j=1 duj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A13) Similarly, the integral in the denominator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A3) is � ∞ −∞ dui+1 � ∞ −∞ dui � r −r i+1 ∏ j=1 g(τA) � 2ui+1 − � i+1 ∑ l=j ul �� i−1 ∏ j=1 duj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A14) Because the only difference between Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A13) and (A14) is the range of the integral for ui, we can cancel the integrals for u1,u2,··· ,ui−1, and the remaining integrals for the numerator and denominator are � ∞ −∞ dui+1 � zk −∞ dui g(τA)(ui+1)g(τA)(ui+1 −ui) (A15) and � ∞ −∞ dui+1 � ∞ −∞ dui g(τA)(ui+1)g(τA)(ui+1 −ui) = 1, (A16) Information-theoretical analysis of statistical measures for multiscale dynamics 14 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Consequently, the resulting PDF is h(τA) 1 (z1) = d dz1 � ∞ −∞ dui+1 � z1 −∞ dui g(τA)(ui+1)g(τA)(ui+1 −ui) = d dz1 � ˜T (τA) 1,1 g(τA)(yi)g(τA)(yi+1)dyidyi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A17) This is equivalent to the PDF of Z1 without any conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Summarizing the results thus far, h(τA) k is the same as the PDF of Zk without any conditions for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' In addition, the calcu- lation does not depend on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Consequently, it is sufficient to discuss the distribution of ¯s(τA)(2)− ¯s(τA)(1) in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' From the above results, Pr(Dm(1,1 + k) < r) can be ex- pressed as follows: Pr(Dm(1,1+k) < r) =Pr(D1(1,1+k) < r) m ∏ i=2 Pr � D1(i,i+k) < r | Di−1(1,1+k) < r � (A18) =Pr(D1(1,1+k) < r) m ∏ i=2 Pr � D1(i,i+k) < r � (A19) =Pr(D1(1,1+k) < r)m (A20) =Pr(D1(1,2) < r)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A21) Here, we obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A18) in the same manner as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A18) to (A19), we ignored the condition term, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' (A19) to(A20) and (A21), we used the fact that the distribution of ¯s(τA)(i+k)− ¯s(τA)(i) is indepen- dent of i and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Consequently, ⟨Cm(r)⟩ = Pr(D1(1,2) < r)m, and MSE ≃ −logPr(D1(1,2) < r) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' It is noteworthy that the variances of ¯s(τA)(1) and ¯s(τA)(2)− ¯s(τA)(1) are 1/τA and 2/τA respectively, with regard to the variance of s(1), as Var(¯s(τA)(1)) = Var � 1 τA τA ∑ t=1 s(t) � = 1 τ2 A τA ∑ t=1 Var(s(t)) = 1 τA Var(s(1)), (A22) and Var(¯s(τA)(2)− ¯s(τA)(1)) = Var(¯s(τA)(2))+Var(¯s(τA)(1)) = 2 τA Var(s(1)) (A23) hold because s(t) and ¯s(τA)(l) are i.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Yorke, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} +page_content=' Casdagli, “Embedology,” Journal of Statisti- cal Physics 65, 579–616 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAzT4oBgHgl3EQf-P4i/content/2301.01930v1.pdf'} diff --git a/7NAyT4oBgHgl3EQf2vm0/content/tmp_files/2301.00757v1.pdf.txt b/7NAyT4oBgHgl3EQf2vm0/content/tmp_files/2301.00757v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a8be5f4f5d595ce61a4b8139b19542134ad4335 --- /dev/null +++ b/7NAyT4oBgHgl3EQf2vm0/content/tmp_files/2301.00757v1.pdf.txt @@ -0,0 +1,1742 @@ +1 +ENGNN: A General Edge-Update Empowered +GNN Architecture for Radio Resource +Management in Wireless Networks +Yunqi Wang, Yang Li, Qingjiang Shi, and Yik-Chung Wu +Abstract +In order to achieve high data rate and ubiquitous connectivity in future wireless networks, a +key task is to efficiently manage the radio resource by judicious beamforming and power allocation. +Unfortunately, the iterative nature of the commonly applied optimization-based algorithms cannot meet +the low latency requirements due to the high computational complexity. For real-time implementations, +deep learning-based approaches, especially the graph neural networks (GNNs), have been demonstrated +with good scalability and generalization performance due to the permutation equivariance (PE) property. +However, the current architectures are only equipped with the node-update mechanism, which prohibits +the applications to a more general setup, where the unknown variables are also defined on the graph +edges. To fill this gap, we propose an edge-update mechanism, which enables GNNs to handle both node +and edge variables and prove its PE property with respect to both transmitters and receivers. Simulation +results on typical radio resource management problems demonstrate that the proposed method achieves +higher sum rate but with much shorter computation time than state-of-the-art methods and generalizes +well on different numbers of base stations and users, different noise variances, interference levels, and +transmit power budgets. +Index Terms +Yunqi Wang is with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong, +and also with Shenzhen Research Institute of Big Data, Shenzhen 518172, China (email: yunqi9@connect.hku.hk). +Yang Li is with Shenzhen Research Institute of Big Data, Shenzhen 518172, China (e-mail: liyang@sribd.cn). +Qingjiang Shi is with the School of Software Engineering, Tongji University, Shanghai 200092, China, and also with Shenzhen +Research Institute of Big Data, Shenzhen 518172, China (email: shiqj@tongji.edu.cn). +Yik-Chung Wu is with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong +(email: ycwu@eee.hku.hk). +arXiv:2301.00757v1 [cs.NI] 14 Dec 2022 + +2 +Beamforming design, power allocation, heterogeneous graph neural network (GNN), edge-update +mechanism. +I. INTRODUCTION +Efficient radio resource management plays a vital role in achieving high data rate and +ubiquitous connectivity of future wireless networks. In particular, beamforming design and power +allocation have been recognized as crucial components to improve the spectrum/energy efficiency +in ultra-dense networks [1], cloud radio access networks [2], [3], and cell-free massive multiple- +input multiple-output systems [4]. +Mathematically, many of the radio resource management problems belong to the challenging +non-convex optimization problems, which are conventionally solved by numerical algorithms +with a lot of iterations [5], [6]. However, due to the fast variation of the wireless environment, +the iterative nature of the commonly applied optimization-based numerical algorithms cannot +satisfy the low-latency requirement in beyond 5G paradigm. For instance, to maximize the sum +rate of a multi-cell wireless system under the maximum transmit power constraint of each base +station (BS), the conventional first-order algorithm, e.g., the gradient projection (GP) method [7], +requires a lot of iterations to converge to a stationary point. To improve the convergence rate, +while more advanced numerical algorithms such as the weighted minimum mean-square error +(WMMSE) [8] algorithm can be applied, the matrix inverse in each iteration still makes it +computationally expensive and hence difficult for the real-time implementation. +To facilitate the real-time implementation, deep learning based methods have become popular +for radio resource management [9]–[16]. Specifically, deep learning based methods utilize neural +networks to learn a mapping function from many problem features to the corresponding solutions. +Once the neural network is well trained, it can infer the solution of any new setting using simple +feed-forward computations, and thus is extremely fast. +Inspired by the successful applications in computer vision, the multi-layer perceptrons (MLPs) +and convolutional neural networks (CNNs) have been applied as typical architectures for +representing the mapping functions in radio resource management. For example, MLPs were used +to learn the mapping function from the wireless channel to the optimal resource management +policy [9]. Moreover, an MLP-based architecture was adopted to learn the optimal power control +for the multi-user interference channels [10]. Similarly, CNNs have also been applied for power +control [11] and beamforming design [12] in the multiple-input single-output downlink systems. + +3 +However, since MLPs and CNNs cannot fully exploit the topology in the wireless networks, they +usually require a large number of training samples while still result in limited performance. For +instance, it is shown in [16] that a CNN trained on a two-user wireless networks can only achieve +the near-optimal performance for two-user wireless networks during the testing phase, but its +performance degrades by 18% in ten-user networks compared to the conventional optimization- +based numerical algorithms. +Recently, attempts to use graph neural networks (GNNs) are on the rise because of their +ability to exploit the topology of wireless networks. By modeling a wireless network as a graph, +the known system parameters can be modeled as the graph features, which are treated as the +input of a GNN, while the unknown variables to be optimized can also be defined on the +graph and are served as the output of a GNN. The advantage of graph modeling lies in its +permutation equivariance (PE) property, where the graph features/variables can be regarded as a +set of elements whose index order does not matter. Consequently, a large number of unnecessary +permuted training samples can be discarded [17]–[19]. Moreover, since the trainable parameters +of GNNs are independent of the graph size, the well-trained GNNs can generalize well to +different problem dimensions [20]–[24]. +Among the existing GNN-based works, homogeneous GNNs, which share the trainable +parameters among different graph nodes, have shown their good scalability and generalization +performance for the radio resource management problems when there is only one type of graph +nodes [17]–[20], [24]–[28]. For example, in [17], a homogeneous GNN named message passing +graph neural network (MPGNN) was proposed for beamforming design in the multi-transceiver +interference channels, where each transceiver pair is modeled as an individual graph node. +By modeling different transceiver pairs as the same type of nodes and sharing their trainable +parameters, the test performance in terms of sum rate is near optimal even when the number +of transceiver pair is twice larger than that in the training samples. Similarly, homogeneous +GNNs have also been demonstrated to generalize well on different numbers of users in multicast +beamforming design [24], link scheduling [25], power control [26], [27], and joint beam selection +and link activation [28]. +While homogeneous GNNs have shown their great success when there exists only one type +of graph nodes in the radio resource management problems, it should be noticed that the more +common scenarios usually consist of different types of graph nodes. For instance, in a general +wireless network, the transmitters and receivers have different physical characteristics, and hence + +4 +should be more naturally modeled as two different types of graph nodes, i.e., TX-nodes and RX- +nodes. By sharing the trainable parameters within only the same type of nodes, heterogeneous +GNNs [29] have shown their superiority for the more complex radio resource management +problems compared with the homogeneous GNNs [30]–[34]. In particular, the pioneer work [30] +designed a heterogeneous GNN called permutation equivariant heterogeneous GNN (PGNN) +for the power allocation in multi-cell downlink systems, and theoretically established the PE +property with respect to different user equipments (UEs) within each cell and also with respect +to different cells. Moreover, heterogeneous GNNs have also been proposed for the beamforming +design in heterogeneous device-to-device networks [31] and multi-user downlink systems [32]. +In [33], a heterogeneous GNN was designed for jointly learning the beamforming vectors and +reflecting phases for an intelligent reflecting surfaces (IRS) assisted multi-user downlink system, +where the users and the IRS are modeled as heterogeneous graph nodes. Similarly, the trajectory +of unmanned aerial vehicles (UAVs) was cooperatively designed by modeling the UAVs and the +ground terminals as heterogeneous graph nodes in [34]. +Despite the successes of the homogeneous or heterogeneous GNNs in the above existing +works, they are only equipped with the node-update mechanism, which restricts the output of +the neural networks, i.e., the unknown variables to be optimized only appear on the graph nodes. +Notable examples are MPGNN [17] and PGNN [30], both of which do not consider the variables +on the graph edges. In particular, MPGNN is proposed for the beamforming design for the multi- +transceiver interference channels, where each transmitter only serves a single receiver. Therefore, +each transceiver pair can be modeled as a single node, and the channel state information of +each direct communication link serves as the corresponding node features. Furthermore, the +interference links among different transceiver pairs are modeled as graph edges, whose channel +state information is treated as the corresponding edge features. Using this graph model, the +beamforming variable of each transceiver pair can be defined only on the corresponding node. +On the other hand, PGNN is proposed for the power allocation in multi-cell systems, where +each BS serves multiple UEs within the same cell. In this scenario, each BS adopts a pre- +designed beamformer, so that a dedicated equivalent single-antenna channel is created for each +UE. Consequently, with each equivalent transmit antenna treated as an individual node, the power +allocation variables can also be defined on the graph nodes. +While all the above pioneering works exemplify the benefits of GNNs in radio resource +management, currently applied architectures prohibit the extension to a more general setting, + +5 +where the unknown variables are also defined on the graph edges. A typical application scenario +is the cooperative beamforming design, where each transmitter serves multiple receivers, while +each receiver is also served by multiple transmitters. These complicated transceiver interactions +cannot be easily modeled by the current GNN architectures that are only equipped with the +node-update mechanism. In fact, for the more general wireless environment, an individual +beamforming or power variable belongs to a transceiver pair, which is represented by two +different nodes. Thus, beamformers or power variables should be more naturally defined on the +graph edges. Unfortunately, without a judiciously designed edge-update mechanism, the current +widely adopted GNN architectures cannot handle such a general setting. +To fill this gap, we propose a novel edge-update mechanism, which enables the GNN +architecture to deal with both the edge and node variables for the radio resource management +problems. The contributions of this paper are summarized as follows. +1) We propose a general problem formulation using the heterogeneous graph for the radio +resource management problems, where the unknown variables to be optimized can be +defined on the graph edges. To learn the edge variables, we design a novel edge-update +mechanism and prove its PE property with respect to both the transmitters and receivers. +Compared with the existing node-update mechanism that gathers the information from +the neighboring nodes, the update of an edge variable is more challenging, since it is +more complicated to define the neighbors of an edge, let alone how to aggregate their +representations. Based on the observation that the neighboring edges can be divided into +two categories according to their connected nodes, we propose an edge-update mechanism +that extracts the information from the two types of neighboring edges in a different manner. +2) Based on the edge-update mechanism, we propose an edge-update empowered neural +network architecture termed as edge-node GNN (ENGNN), which can represent the +mapping function from the graph features to the edge/node variables for the radio resource +management problems. We prove that the proposed ENGNN is permutation equivariant with +respect to both transmitters and receivers. Moreover, since the trainable parameters of the +proposed ENGNN are independent of the graph size, it can generalize to different numbers +of transmitters and receivers. Last but not the least, the proposed ENGNN can be applied in +a wide range of radio resource management problems, where the variables occur between +any pair of the TX-nodes and RX-nodes. +3) Simulation results demonstrate the superiority of the proposed ENGNN for typical radio + +6 +resource management problems, including the beamforming design in the interference +channels, the power allocation in the interference broadcast channels, and the cooperative +beamforming design, respectively. It is shown that the proposed ENGNN achieves higher +sum rate with much shorter computation time than state-of-the-art methods and generalizes +well on different numbers of BSs and UEs, different noise variances, interference levels, +and transmit power budgets. +Notations: In this paper, we use bold lowercase letters, bold uppercase letters, and bold +italicized uppercase letters to represent vectors, matrices, and tensors, respectively. The sets +are represented by stylized uppercase letters. The notations (·)T and (·)H refer to transpose and +Hermitian transpose, respectively. Moreover, |·|2 denotes the l2-norm operation, and |·| computes +the magnitude of a complex number or the cardinality of a set. +The rest of the paper is organized as follows. In Section II, we propose a general problem +formulation on the heterogeneous graph for the radio resource management problems. In +Section III, we design a novel neural network architecture named ENGNN with both the edge- +update and node-update mechanisms. Then, in Section IV, numerical results are presented to +demonstrate the superiority of the proposed ENGNN on three typical scenarios. Finally, the +conclusion is drawn in Section V. +II. PROBLEM FORMULATION ON HETEROGENEOUS GRAPH +A. General Graph Modeling +Consider a wireless network with M transmitters and K receivers, which can be modeled by +a heterogeneous graph. Specifically, the transmitters and receivers can be viewed as two types +of nodes, i.e., TX-nodes and RX-nodes, respectively. Moreover, an edge is drawn between a +TX-node and an RX-node if there exists a direct communication or interference link between +them. Such a heterogeneous graph can be expressed as G = {M, K, E}, where M ≜ {1, . . . , M} +is the set of TX-nodes, K ≜ {1, . . . , K} is the set of RX-nodes, and E ⊆ {(m, k)}m∈M,k∈K is +the set of edges, respectively. +The TX-nodes, RX-nodes, and edges may contain features and/or variables. Specifically, +features are known system parameters. For example, features on the nodes can be position +coordinates, maximum transmit power budgets, and/or noise variances, while features on the +edges can be channel state information and/or indicators of direct communication or interference + +7 +links. On the other hand, variables are unknown beamformers and/or allocated powers to be +designed. +B. Problem Formulation +Denote the feature vectors on the m-th TX-node and k-th RX-node as fTX,m ∈ CdTX +and fRX,k ∈ CdRX, where dTX and dRX denote the corresponding feature dimensions, respec- +tively. Consequently, the feature matrices of TX-nodes and RX-nodes can be expressed as +FTX = [fTX,1, · · · , fTX,M]T ∈ CM×dTX and FRX = [fRX,1, · · · , fRX,K]T ∈ CK×dRX. Similarly, +we can express the edge features as a tensor E ∈ CM×K×dE, where dE is the edge feature +dimension. In particular, the (m, k, :)-th fiber, E(m,k,:), takes values if (m, k) ∈ E, and is an +all-zero vector otherwise. On the other hand, the variables on TX-nodes and RX-nodes can be +expressed as STX = [sTX,1, · · · , sTX,M]T and SRX = [sRX,1, · · · , sRX,K]T, where sTX,m ∈ Cd′ +TX and +sRX,k ∈ Cd′ +RX denote the variables on the m-th TX-node and k-th RX-node, d′ +TX and d′ +RX denote +the corresponding variable dimensions, respectively. Similarly, edge variables can be expressed +as Ξ ∈ CM×K×d′ +E. +Based on the above notations, a beamforming design or power allocation problem can be +formulated as +max +φ(·,·,·) +f (STX, SRX, Ξ ; FTX, FRX, E) , +(1a) +s.t. +(STX, SRX, Ξ ) = φ (FTX, FRX, E) , +(1b) +where (1a) is the utility function, and φ(·, ·, ·) denotes the mapping function from the features +(FTX, FRX, E) to the variables (STX, SRX, Ξ ). Next, we show three typical examples under the +general problem formulation (1). +Example 1: Beamforming Design for Interference Channels. Consider a wireless network with +K BS-UE pairs, where the k-th UE is served by the m1(k)-th BS, and m1(·) is any one-to-one +mapping from K to M. Each BS is equipped with N antennas, and each UE is equipped with +a single antenna. The beamforming vector of the k-th UE is denoted as vk ∈ CN, while the +channel between the m1(k′)-th BS and the k-th UE is denoted as hm1(k′),k ∈ CN. Then the +received signal at the k-th UE is given by +yk = hH +m1(k),kvksk + +K +� +k′=1,k′̸=k +hH +m1(k′),kvk′sk′ + nk, ∀k ∈ K, +(2) + +8 +(a) Example 1: beamforming design for +interference channels. +(b) Example 2: power allocation for inter- +ference broadcast channels. +(c) Example 3: cooperative beamforming +design. +Fig. 1. Graph modeling for three typical examples. +where sk is the desired symbol of the k-th UE, and nk ∼ CN(0, σ2 +k) is the additive complex +Gaussian noise. Consequently, the signal-to-interference-plus-noise ratio (SINR) at the k-th UE +can be written as +SINRk = +���hH +m1(k),kvk +��� +2 +�K +k′=1,k′̸=k +���hH +m1(k′),kvk′ +��� +2 ++ σ2 +k +, ∀k ∈ K. +(3) +The beamforming design problem for sum rate maximization can be formulated as: +max +{vk}k∈K +K +� +k=1 +log2 (1 + SINRk) , +(4a) +s.t. +∥vk∥2 ≤ Pm1(k), ∀k ∈ K, +(4b) +where (4b) represents the maximum transmit power constraint at each BS. +As shown in Fig. 1(a), by modeling the BSs and UEs as TX-nodes and RX-nodes respectively, +we can incorporate the maximum transmit power budget p = [P1, · · · , PK]T and the noise +standard deviation σ += [σ1, · · · , σK]T as the features on the TX-nodes and RX-nodes, +respectively. Moreover, the feature vector on the (m, k)-th edge contains both the channel state +information hm,k and the indicator of direct communication or interference link: +H(m,k,:) = +� +� +� +� +� +[hT +m,k, 0T]T, if m = m1(k), +[0T, hT +m,k]T, otherwise, +(5) + +TXm1(1) +RX1 +TXm1(2) +RX2 +TXm1(K) +RXK +desired signal +interferenceTXm1(1) +RX1 +TXm1(2) +RX2 +TXm1(K-1) +RXK-1 +TXm1(K) +RXK +desired signal +inter-cell interference +intra-cell interferenceRX1 +TX1 +RX2 +TX2 +RX3 +RX4 +TXM +RXK9 +where we adopt the idea of one-hot encoding to embed the information of direct communication +or interference links. On the other hand, since vk is the beamforming variable corresponding to +the (m1(k), k, :)-th TX-RX pair and m1(·) is a one-to-one mapping, we can either define vk on +the (m1(k), k, :)-th edge, the k-th RX-node, or the m1(k)-th TX-node. Without loss of generality, +we put vk on the (m1(k), k, :)-th fiber of the edge variable tensor V , and problem (4) can be +reformulated on the heterogeneous graph as +max +φ(·,·,·) +K +� +k=1 +log2 (1 + SINRk) , +(6a) +s.t. +V = φ (p, σ, H) , with ∥vk∥2 ≤ Pm1(k), ∀k ∈ K. +(6b) +Comparing (6) with (1), the objective function (6a) is a specification of (1a). Particularly, +the features p, σ, and H correspond to FTX, FRX, and E, respectively, and the variable V +corresponds to Ξ . +Example 2: Power Allocation for Interference Broadcast Channels. Consider a B-cell +downlink cellular network, where each BS serves Q UEs. Each BS is equipped with N antennas +and each UE is equipped with a single antenna. The normalized beamforming vector of UE +q ∈ Q ≜ {1, · · · , Q} in cell b ∈ B ≜ {1, · · · , B} is denoted as wqb ∈ CN, while the channel +between BS b′ and UE q in cell b is denoted as hb′,qb ∈ CN. Then the received signal at UE q +in cell b is given by +yqb = √pqbhH +b,qbwqbsqb + +Q +� +q′=1,q′̸=q +�pq′ +bhH +b,qbwq′ +bsq′ +b ++ +B +� +b′=1,b′̸=b +Q +� +q′=1 +�pq′ +b′hH +b′,qbwq′ +b′sq′ +b′ + nqb, ∀q ∈ Q, ∀b ∈ B, +(7) +where sqb and pqb are the desired symbol and transmit power, respectively, the second term is +the intra-cell interference, the third term is the inter-cell interference, and nqb ∼ CN(0, σ2 +qb) is +the additive complex Gaussian noise. Correspondingly, the SINR is given by +SINRqb = +��hH +b,qbwqb +��2 pqb +�Q +q′=1,q′̸=q +��hH +b,qbwq′ +b +��2 pq′ +b + �B +b′=1,b′̸=b +�Q +q′=1 +���hH +b′,qbwq′ +b′ +��� +2 +pq′ +b′ + σ2 +qb +, ∀q ∈ Q, ∀b ∈ B. +(8) + +10 +The power allocation problem for sum rate maximization can be formulated as: +max +{pqb}q∈Q,b∈B +B +� +b=1 +Q +� +q=1 +log2 (1 + SINRqb) , +(9a) +s.t. +0 ≤ +Q +� +q=1 +pqb ≤ Pb, ∀b ∈ B, +(9b) +where Pb is the maximum transmit power budget of BS b. +As observed in (8), the inner product of hb,qb and wqb has BQ×BQ combinations, leading to +K = BQ equivalent TX-nodes and K RX-nodes, respectively. By modeling each BS as Q TX- +nodes, and each UE as an RX-node, we define two one-to-one mappings ψTX(·, ·) and ψRX(·, ·) +from B × Q to M and K, respectively. Let k = ψRX(b, q), and then for RXk, the TX-nodes can +be divided into three types as shown in Fig. 1(b). The first type is the TX-node that serves RXk, +denoted as TXm1(k), where m1(k) ≜ ψTX(b, q). The second type consists of other TX-nodes in the +same cell as RXk, denoted as TXm2(k), where m2(k) ∈ M2(k) ≜ {ψTX(b, q′)|∀q′ ∈ Q, q′ ̸= q}. +The third type consists of TX-nodes in other cells, denoted as TXm3(k), where m3(k) ∈ M3(k) ≜ +{ψTX(b′, q′)|∀b′ ∈ B, b′ ̸= b, ∀q′ ∈ Q}. Accordingly, the equivalent channel gain between different +TX-nodes and RXk can be written as three types: +gm1(k),k ≜ +��hH +b,qbwqb +�� , +(10a) +gm2(k),k ≜ +��hH +b,qbwq′ +b +�� , ∀q′ ∈ Q, q′ ̸= q, +(10b) +gm3(k),k ≜ +���hH +b′,qbwq′ +b′ +��� , ∀b′ ∈ B, b′ ̸= b, ∀q′ ∈ Q. +(10c) +We incorporate the maximum transmit power budget p = +� +˜P1, · · · , ˜PK +�T +and the noise +standard deviation σ += [σ1, · · · , σK]T as the features on the TX-nodes and RX-nodes, +respectively, where ˜Pm1(k) = Pb. Moreover, the feature vector on the (m, k)-th edge contains both +the equivalent channel gain gm,k and the indicator of direct communication, intra-cell interference, +or inter-cell interference link: +G(m,k,:) += +� +� +� +� +� +� +� +� +� +� +� +[gm,k, 0, 0]T, +if m = m1(k), +[0, gm,k, 0]T, +if m = m2(k), +[0, 0, gm,k]T, +otherwise, +(11) +where we adopt the idea of one-hot encoding to embed the information of direct communication, +inter-cell interference, or intra-cell interference links. On the other hand, since pqb is the power + +11 +allocation variable corresponding to the (m1(k), k, :)-th TX-RX pair and m1(·) is a one-to-one +mapping, we can either define pqb on the (m1(k), k, :)-th edge, the k-th RX-node, or the m1(k)- +th TX-node. Without loss of generality, putting the unknown power allocation variable pqb on +the (m1(k), k, :)-th fiber of the edge variable tensor P , problem (9) can be reformulated on the +heterogeneous graph as +max +φ(·,·,·) +B +� +b=1 +Q +� +q=1 +log2 (1 + SINRqb) , +(12a) +s.t. +P = φ (p, σ, G) , with 0 ≤ +Q +� +q=1 +pqb ≤ Pb, ∀b ∈ B. +(12b) +Comparing (12) with (1), the objective function (12a) is a specification of (1a). In particular, +the features p, σ, and G correspond to FTX, FRX, and E, respectively, and the variable P +corresponds to Ξ . +Example 3: Cooperative Beamforming Design. Consider a downlink system where M BSs +serve K UEs cooperatively. Each BS is equipped with N antennas and serves all UEs, while +each UE is equipped with a single antenna and served by all BSs. The channel between the m-th +BS and the k-th UE can be defined as hm,k ∈ CN. The beamforming vector used by the m-th +BS for serving the k-th UE is denoted as vm,k ∈ CN. With sk denoting the desired symbol of +the k-th UE, the received signal at the k-th UE is expressed as +yk = +M +� +m=1 +hH +m,kvm,ksk + +K +� +k′=1,k′̸=k +M +� +m=1 +hH +m,kvm,k′sk′ + nk, ∀k ∈ K, +(13) +where nk ∼ CN(0, σ2 +k) is the additive complex Gaussian noise. The SINR can be written as +SINRk = +����M +m=1 hH +m,kvm,k +��� +2 +�K +k′=1,k′̸=k +��� +�M +m=1 hH +m,kvm,k′ +��� +2 ++ σ2 +k +, ∀k ∈ K. +(14) +The cooperative beamforming design problem for sum rate maximization can be formulated as +max +{vm,k}m∈M,k∈K +K +� +k=1 +log2 (1 + SINRk) , +(15a) +s.t. +K +� +k=1 +∥vm,k∥2 ≤ Pm, ∀m ∈ M, +(15b) +where Pm denotes the maximum power budget of BS m. +As illustrated in Fig. 1(c), by modeling the BSs and UEs as TX-nodes and RX-nodes +respectively, we can incorporate the maximum transmit power budget p = [P1, · · · , PM]T + +12 +and the noise standard deviation σ = [σ1, · · · , σK]T as the features on the TX-nodes and +RX-nodes, respectively. Moreover, the feature vector on the (m, k)-th edge can be defined as +H(m,k,:) = hm,k. Unlike the previous examples, the unknown beamforming variable vm,k is not +a variable corresponding to a TX/RX-node, but rather corresponding to the (m, k)-th TX-RX +pair. Thus, vm,k can only be defined on the (m, k)-th edge. Putting vm,k on the (m, k, :)-th fiber +of the edge variable tensor V , problem (15) can be reformulated on the heterogeneous graph as +max +φ(·,·,·) +K +� +k=1 +log2 (1 + SINRk) , +(16a) +s.t. +V = φ (p, σ, H) , with +K +� +k=1 +∥vm,k∥2 ≤ Pm, ∀m ∈ M. +(16b) +Comparing (16) with (1), the objective function (16a) is a specification of (1a). Particularly, +the features p, σ, and H correspond to FTX, FRX, and E, respectively, and the variable V +corresponds to Ξ . +Notice that in the cooperative beamforming application, beamforming variables exist on the +communication links between TX-nodes and RX-nodes and should therefore be defined on the +edges. However, existing GNNs [17] [30] are only equipped with the node-update mechanism, +which cannot cope with the more complicated problem of cooperative beamforming design, +where each TX-node serves multiple RX-nodes and each RX-node is also served by multiple +TX-nodes. +C. PE Property +A unique property of beamforming design and power allocation is that the optimized strategy +is independent of the indices of TX-nodes and RX-nodes. In particular, the learned mapping +function φ(·, ·, ·) is inherently permutation equivariant with respect to TX-nodes and RX-nodes, +i.e., if the indices of any two TX-nodes or RX-nodes are exchanged, φ(·, ·, ·) should output a +corresponding permutation. +To visualize this, we show a heterogeneous graph before and after permutations of TX-nodes +and RX-nodes in Fig. 2. Define two permutations πTX(·) and πRX(·), and let TXm and RXk in + +13 +(a) Original heterogeneous graph. +(b) Heterogeneous graph after permutations of TX-nodes and +RX-nodes. +Fig. 2. Permutation equivariance illustration. +Fig. 2(a) be re-ordered as +˙ +TXπTX(m) and +˙ +RXπRX(k) in Fig. 2(b), where πTX(1) = 2, πTX(2) = 1, +πRX(1) = 3, πRX(2) = 1, and πRX(3) = 2. Accordingly, the graph features satisfy +˙fTX,πTX(m) = fTX,m, ∀m ∈ M, +(17a) +˙fRX,πRX(k) = fRX,k, ∀k ∈ K, +(17b) +˙E (πTX(m),πRX(k),:) = E (m,k,:), ∀(m, k) ∈ E. +(17c) +Let +� +˙STX, ˙SRX, ˙Ξ +� += φ +� +˙FTX, ˙FRX, ˙E +� +and (STX, SRX, Ξ ) = φ (FTX, FRX, E) be the corre- +sponding outputs of the mapping function φ(·, ·, ·), respectively. Since +� +˙FTX, ˙FRX, ˙E +� +is just a +re-ordering of the TX-nodes and RX-nodes in (FTX, FRX, E), the corresponding outputs of the +mapping function φ(·, ·, ·) should satisfy +˙sTX,πTX(m) = sTX,m, ∀m ∈ M, +(18a) +˙sRX,πRX(k) = sRX,k, ∀k ∈ K, +(18b) +˙Ξ (πTX(m),πRX(k),:) = Ξ (m,k,:), ∀(m, k) ∈ E. +(18c) +We will show in the next section that (18) can be guaranteed by the proposed ENGNN with +properly designed edge/node-update mechanisms. +III. THE PROPOSED ENGNN +In this section, we propose a customized neural network architecture to represent the mapping +function φ(·, ·, ·) from (FTX, FRX, E) to (STX, SRX, Ξ ) in problem (1). The proposed neural + +E/E(1,1,;) +RX1 +TX1 +E /E(1,2,:) +E /E(1,3,:) +RX2 +E /E(2,1,:) +E /E(2,2,) +TX2 +RX +E /E(2,3,:)E /三 (2,3.) +RX3 +TX2 +E/E(2,1,:) +E /三(2,2,:) +E /三(1,3;) +RX1 +E /E(1,1,:) +TX1 +E /三(1,2,:)14 +Fig. 3. +The overall architecture of the proposed ENGNN, which contains a preprocessing layer, L updating layers, and a +postprocessing layer. +network architecture incorporates both an edge-update mechanism and a node-update mechanism +into a GNN and hence we name it as ENGNN. We will show that the proposed ENGNN enjoys +the PE property given by (18). +A. Overall Architecture +The +proposed +ENGNN +consists +of +a +preprocessing +layer, +L +updating +layers, +and +a +postprocessing +layer +as +illustrated +in +Fig. +3. +The +preprocessing +layer +transforms +the +input +features +(FTX, FRX, E) +into +the +initial +node- +and +edge-representations +� +F(0) +TX ∈ RM× ˘dTX, F(0) +RX ∈ RK× ˘dRX, E(0) ∈ RM×K× ˘dE +� +, +where +˘dTX, +˘dRX, +and +˘dE +denote +the +dimensions of the representations on TX-nodes, RX-nodes, and edges, respectively. These +representations will be updated according to node- and edge-update mechanisms in the L +updating layers, where the l-th updating layer takes +� +F(l−1) +TX +, F(l−1) +RX , E(l−1)� +as the inputs and +then outputs the updated representations +� +F(l) +TX, F(l) +RX, E(l)� +. The dimensions of the representations +will not change in the updating layers. Finally, the postprocessing layer transforms the graph +representations +� +F(L) +TX , F(L) +RX, E(L)� +into variables (STX, SRX, Ξ ). +B. Preprocessing Layer +The preprocessing layer converts complex-valued features (if any) on TX-nodes, RX-nodes, +and edges into the real-valued form that can be processed by neural networks, and then transforms +the real-valued features into initial representations. Specifically, the inputs of the preprocessing + +Preprocessing Layer +Postprocessing Layer +1st Updating Layer +Lth Updating Layer +(0) +(L) +TX +TX +.(0) +(L) +RX +RX +RX +RX +(0) +(L +E15 +layer (FTX, FRX, E) are converted into their corresponding real-valued forms +� +ˆFTX, ˆFRX, ˆE +� +, +where the m-th row of FTX, the k-th row of FRX, and the (m, k)-th fiber of E are given by +ˆfTX,m = +� +ℜ {fTX,m}T , ℑ {fTX,m}T�T +, ∀m ∈ M, +(19a) +ˆfRX,k = +� +ℜ {fRX,k}T , ℑ {fRX,k}T�T +, ∀k ∈ K, +(19b) +ˆ +E(m,k,:) = +� +ℜ +� +E(m,k,:) +�T , ℑ +� +E(m,k,:) +�T�T +, ∀(m, k) ∈ E. +(19c) +Then, the initial representations of TX-nodes, RX-nodes, and edges are transformed by a one- +layer MLP with rectified linear unit (ReLU) as the activation function: +f (0) +TX,m = ReLU +� +Wpre +TXˆfTX,m + bpre +TX +� +, ∀m ∈ M, +(20a) +f (0) +RX,k = ReLU +� +Wpre +RXˆfRX,k + bpre +RX +� +, ∀k ∈ K, +(20b) +E(0) +(m,k,:) = ReLU +� +Wpre ˆ +E(m,k,:) + bpre� +, ∀(m, k) ∈ E, +(20c) +where Wpre +TX ∈ R +˘dTX×2dTX, bpre +TX ∈ R +˘dTX, Wpre +RX ∈ R +˘dRX×2dRX, bpre +RX ∈ R +˘dRX, Wpre ∈ R +˘d E×2dE, and +bpre ∈ R +˘dE are trainable parameters. +C. Updating Layer +The inputs and outputs of the updating layer l ∈ {1, · · · , L} are +� +F(l−1) +TX +, F(l−1) +RX , E(l−1)� +and +� +F(l) +TX, F(l) +RX, E(l)� +, respectively. We next show the node- and edge-update mechanisms in the l-th +updating layer. +1) Node-Update Mechanism: The update of node representations in the l-th updating layer +takes +� +F(l−1) +TX +, F(l−1) +RX , E(l−1)� +as the inputs, and then outputs the updated node representations +� +F(l) +TX, F(l) +RX +� +. In particular, when updating the representation of TXm, the inputs are composed of +the previous layer’s representations of TXm, the neighboring RX-nodes RXk, and edges (m, k) +for all k ∈ N TX +m , where N TX +m +is the set of neighboring RX-nodes of TXm. First, the input +representations on RXk and edge (m, k) are concatenated and then processed by an MLP. Next, +the processing results from all RXk with k ∈ N TX +m +are combined by an aggregation function +(e.g., mean or max aggregators), which extracts information from all the neighboring RX-nodes +regardless of their input order. Finally, the input representation of TXm and the aggregated result +are concatenated and then processed by another MLP. The above procedure gives the following +TX-update mechanism in the l-th updating layer: +f (l) +TX,m = MLP(l) +2 +� +f (l−1) +TX,m, AGG(l) +TX +� +MLP(l) +1 +� +f (l−1) +RX,k , E(l−1) +(m,k,:) +�� +k∈N TX +m +� +, ∀m ∈ M, +(21) + +16 +where f (l−1) +TX,m is the m-th row of F(l−1) +TX +, f (l−1) +RX,k is the k-th row of F(l−1) +RX , E(l−1) +(m,k,:) is the (m, k)- +th fiber of E(l−1), MLP(l) +1 +and MLP(l) +2 +are two MLPs, and AGG(l) +TX is an aggregation function. +Similarly, the RX-update mechanism in the l-th updating layer reverses the roles of TX-nodes +and UE-nodes in (21): +f (l) +RX,k = MLP(l) +4 +� +f (l−1) +RX,k , AGG(l) +RX +� +MLP(l) +3 +� +f (l−1) +TX,m, E(l−1) +(m,k,:) +�� +m∈N RX +k +� +, ∀k ∈ K, +(22) +where N RX +k +is the set of neighboring TX-nodes of RXk, MLP(l) +3 and MLP(l) +4 are two MLPs, and +AGG(l) +RX is an aggregation function. +Notice that the representations of TX-nodes and RX-nodes are updated differently in the +proposed ENGNN. This is different from the previous work MPGNN [17] in which the node +representations are updated homogeneously. Moreover, in the proposed ENGNN, the input edge +representations E(l−1) +(m,k,:) in (21) and (22) are with superscript (l − 1) and hence are also updated +(see the edge-update mechanism). Taking a similar analysis in [17], we can show that (21) and +(22) satisfy the following PE property: +Property 1 (PE in Node-Update Mechanism): The node-update mechanism (21) and (22) +are permutation equivariant with respect to TX-nodes and RX-nodes, respectively. Specifically, +for any permutations πTX(·) and πRX(·), we have +f (l) +TX,πTX(m) = +MLP(l) +2 +� +f (l−1) +TX,πTX(m), AGG(l) +TX +� +MLP(l) +1 +� +f (l−1) +RX,k , E(l−1) +(πTX(m),k,:) +�� +k∈N TX +πTX(m) +� +, ∀m ∈ M, +(23a) +f (l) +RX,πRX(k) = +MLP(l) +4 +� +f (l−1) +RX,πRX(k), AGG(l) +RX +� +MLP(l) +3 +� +f (l−1) +TX,m, E(l−1) +(m,πRX(k),:) +�� +m∈N RX +πRX(k) +� +, ∀k ∈ K. +(23b) +2) Edge-Update Mechanism: The update of edge representations in the l-th updating layer +takes +� +F(l−1) +TX +, F(l−1) +RX , E(l−1)� +as the inputs, and then outputs the updated edge representations +E(l). Different from the node-update mechanism, where the neighbors of a TX-node (or RX- +node) are clearly defined as the connecting RX-nodes (or TX-nodes), it is more complicated to +define the neighbors of an edge, let alone how to aggregate their representations. Notice that an +edge may connect with other edges through either a TX-node or an RX-node. In particular, for +the edge (m, k) ∈ E, its neighboring edges through TXm are (m, k1), ∀k1 ∈ N TX +m \ {k}, while +the neighboring edges through RXk are (m1, k), ∀m1 ∈ N RX +k +\ {m}. For example, in Fig. 4, the + +17 +Fig. 4. The neighbors that share the connection with edge (1, 1) through TX1 are edge (1, 2) and edge (1, 3), which are denoted +by dotted lines. The neighbor that shares the connection with edge (1, 1) through RX1 is edge (2, 1), which is denoted by a +dashed line. +neighboring edges of edge (1, 1) through TX1 are edge (1, 2) and edge (1, 3). On the other hand, +the neighboring edge of edge (1, 1) through RX1 is edge (2, 1). This causes the neighbors of +an edge to be innately divided into two categories based on the connecting node. Consequently, +different from the node-update mechanism (21) and (22), the edge-update mechanism should +provide two different aggregations for the two types of neighboring edges. +Specifically, when updating the representation of edge (m, k), the inputs are composed of +the previous representations of edge (m, k), TXm, RXk, the neighboring edges (m, k1), ∀k1 ∈ +N TX +m \ {k}, and neighboring edges (m1, k), ∀m1 ∈ N RX +k +\ {m}. The input representations of +neighboring edges (m, k1), ∀k1 ∈ N TX +m \{k} and the connecting node TXm are concatenated and +then processed by an MLP, while the input representations of neighboring edges (m1, k), ∀m1 ∈ +N RX +k +\{m} and the connecting node RXk are concatenated and then processed by another MLP. +The processing results of all the neighboring edges are aggregated and then concatenated with +the input representation of edge (m, k). An MLP is finally applied to produce the updated +representation of edge (m, k). We can express the above edge-update procedure in the l-th + +(1,2.:) +RX1 +(1,1,) +(1,3,) +(1,2.) +RX2 +E(2,1,;) +Edge (1,1) +Edge (1,1) +TX2 +E +(2,2,) +and its +and its +E(2,3;) +neighbours +neighbours +RX3 +through TXi +through RXi18 +updating layer as +E(l) +(m,k,:) = MLP(l) +7 +� +E(l−1) +(m,k,:), AGG(l) +E +� +MLP(l) +5 +� +E(l−1) +(m,k1,:), f (l−1) +TX,m +� +, +MLP(l) +6 +� +E(l−1) +(m1,k,:), f (l−1) +RX,k +�� +k1∈N TX +m \{k},m1∈N RX +k \{m} +� +, ∀(m, k) ∈ E, +(24) +where E(l−1) +(m,k,:) is the (m, k)-th fiber of E(l−1), MLP(l) +5 , MLP(l) +6 , and MLP(l) +7 are three MLPs, and +AGG(l) +E is an aggregation function. +Compared with the node-update mechanism (21) and (22), the edge-update mechanism (24) +is more complicated, since the definition of neighbors in edge-update mechanism is more +complex than that in the node-update one. In particular, the edge-update mechanism faces a +more complicated situation where the neighboring edges are innately divided into two categories +according to the two possible connected nodes. Consequently, different from the node-update +mechanism (21) and (22), where the information from neighboring nodes are gathered by one +MLP, the proposed edge-update mechanism applies two different transformations to extract the +information from two different types of neighboring edges. We next show that (24) enjoys the +following PE property: +Property 2 (PE in Edge-Update Mechanism): The edge-update mechanism (24) is permu- +tation equivariant with respect to TX-nodes and RX-nodes. Specifically, for any permutations +πTX(·) and πRX(·), we have +E(l) +(πTX(m),πRX(k),:) = MLP(l) +7 +� +E(l−1) +(πTX(m),πRX(k),:), AGG(l) +E +� +MLP(l) +5 +� +E(l−1) +(πTX(m),k1,:), f (l−1) +TX,πTX(m) +� +, +MLP(l) +6 +� +E(l−1) +(m1,πRX(k),:), f (l−1) +RX,πRX(k) +�� +k1∈N TX +πTX(m)\{πRX(k)},m1∈N RX +πRX(k)\{πTX(m)} +� +, ∀(m, k) ∈ E. +(25) +Proof: See Appendix A. +D. Postprocessing Layer +The postprocessing layer converts the graph representations +� +F(L) +TX , F(L) +RX, E(L)� +into the final +output (STX, SRX, Ξ ). First, if the variables are complex, +� +F(L) +TX , F(L) +RX, E(L)� +are transformed into + +19 +the complex form +� +˜STX, ˜SRX, ˜Ξ +� +by +� +ℜ {˜sTX,m}T , ℑ {˜sTX,m}T�T += Wpost +TX f (L) +TX,m + bpost +TX , ∀m ∈ M, +(26a) +� +ℜ {˜sRX,k}T , ℑ {˜sRX,k}T�T += Wpost +RX f (L) +RX,k + bpost +RX , ∀k ∈ K, +(26b) +� +ℜ +� +˜Ξ (m,k,:) +�T +, ℑ +� +˜Ξ (m,k,:) +�T�T += WpostE(L) +(m,k.:) + bpost, ∀(m, k) ∈ E, +(26c) +where Wpost +TX ∈ R2d′ +TX× ˘dTX, bpost +TX ∈ R2d′ +TX, Wpost +RX ∈ R2d′ +RX× ˘dRX, bpost +RX ∈ R2d′ +RX, Wpost ∈ R2d′ +E× ˘dE, +and bpost ∈ R2d′ +E are trainable parameters. Next, +� +˜STX, ˜SRX, ˜Ξ +� +are normalized to satisfy the +constraints (if any), obtaining the final output (STX, SRX, Ξ ). +E. Key Insights +The proposed ENGNN for representing φ(·, ·, ·) has been specified as a preprocessing layer, L +updating layers, and a postprocessing layer, where the preprocessing and postprocessing layers +utilize edge/node-wise MLPs, and the L updating layers are built on node- and edge-update +mechanisms (21), (22), and (24). Next, we provide some key insights of the proposed ENGNN +for learning the beamforming design and power allocation as follows. +1) Permutation Equivariant with Respect to TX-nodes and RX-nodes: +Proposition 1 (PE in ENGNN): The proposed ENGNN is permutation equivariant with +respect to TX-nodes and RX-nodes. Specifically, for any permutations πTX(·) and πRX(·), denote +a permuted problem instance of (FTX, FRX, E) as +� +˙FTX, ˙FRX, ˙E +� +, whose entries satisfy (17). +The corresponding outputs of the proposed ENGNN, +� +˙STX, ˙SRX, ˙Ξ +� += φ +� +˙FTX, ˙FRX, ˙E +� +and +(STX, SRX, Ξ ) = φ (FTX, FRX, E), always satisfy (18). +Proof: See Appendix B. +Proposition 1 implies that the proposed ENGNN is inherently incorporated with the PE +property. This is in sharp contrast to the generic MLPs, which require all permutations of each +training sample to approximate this property. Thus, the proposed ENGNN can reduce the sample +complexity and training difficulty. +2) Generalization on Different Numbers of TX-nodes and RX-nodes: In all the layers of the +proposed ENGNN, the representations on different edges/TX-nodes/RX-nodes are transformed +by the same architecture using the same trainable parameters. Therefore, the dimensions of the +trainable parameters are independent of the numbers of TX-nodes and RX-nodes. This scale + +20 +adaptability empowers ENGNN to be trained in a setup with a small graph size, while being +deployed to a much larger wireless network for the inference. +3) Tackling Edge Variables: The proposed ENGNN is equipped with an edge-update mech- +anism, which facilitates the update of the variables on graph edges. This allows ENGNN to be +applied in a wider range of scenarios, where variables are defined between a pair of nodes. +IV. SIMULATION RESULTS +In this section, we demonstrate the superiority of the proposed ENGNN on the three examples +introduced in Section II-B via simulations. We consider a downlink wireless network in a 2 × 2 +km2 area, where the BSs and UEs are uniformly distributed. Each BS has a maximum transmit +power of 33 dBm. The path loss is 30.5 + 36.7 log10 d in dB, where d is the distance in meters. +The small scale channels follow Rayleigh fading and the noise power is −99 dBm. +For the proposed ENGNN, each aggregation function in (21), (22), and (24) is implemented by +a max aggregator, which returns the element-wise maximum value of the inputs. All the MLPs +in (21), (22), and (24) are implemented by 3 linear layers, each followed by a ReLU activation +function. In the training procedure, the number of epochs is set to 500. Each epoch consists of +100 mini-batches of training samples with a batch size of 256. For each training sample, the +BSs’ and UEs’ locations, and the small scale channels are randomly generated. A learning rate +γ = 10−4 is adopted to update the trainable parameters of ENGNN by maximizing (1a) using +RMSProp [35] in an unsupervised manner. After training, we test the average performance of +100 samples. All the experiments are implemented using Pytorch on one NVIDIA V100 GPU +(32 GB, SMX2). +A. Beamforming Design for Interference Channels +First, we demonstrate the performance of the proposed ENGNN on the problem of beam- +forming design for interference channels. During the training procedure, we set the wireless +network with 20 BS-UE pairs, where each BS is equipped with 2 antennas. An ENGNN with +1 updating layer is adopted, and the dimension of edge features ˘dE is set to 8. As explained +before (6), the beamforming variables in this scenario can be defined on either the edges or the +nodes, which only affects the postprocessing layers. The corresponding simulation results are +termed as ENGNN-E and ENGNN-N, respectively. For performance comparison, we include +two state-of-the-art methods: + +21 +Fig. 5. Generalization on number of BS-UE pairs. +(a) WMMSE: a widely used benchmark algorithm for sum rate maximization [8]. +(b) MPGNN: the latest learning based method for sum rate maximization in interference +channels [17]. +1) Generalization on Number of BS-UE Pairs: We first compare the performance of different +approaches as the number of BS-UE pairs increases. In particular, ENGNN-E, ENGNN-N, +MPGNN are trained on 20 BS-UE pairs, while we test their performance in terms of sum rate +on larger problem scales from 20 to 100 BS-UE pairs in Fig. 5. It can be seen that ENGNN-E, +ENGNN-N, and MPGNN generalize well as the number of BS-UE pairs increases from 20 to +40. However, as the number of BS-UE pairs increases from 40 to 100, both ENGNN-E and +ENGNN-N can still generalize very well, while the performance of MPGNN becomes worse +than that of WMMSE. The superiority of ENGNN-E and ENGNN-N is owing to the proposed +edge-update mechanism, which better extracts the features from channel states and hence further +empowers the original node-update mechanism in MPGNN. Since ENGNN-E and ENGNN-N +result in similar performance, we only show ENGNN-E in the rest of simulations and term it as +ENGNN for simplicity. +2) Generalization on Noise Power: To demonstrate the generalization performance on noise +power, we set the noise power during the training procedure as −99 dBm, while we test the +generalization performance under different noise powers from −99 dBm to −89 dBm. It can +be seen from Fig. 6 that ENGNN consistently achieves higher sum rate than those of MPGNN +and WMMSE, which demonstrates the superiority of ENGNN in generalizing to different noise + +ENGNN-E +220 +MPGNN +WMMSE +200 +ENGNN-N +Sum rate (bps/Hz) +180 +160 +140 +120 +100 +80 +20 +40 +60 +80 +100 +Number of BS-UE pairs22 +Fig. 6. Generalization on noise power. +Fig. 7. Generalization on different levels of interference among BS-UE pairs. +powers. +3) Generalization on Different Levels of Interference: In this experiment, we fix the field size +during the training procedure as 2 × 2 km2, while we test the performance by varying the field +size from 2 × 2 km2 to 4.5 × 4.5 km2. By keeping the distance between each BS and its serving +UE within 50-250 meters, the interference among different BS-UE pairs becomes weaker as the +field size increases, and hence the sum rates of different approaches become higher in Fig. 7. +Moreover, ENGNN consistently achieves higher sum rate than those of MPGNN and WMMSE +as the field size increases, which demonstrates that ENGNN generalizes well on different levels +of interference. + +150 +ENGNN +MPGNN +140 +WMMSE +130 +120 +110 +100 +90 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +Field side length (km)ENGNN +95 +MPGNN +WMMSE +Sum rate (bps/Hz) +90 +85 +80 +75 +-99 +-97 +-95 +-93 +-91 +-89 +Noise power (dBm)23 +Fig. 8. Sample complexity comparison between ENGNN and MPGNN. +4) Sample Complexity Comparison: We further compare the performance of ENGNN and +MPGNN when trained on different numbers of training samples in Fig. 8. It can be seen that +ENGNN outperforms MPGNN especially when the number of training samples is small. In +particular, when the number of training samples decreases to 10-100, the sum rate of ENGNN +only decreases to 83.39-89.68 bps/Hz, while that of MPGNN decreases sharply to 68.60-82.10 +bps/Hz. The required number of training samples of ENGNN is less than 10% of that of MPGNN +when they achieve the same sum rate. This demonstrates the advantage of the proposed edge- +update mechanism in sample complexity. +B. Power Allocation for Interference Broadcast Channels +Next, we demonstrate the performance of ENGNN on the problem of power allocation for +interference broadcast channels. In the training procedure, we set the wireless network with +5 BSs, with a minimum distance of 500 meters between BSs. Each BS is equipped with 16 +antennas and serves 2 UEs. We use zero-forcing beamforming to avoid multi-user interference. +The ENGNN is set with 1 updating layer and a dimension of ˘dE = 32 for the edge features. For +performance comparison, we provide the simulation results of WMMSE and PGNN [30], which +is the latest learning-based method for power allocation in interference broadcast channels. +1) Generalization on Number of UEs: We first compare the performance of different +approaches as the number of UEs increases. In particular, both ENGNN and PGNN are trained +under 10 UEs, while we test their performance on larger problem scales from 10 to 50 UEs +by varying the number of UEs in each cell from 2 to 10 in Fig. 9. It can be seen that both + +ENGNN +95 +MPGNN +90 +Sum rate (bps/Hz) +85 +80 +75 +70 +101 +102 +103 +104 +Number of training samples24 +Fig. 9. Generalization on number of UEs for power allocation in interference broadcast channels. +Fig. 10. Generalization on different levels of inter-cell interference. +ENGNN and PGNN generalize well as the number of UEs increases from 10 to 50. However, +ENGNN always outperforms PGNN, and achieves competitive performance compared to that of +WMMSE under different numbers of UEs. +2) Generalization on Different Levels of Interference: To demonstrate the generalization +ability on different levels of interference, during the training procedure, the field size is fixed +as 2 × 2 km2, while we test the performance by varying the field size from 2 × 2 km2 to +4.5 × 4.5 km2. By keeping the distance between each BS and its serving UE within 50-250 +meters, the inter-cell interference becomes weaker as the field size increases, and hence the sum +rate becomes higher in Fig. 10. It can be observed that the advantage of ENGNN over PGNN +is stable as the field size increases, which demonstrates its superior generalization capability on + +220 +ENGNN +PGNN +200 +WMMSE +Sum rate (bps/Hz) +180 +160 +140 +120 +100 +80 +10 +20 +30 +40 +50 +Number of UEs85 +ENGNN +PGNN +84 +WMMSE +83 +82 +81 +80 +79 +78 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +Field side length (km)25 +Fig. 11. Generalization on power budget. +Fig. 12. Sample complexity comparison between ENGNN and PGNN. +different levels of interference. +3) Generalization on Power Budget: To demonstrate the generalization capability of ENGNN +on different power budgets, we set the power budget at each BS as 33 dBm during the training +procedure, while we test the generalization performance under different power budgets from 21 +dBm to 33 dBm in Fig. 11. It can be seen that both ENGNN and PGNN generalize well as +the power budgets decreases from 33 dBm to 21 dBm. However, ENGNN always outperforms +PGNN under different power budgets. +4) Sample Complexity Comparison: We further compare the performance of ENGNN and +PGNN when trained on different numbers of samples in Fig. 12. It can be seen that ENGNN +outperforms PGNN especially when the number of training samples is small. Particularly, when + +ENGNN +75 +PGNN +WMMSE +70 +Sum rate (bps/Hz) +65 +60 +55 +50 +45 +21 +23 +25 +27 +29 +31 +33 +Power budget (dBm)78.0 +77.5 +77.0 +Sum rate (bps/Hz) +76.5 +76.0 +75.5 +75.0 +74.5 +ENGNN +74.0 +PGNN +2 +101 +5 +102 +103 +Number of training samples26 +(a) Sum rate comparison +(b) Computation time comparison +Fig. 13. Generalization on number of UEs for cooperative beamforming. +the number of training samples decreases to 2-5, the sum rate of ENGNN only drops to 77.30- +77.52 bps/Hz, while that of PGNN drops to 73.89-75.12 bps/Hz. The required number of training +samples of ENGNN is about 1% of that of MPGNN when they achieve the same sum rate. This +demonstrates the advantage of the proposed edge-update mechanism in sample complexity. +C. Cooperative Beamforming Design +Finally, we demonstrate the performance of ENGNN on the problem of cooperative beam- +forming design. During the training procedure, we set the wireless network with 5 BSs and 2 +UEs. Each BS is equipped with 2 antennas and the minimum distance between BSs is 500 m. An +ENGNN with 2 updating layers is adopted, and the dimension of edge features ˘dE is set to 64. For +performance comparison, we include WMMSE and GP [7], the latter being a computationally +efficient first-order algorithm for solving simply constrained optimization problems. +1) Generalization on Number of UEs: To demonstrate the generalization ability of ENGNN +with respect to different numbers of UEs, during the training procedure, the number of UEs is +fixed as 2, while we test the performance of the trained ENGNN by varying the number of UEs +from 2 to 8. The performance comparison in terms of sum rate and computation time is shown +in Fig. 13. We observe from Fig. 13(a) that as the number of UEs increases, ENGNN always +outperforms GP and WMMSE in terms of sum rate, which demonstrates its generalization ability +with respect to different numbers of UEs. On the other hand, Fig. 13(b) shows that ENGNN + +ENGNN +GP +30 +WMMSE +n rate (bps/Hz) +25 +Sum +20 +15 +2 +4 +6 +8 +Number of UEs101 +(s) +Average computation time +100 +ENGNN +GP +10-1 +WMMSE +2 +4 +6 +8 +Number of UEs27 +(a) Sum rate comparison +(b) Computation time comparison +Fig. 14. Generalization on numbers of BSs for cooperative beamforming. +achieves a remarkable running speed, with over 100 times faster than that of GP and over 1000 +times faster than that of WMMSE due to the computationally efficient feed forward computations. +2) Generalization on Number of BSs: We further demonstrate the generalization ability of +ENGNN with respect to different numbers of BSs. Specifically, the number of BSs is fixed as 5 +during the training procedure, while we test the performance of the trained ENGNN by varying +the number of BSs from 5 to 8. The performance comparison is shown in Fig. 14. We observe +from Fig. 14(a) that ENGNN achieves higher sum rate than those of GP and WMMSE under +different numbers of BSs. Moreover, Fig. 14(b) shows that ENGNN achieves a much faster +running speed than that of GP and WMMSE under different numbers of BSs. +V. CONCLUSION +This paper proposed a general problem formulation on the heterogeneous graph for the radio +resource management problems, where the unknown variables to be designed can be defined +on both the graph nodes and edges. A novel edge-update mechanism with desired PE property +was incorporated, and a general neural network architecture was designed based on it, which +can represent the mapping function from the node/edge features to variables for the radio +resource management problems. Simulation results demonstrated the superiority of the proposed +architecture on three typical problems, with higher sum rate and much shorter computation time +compared with state-of-the-art methods. Moreover, the proposed architecture generalizes well on + +17 +16 +(zZH/sdq) +15 +rate +Sum +14 +13 +ENGNN +GP +WMMSE +12 +5 +6 +7 +8 +Number of BSsAverage computation time (s) +100 +ENGNN +10-1 +GP +WMMSE +10-2 +5 +6 +7 +8 +NumberofBSs28 +different numbers of BSs and UEs, different noise variances, interference levels, and transmit +power budgets. +APPENDIX A +PROOF OF PROPERTY 2 +Let m′ = πTX(m) and k′ = πRX(k). Substituting these two equations into (24), we have +E(l) +(m′,k′,:) = MLP(l) +7 +� +E(l−1) +(m′,k′,:), AGG(l) +E +� +MLP(l) +5 +� +E(l−1) +(m′,k′ +1,:), f (l−1) +TX,m′ +� +, +MLP(l) +6 +� +E(l−1) +(m′ +1,k′,:), f (l−1) +RX,k′ +�� +k′ +1∈N TX +m′\{k′},m′ +1∈N RX +k′ \{m′} +� +, ∀(m′, k′) ∈ E, +(27) +which implies that for any πTX(·) and πRX(·), we always have (25). +APPENDIX B +PROOF OF PROPOSITION 1 +Substituting (17) into (19) and (20) we have +˙f (0) +TX,πTX(m) = f (0) +TX,m, ∀m ∈ M, +(28a) +˙f (0) +RX,πRX(k) = f (0) +RX,k, ∀k ∈ K, +(28b) +˙E +(0) +(πTX(m),πRX(k),:) = E(0) +(m,k,:), ∀(m, k) ∈ E. +(28c) +Next, we substitute (28) into (21), (22), and (24). According to Property 1 and Property 2, we +have +˙f (l) +TX,πTX(m) = f (l) +TX,m, ∀m ∈ M, ∀l = 1, · · · , L, +(29a) +˙f (l) +RX,πRX(k) = f (l) +RX,k, ∀k ∈ K, ∀l = 1, · · · , L, +(29b) +˙E +(l) +(πTX(m),πRX(k),:) = E(L) +(m,k.:), ∀(m, k) ∈ E, ∀l = 1, · · · , L. +(29c) +Substituting (29) into (26), we obtain +˙˜sTX,πTX(m) = ˜sTX,m, ∀m ∈ M, +(30a) +˙˜sRX,πRX(k) = ˜sRX,k, ∀k ∈ K, +(30b) +˙˜Ξ (πTX(m),πRX(k),:) = ˜Ξ (m,k,:), ∀(m, k) ∈ E. +(30c) +Finally, since the normalization of +� +˜STX, ˜SRX, ˜Ξ +� +is an edge/node-wise computation, the final +outputs satisfy ˙sTX,πTX(m) = sTX,m, ∀m ∈ M, ˙sRX,πRX(k) = sRX,k, ∀k ∈ K, and ˙Ξ (πTX(m),πRX(k),:) = +Ξ (m,k,:), ∀(m, k) ∈ E. + +29 +REFERENCES +[1] H. Zhang and H. Dai, “Cochannel interference mitigation and cooperative processing in downlink multicell multiuser +MIMO networks,” EURASIP Journal on Wireless Communications and Networking, vol. 2004, no. 2, pp. 1–14, 2004. +[2] Y. Shi, J. Zhang, K. B. Letaief, B. Bai, and W. 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Hinton et al., “Lecture 6.5-RMSProp: Divide the gradient by a running average of its recent magnitude,” +COURSERA: Neural networks for machine learning, vol. 4, no. 2, pp. 26–31, 2012. + diff --git a/7NAyT4oBgHgl3EQf2vm0/content/tmp_files/load_file.txt b/7NAyT4oBgHgl3EQf2vm0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8828a9920706189f8e801bb69a4e9c6164223f72 --- /dev/null +++ b/7NAyT4oBgHgl3EQf2vm0/content/tmp_files/load_file.txt @@ -0,0 +1,844 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf,len=843 +page_content='1 ENGNN: A General Edge-Update Empowered GNN Architecture for Radio Resource Management in Wireless Networks Yunqi Wang, Yang Li, Qingjiang Shi, and Yik-Chung Wu Abstract In order to achieve high data rate and ubiquitous connectivity in future wireless networks, a key task is to efficiently manage the radio resource by judicious beamforming and power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Unfortunately, the iterative nature of the commonly applied optimization-based algorithms cannot meet the low latency requirements due to the high computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' For real-time implementations, deep learning-based approaches, especially the graph neural networks (GNNs), have been demonstrated with good scalability and generalization performance due to the permutation equivariance (PE) property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' However, the current architectures are only equipped with the node-update mechanism, which prohibits the applications to a more general setup, where the unknown variables are also defined on the graph edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' To fill this gap, we propose an edge-update mechanism, which enables GNNs to handle both node and edge variables and prove its PE property with respect to both transmitters and receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Simulation results on typical radio resource management problems demonstrate that the proposed method achieves higher sum rate but with much shorter computation time than state-of-the-art methods and generalizes well on different numbers of base stations and users, different noise variances, interference levels, and transmit power budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Index Terms Yunqi Wang is with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong, and also with Shenzhen Research Institute of Big Data, Shenzhen 518172, China (email: yunqi9@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='hku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Yang Li is with Shenzhen Research Institute of Big Data, Shenzhen 518172, China (e-mail: liyang@sribd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Qingjiang Shi is with the School of Software Engineering, Tongji University, Shanghai 200092, China, and also with Shenzhen Research Institute of Big Data, Shenzhen 518172, China (email: shiqj@tongji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Yik-Chung Wu is with the Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong (email: ycwu@eee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='hku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='00757v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='NI] 14 Dec 2022 2 Beamforming design, power allocation, heterogeneous graph neural network (GNN), edge-update mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' INTRODUCTION Efficient radio resource management plays a vital role in achieving high data rate and ubiquitous connectivity of future wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In particular, beamforming design and power allocation have been recognized as crucial components to improve the spectrum/energy efficiency in ultra-dense networks [1], cloud radio access networks [2], [3], and cell-free massive multiple- input multiple-output systems [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Mathematically, many of the radio resource management problems belong to the challenging non-convex optimization problems, which are conventionally solved by numerical algorithms with a lot of iterations [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' However, due to the fast variation of the wireless environment, the iterative nature of the commonly applied optimization-based numerical algorithms cannot satisfy the low-latency requirement in beyond 5G paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' For instance, to maximize the sum rate of a multi-cell wireless system under the maximum transmit power constraint of each base station (BS), the conventional first-order algorithm, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=', the gradient projection (GP) method [7], requires a lot of iterations to converge to a stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' To improve the convergence rate, while more advanced numerical algorithms such as the weighted minimum mean-square error (WMMSE) [8] algorithm can be applied, the matrix inverse in each iteration still makes it computationally expensive and hence difficult for the real-time implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' To facilitate the real-time implementation, deep learning based methods have become popular for radio resource management [9]–[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Specifically, deep learning based methods utilize neural networks to learn a mapping function from many problem features to the corresponding solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Once the neural network is well trained, it can infer the solution of any new setting using simple feed-forward computations, and thus is extremely fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Inspired by the successful applications in computer vision, the multi-layer perceptrons (MLPs) and convolutional neural networks (CNNs) have been applied as typical architectures for representing the mapping functions in radio resource management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' For example, MLPs were used to learn the mapping function from the wireless channel to the optimal resource management policy [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Moreover, an MLP-based architecture was adopted to learn the optimal power control for the multi-user interference channels [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Similarly, CNNs have also been applied for power control [11] and beamforming design [12] in the multiple-input single-output downlink systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 3 However, since MLPs and CNNs cannot fully exploit the topology in the wireless networks, they usually require a large number of training samples while still result in limited performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' For instance, it is shown in [16] that a CNN trained on a two-user wireless networks can only achieve the near-optimal performance for two-user wireless networks during the testing phase, but its performance degrades by 18% in ten-user networks compared to the conventional optimization- based numerical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Recently, attempts to use graph neural networks (GNNs) are on the rise because of their ability to exploit the topology of wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' By modeling a wireless network as a graph, the known system parameters can be modeled as the graph features, which are treated as the input of a GNN, while the unknown variables to be optimized can also be defined on the graph and are served as the output of a GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The advantage of graph modeling lies in its permutation equivariance (PE) property, where the graph features/variables can be regarded as a set of elements whose index order does not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Consequently, a large number of unnecessary permuted training samples can be discarded [17]–[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Moreover, since the trainable parameters of GNNs are independent of the graph size, the well-trained GNNs can generalize well to different problem dimensions [20]–[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Among the existing GNN-based works, homogeneous GNNs, which share the trainable parameters among different graph nodes, have shown their good scalability and generalization performance for the radio resource management problems when there is only one type of graph nodes [17]–[20], [24]–[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' For example, in [17], a homogeneous GNN named message passing graph neural network (MPGNN) was proposed for beamforming design in the multi-transceiver interference channels, where each transceiver pair is modeled as an individual graph node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' By modeling different transceiver pairs as the same type of nodes and sharing their trainable parameters, the test performance in terms of sum rate is near optimal even when the number of transceiver pair is twice larger than that in the training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Similarly, homogeneous GNNs have also been demonstrated to generalize well on different numbers of users in multicast beamforming design [24], link scheduling [25], power control [26], [27], and joint beam selection and link activation [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' While homogeneous GNNs have shown their great success when there exists only one type of graph nodes in the radio resource management problems, it should be noticed that the more common scenarios usually consist of different types of graph nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' For instance, in a general wireless network, the transmitters and receivers have different physical characteristics, and hence 4 should be more naturally modeled as two different types of graph nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=', TX-nodes and RX- nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' By sharing the trainable parameters within only the same type of nodes, heterogeneous GNNs [29] have shown their superiority for the more complex radio resource management problems compared with the homogeneous GNNs [30]–[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In particular, the pioneer work [30] designed a heterogeneous GNN called permutation equivariant heterogeneous GNN (PGNN) for the power allocation in multi-cell downlink systems, and theoretically established the PE property with respect to different user equipments (UEs) within each cell and also with respect to different cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Moreover, heterogeneous GNNs have also been proposed for the beamforming design in heterogeneous device-to-device networks [31] and multi-user downlink systems [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In [33], a heterogeneous GNN was designed for jointly learning the beamforming vectors and reflecting phases for an intelligent reflecting surfaces (IRS) assisted multi-user downlink system, where the users and the IRS are modeled as heterogeneous graph nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Similarly, the trajectory of unmanned aerial vehicles (UAVs) was cooperatively designed by modeling the UAVs and the ground terminals as heterogeneous graph nodes in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Despite the successes of the homogeneous or heterogeneous GNNs in the above existing works, they are only equipped with the node-update mechanism, which restricts the output of the neural networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=', the unknown variables to be optimized only appear on the graph nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Notable examples are MPGNN [17] and PGNN [30], both of which do not consider the variables on the graph edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In particular, MPGNN is proposed for the beamforming design for the multi- transceiver interference channels, where each transmitter only serves a single receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Therefore, each transceiver pair can be modeled as a single node, and the channel state information of each direct communication link serves as the corresponding node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Furthermore, the interference links among different transceiver pairs are modeled as graph edges, whose channel state information is treated as the corresponding edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Using this graph model, the beamforming variable of each transceiver pair can be defined only on the corresponding node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' On the other hand, PGNN is proposed for the power allocation in multi-cell systems, where each BS serves multiple UEs within the same cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In this scenario, each BS adopts a pre- designed beamformer, so that a dedicated equivalent single-antenna channel is created for each UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Consequently, with each equivalent transmit antenna treated as an individual node, the power allocation variables can also be defined on the graph nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' While all the above pioneering works exemplify the benefits of GNNs in radio resource management, currently applied architectures prohibit the extension to a more general setting, 5 where the unknown variables are also defined on the graph edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' A typical application scenario is the cooperative beamforming design, where each transmitter serves multiple receivers, while each receiver is also served by multiple transmitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' These complicated transceiver interactions cannot be easily modeled by the current GNN architectures that are only equipped with the node-update mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In fact, for the more general wireless environment, an individual beamforming or power variable belongs to a transceiver pair, which is represented by two different nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Thus, beamformers or power variables should be more naturally defined on the graph edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Unfortunately, without a judiciously designed edge-update mechanism, the current widely adopted GNN architectures cannot handle such a general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' To fill this gap, we propose a novel edge-update mechanism, which enables the GNN architecture to deal with both the edge and node variables for the radio resource management problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The contributions of this paper are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 1) We propose a general problem formulation using the heterogeneous graph for the radio resource management problems, where the unknown variables to be optimized can be defined on the graph edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' To learn the edge variables, we design a novel edge-update mechanism and prove its PE property with respect to both the transmitters and receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Compared with the existing node-update mechanism that gathers the information from the neighboring nodes, the update of an edge variable is more challenging, since it is more complicated to define the neighbors of an edge, let alone how to aggregate their representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Based on the observation that the neighboring edges can be divided into two categories according to their connected nodes, we propose an edge-update mechanism that extracts the information from the two types of neighboring edges in a different manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 2) Based on the edge-update mechanism, we propose an edge-update empowered neural network architecture termed as edge-node GNN (ENGNN), which can represent the mapping function from the graph features to the edge/node variables for the radio resource management problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' We prove that the proposed ENGNN is permutation equivariant with respect to both transmitters and receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Moreover, since the trainable parameters of the proposed ENGNN are independent of the graph size, it can generalize to different numbers of transmitters and receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Last but not the least, the proposed ENGNN can be applied in a wide range of radio resource management problems, where the variables occur between any pair of the TX-nodes and RX-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 3) Simulation results demonstrate the superiority of the proposed ENGNN for typical radio 6 resource management problems, including the beamforming design in the interference channels, the power allocation in the interference broadcast channels, and the cooperative beamforming design, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' It is shown that the proposed ENGNN achieves higher sum rate with much shorter computation time than state-of-the-art methods and generalizes well on different numbers of BSs and UEs, different noise variances, interference levels, and transmit power budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Notations: In this paper, we use bold lowercase letters, bold uppercase letters, and bold italicized uppercase letters to represent vectors, matrices, and tensors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The sets are represented by stylized uppercase letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The notations (·)T and (·)H refer to transpose and Hermitian transpose, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Moreover, |·|2 denotes the l2-norm operation, and |·| computes the magnitude of a complex number or the cardinality of a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In Section II, we propose a general problem formulation on the heterogeneous graph for the radio resource management problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In Section III, we design a novel neural network architecture named ENGNN with both the edge- update and node-update mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Then, in Section IV, numerical results are presented to demonstrate the superiority of the proposed ENGNN on three typical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Finally, the conclusion is drawn in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' PROBLEM FORMULATION ON HETEROGENEOUS GRAPH A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' General Graph Modeling Consider a wireless network with M transmitters and K receivers, which can be modeled by a heterogeneous graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Specifically, the transmitters and receivers can be viewed as two types of nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=', TX-nodes and RX-nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Moreover, an edge is drawn between a TX-node and an RX-node if there exists a direct communication or interference link between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Such a heterogeneous graph can be expressed as G = {M, K, E}, where M ≜ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' , M} is the set of TX-nodes, K ≜ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' , K} is the set of RX-nodes, and E ⊆ {(m, k)}m∈M,k∈K is the set of edges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The TX-nodes, RX-nodes, and edges may contain features and/or variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Specifically, features are known system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' For example, features on the nodes can be position coordinates, maximum transmit power budgets, and/or noise variances, while features on the edges can be channel state information and/or indicators of direct communication or interference 7 links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' On the other hand, variables are unknown beamformers and/or allocated powers to be designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Problem Formulation Denote the feature vectors on the m-th TX-node and k-th RX-node as fTX,m ∈ CdTX and fRX,k ∈ CdRX, where dTX and dRX denote the corresponding feature dimensions, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Consequently, the feature matrices of TX-nodes and RX-nodes can be expressed as FTX = [fTX,1, · · · , fTX,M]T ∈ CM×dTX and FRX = [fRX,1, · · · , fRX,K]T ∈ CK×dRX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Similarly, we can express the edge features as a tensor E ∈ CM×K×dE, where dE is the edge feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In particular, the (m, k, :)-th fiber, E(m,k,:), takes values if (m, k) ∈ E, and is an all-zero vector otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' On the other hand, the variables on TX-nodes and RX-nodes can be expressed as STX = [sTX,1, · · · , sTX,M]T and SRX = [sRX,1, · · · , sRX,K]T, where sTX,m ∈ Cd′ TX and sRX,k ∈ Cd′ RX denote the variables on the m-th TX-node and k-th RX-node, d′ TX and d′ RX denote the corresponding variable dimensions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Similarly, edge variables can be expressed as Ξ ∈ CM×K×d′ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Based on the above notations, a beamforming design or power allocation problem can be formulated as max φ(·,·,·) f (STX, SRX, Ξ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' FTX, FRX, E) , (1a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (STX, SRX, Ξ ) = φ (FTX, FRX, E) , (1b) where (1a) is the utility function, and φ(·, ·, ·) denotes the mapping function from the features (FTX, FRX, E) to the variables (STX, SRX, Ξ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Next, we show three typical examples under the general problem formulation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Example 1: Beamforming Design for Interference Channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Consider a wireless network with K BS-UE pairs, where the k-th UE is served by the m1(k)-th BS, and m1(·) is any one-to-one mapping from K to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Each BS is equipped with N antennas, and each UE is equipped with a single antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The beamforming vector of the k-th UE is denoted as vk ∈ CN, while the channel between the m1(k′)-th BS and the k-th UE is denoted as hm1(k′),k ∈ CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Then the received signal at the k-th UE is given by yk = hH m1(k),kvksk + K � k′=1,k′̸=k hH m1(k′),kvk′sk′ + nk, ∀k ∈ K, (2) 8 (a) Example 1: beamforming design for interference channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (b) Example 2: power allocation for inter- ference broadcast channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (c) Example 3: cooperative beamforming design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Graph modeling for three typical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' where sk is the desired symbol of the k-th UE, and nk ∼ CN(0, σ2 k) is the additive complex Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Consequently, the signal-to-interference-plus-noise ratio (SINR) at the k-th UE can be written as SINRk = ���hH m1(k),kvk ��� 2 �K k′=1,k′̸=k ���hH m1(k′),kvk′ ��� 2 + σ2 k , ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (3) The beamforming design problem for sum rate maximization can be formulated as: max {vk}k∈K K � k=1 log2 (1 + SINRk) , (4a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' ∥vk∥2 ≤ Pm1(k), ∀k ∈ K, (4b) where (4b) represents the maximum transmit power constraint at each BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 1(a), by modeling the BSs and UEs as TX-nodes and RX-nodes respectively, we can incorporate the maximum transmit power budget p = [P1, · · · , PK]T and the noise standard deviation σ = [σ1, · · · , σK]T as the features on the TX-nodes and RX-nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' the feature vector on the (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' k)-th edge contains both the channel state information hm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='k and the indicator of direct communication or interference link: H(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=':) = � � � � � [hT m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 0T]T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' if m = m1(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' [0T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' hT m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='k]T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (5) TXm1(1) RX1 TXm1(2) RX2 TXm1(K) RXK desired signal interferenceTXm1(1) RX1 TXm1(2) RX2 TXm1(K-1) RXK-1 TXm1(K) RXK desired signal inter-cell interference intra-cell interferenceRX1 TX1 RX2 TX2 RX3 RX4 TXM RXK9 where we adopt the idea of one-hot encoding to embed the information of direct communication or interference links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' On the other hand, since vk is the beamforming variable corresponding to the (m1(k), k, :)-th TX-RX pair and m1(·) is a one-to-one mapping, we can either define vk on the (m1(k), k, :)-th edge, the k-th RX-node, or the m1(k)-th TX-node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Without loss of generality, we put vk on the (m1(k), k, :)-th fiber of the edge variable tensor V , and problem (4) can be reformulated on the heterogeneous graph as max φ(·,·,·) K � k=1 log2 (1 + SINRk) , (6a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' V = φ (p, σ, H) , with ∥vk∥2 ≤ Pm1(k), ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (6b) Comparing (6) with (1), the objective function (6a) is a specification of (1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Particularly, the features p, σ, and H correspond to FTX, FRX, and E, respectively, and the variable V corresponds to Ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Example 2: Power Allocation for Interference Broadcast Channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Consider a B-cell downlink cellular network, where each BS serves Q UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Each BS is equipped with N antennas and each UE is equipped with a single antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The normalized beamforming vector of UE q ∈ Q ≜ {1, · · · , Q} in cell b ∈ B ≜ {1, · · · , B} is denoted as wqb ∈ CN, while the channel between BS b′ and UE q in cell b is denoted as hb′,qb ∈ CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Then the received signal at UE q in cell b is given by yqb = √pqbhH b,qbwqbsqb + Q � q′=1,q′̸=q �pq′ bhH b,qbwq′ bsq′ b + B � b′=1,b′̸=b Q � q′=1 �pq′ b′hH b′,qbwq′ b′sq′ b′ + nqb, ∀q ∈ Q, ∀b ∈ B, (7) where sqb and pqb are the desired symbol and transmit power, respectively, the second term is the intra-cell interference, the third term is the inter-cell interference, and nqb ∼ CN(0, σ2 qb) is the additive complex Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Correspondingly, the SINR is given by SINRqb = ��hH b,qbwqb ��2 pqb �Q q′=1,q′̸=q ��hH b,qbwq′ b ��2 pq′ b + �B b′=1,b′̸=b �Q q′=1 ���hH b′,qbwq′ b′ ��� 2 pq′ b′ + σ2 qb , ∀q ∈ Q, ∀b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (8) 10 The power allocation problem for sum rate maximization can be formulated as: max {pqb}q∈Q,b∈B B � b=1 Q � q=1 log2 (1 + SINRqb) , (9a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 0 ≤ Q � q=1 pqb ≤ Pb, ∀b ∈ B, (9b) where Pb is the maximum transmit power budget of BS b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' As observed in (8), the inner product of hb,qb and wqb has BQ×BQ combinations, leading to K = BQ equivalent TX-nodes and K RX-nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' By modeling each BS as Q TX- nodes, and each UE as an RX-node, we define two one-to-one mappings ψTX(·, ·) and ψRX(·, ·) from B × Q to M and K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Let k = ψRX(b, q), and then for RXk, the TX-nodes can be divided into three types as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The first type is the TX-node that serves RXk, denoted as TXm1(k), where m1(k) ≜ ψTX(b, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The second type consists of other TX-nodes in the same cell as RXk, denoted as TXm2(k), where m2(k) ∈ M2(k) ≜ {ψTX(b, q′)|∀q′ ∈ Q, q′ ̸= q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The third type consists of TX-nodes in other cells, denoted as TXm3(k), where m3(k) ∈ M3(k) ≜ {ψTX(b′, q′)|∀b′ ∈ B, b′ ̸= b, ∀q′ ∈ Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Accordingly, the equivalent channel gain between different TX-nodes and RXk can be written as three types: gm1(k),k ≜ ��hH b,qbwqb �� , (10a) gm2(k),k ≜ ��hH b,qbwq′ b �� , ∀q′ ∈ Q, q′ ̸= q, (10b) gm3(k),k ≜ ���hH b′,qbwq′ b′ ��� , ∀b′ ∈ B, b′ ̸= b, ∀q′ ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (10c) We incorporate the maximum transmit power budget p = � ˜P1, · · · , ˜PK �T and the noise standard deviation σ = [σ1, · · · , σK]T as the features on the TX-nodes and RX-nodes, respectively, where ˜Pm1(k) = Pb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Moreover, the feature vector on the (m, k)-th edge contains both the equivalent channel gain gm,k and the indicator of direct communication, intra-cell interference, or inter-cell interference link: G(m,k,:) = � � � � � � � � � � � [gm,k, 0, 0]T, if m = m1(k), [0, gm,k, 0]T, if m = m2(k), [0, 0, gm,k]T, otherwise, (11) where we adopt the idea of one-hot encoding to embed the information of direct communication, inter-cell interference, or intra-cell interference links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' On the other hand, since pqb is the power 11 allocation variable corresponding to the (m1(k), k, :)-th TX-RX pair and m1(·) is a one-to-one mapping, we can either define pqb on the (m1(k), k, :)-th edge, the k-th RX-node, or the m1(k)- th TX-node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Without loss of generality, putting the unknown power allocation variable pqb on the (m1(k), k, :)-th fiber of the edge variable tensor P , problem (9) can be reformulated on the heterogeneous graph as max φ(·,·,·) B � b=1 Q � q=1 log2 (1 + SINRqb) , (12a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' P = φ (p, σ, G) , with 0 ≤ Q � q=1 pqb ≤ Pb, ∀b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (12b) Comparing (12) with (1), the objective function (12a) is a specification of (1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In particular, the features p, σ, and G correspond to FTX, FRX, and E, respectively, and the variable P corresponds to Ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Example 3: Cooperative Beamforming Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Consider a downlink system where M BSs serve K UEs cooperatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Each BS is equipped with N antennas and serves all UEs, while each UE is equipped with a single antenna and served by all BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The channel between the m-th BS and the k-th UE can be defined as hm,k ∈ CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The beamforming vector used by the m-th BS for serving the k-th UE is denoted as vm,k ∈ CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' With sk denoting the desired symbol of the k-th UE, the received signal at the k-th UE is expressed as yk = M � m=1 hH m,kvm,ksk + K � k′=1,k′̸=k M � m=1 hH m,kvm,k′sk′ + nk, ∀k ∈ K, (13) where nk ∼ CN(0, σ2 k) is the additive complex Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The SINR can be written as SINRk = ����M m=1 hH m,kvm,k ��� 2 �K k′=1,k′̸=k ��� �M m=1 hH m,kvm,k′ ��� 2 + σ2 k , ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (14) The cooperative beamforming design problem for sum rate maximization can be formulated as max {vm,k}m∈M,k∈K K � k=1 log2 (1 + SINRk) , (15a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' K � k=1 ∥vm,k∥2 ≤ Pm, ∀m ∈ M, (15b) where Pm denotes the maximum power budget of BS m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 1(c), by modeling the BSs and UEs as TX-nodes and RX-nodes respectively, we can incorporate the maximum transmit power budget p = [P1, · · · , PM]T 12 and the noise standard deviation σ = [σ1, · · · , σK]T as the features on the TX-nodes and RX-nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Moreover, the feature vector on the (m, k)-th edge can be defined as H(m,k,:) = hm,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Unlike the previous examples, the unknown beamforming variable vm,k is not a variable corresponding to a TX/RX-node, but rather corresponding to the (m, k)-th TX-RX pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Thus, vm,k can only be defined on the (m, k)-th edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Putting vm,k on the (m, k, :)-th fiber of the edge variable tensor V , problem (15) can be reformulated on the heterogeneous graph as max φ(·,·,·) K � k=1 log2 (1 + SINRk) , (16a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' V = φ (p, σ, H) , with K � k=1 ∥vm,k∥2 ≤ Pm, ∀m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (16b) Comparing (16) with (1), the objective function (16a) is a specification of (1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Particularly, the features p, σ, and H correspond to FTX, FRX, and E, respectively, and the variable V corresponds to Ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Notice that in the cooperative beamforming application, beamforming variables exist on the communication links between TX-nodes and RX-nodes and should therefore be defined on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' However, existing GNNs [17] [30] are only equipped with the node-update mechanism, which cannot cope with the more complicated problem of cooperative beamforming design, where each TX-node serves multiple RX-nodes and each RX-node is also served by multiple TX-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' PE Property A unique property of beamforming design and power allocation is that the optimized strategy is independent of the indices of TX-nodes and RX-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In particular, the learned mapping function φ(·, ·, ·) is inherently permutation equivariant with respect to TX-nodes and RX-nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=', if the indices of any two TX-nodes or RX-nodes are exchanged, φ(·, ·, ·) should output a corresponding permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' To visualize this, we show a heterogeneous graph before and after permutations of TX-nodes and RX-nodes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Define two permutations πTX(·) and πRX(·), and let TXm and RXk in 13 (a) Original heterogeneous graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (b) Heterogeneous graph after permutations of TX-nodes and RX-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Permutation equivariance illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 2(a) be re-ordered as ˙ TXπTX(m) and ˙ RXπRX(k) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 2(b), where πTX(1) = 2, πTX(2) = 1, πRX(1) = 3, πRX(2) = 1, and πRX(3) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Accordingly, the graph features satisfy ˙fTX,πTX(m) = fTX,m, ∀m ∈ M, (17a) ˙fRX,πRX(k) = fRX,k, ∀k ∈ K, (17b) ˙E (πTX(m),πRX(k),:) = E (m,k,:), ∀(m, k) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (17c) Let � ˙STX, ˙SRX, ˙Ξ � = φ � ˙FTX, ˙FRX, ˙E � and (STX, SRX, Ξ ) = φ (FTX, FRX, E) be the corre- sponding outputs of the mapping function φ(·, ·, ·), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Since � ˙FTX, ˙FRX, ˙E � is just a re-ordering of the TX-nodes and RX-nodes in (FTX, FRX, E), the corresponding outputs of the mapping function φ(·, ·, ·) should satisfy ˙sTX,πTX(m) = sTX,m, ∀m ∈ M, (18a) ˙sRX,πRX(k) = sRX,k, ∀k ∈ K, (18b) ˙Ξ (πTX(m),πRX(k),:) = Ξ (m,k,:), ∀(m, k) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (18c) We will show in the next section that (18) can be guaranteed by the proposed ENGNN with properly designed edge/node-update mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' THE PROPOSED ENGNN In this section, we propose a customized neural network architecture to represent the mapping function φ(·, ·, ·) from (FTX, FRX, E) to (STX, SRX, Ξ ) in problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The proposed neural E/E(1,1,;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=') RX1 TX1 E /E(1,2,:) E /E(1,3,:) RX2 E /E(2,1,:) E /E(2,2,) TX2 RX E /E(2,3,:)E /三 (2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=') RX3 TX2 E/E(2,1,:) E /三(2,2,:) E /三(1,3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=') RX1 E /E(1,1,:) TX1 E /三(1,2,:)14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The overall architecture of the proposed ENGNN, which contains a preprocessing layer, L updating layers, and a postprocessing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' network architecture incorporates both an edge-update mechanism and a node-update mechanism into a GNN and hence we name it as ENGNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' We will show that the proposed ENGNN enjoys the PE property given by (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Overall Architecture The proposed ENGNN consists of a preprocessing layer, L updating layers, and a postprocessing layer as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The preprocessing layer transforms the input features (FTX, FRX, E) into the initial node- and edge-representations � F(0) TX ∈ RM× ˘dTX, F(0) RX ∈ RK× ˘dRX, E(0) ∈ RM×K× ˘dE � , where ˘dTX, ˘dRX, and ˘dE denote the dimensions of the representations on TX-nodes, RX-nodes, and edges, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' These representations will be updated according to node- and edge-update mechanisms in the L updating layers, where the l-th updating layer takes � F(l−1) TX , F(l−1) RX , E(l−1)� as the inputs and then outputs the updated representations � F(l) TX, F(l) RX, E(l)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The dimensions of the representations will not change in the updating layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Finally, the postprocessing layer transforms the graph representations � F(L) TX , F(L) RX, E(L)� into variables (STX, SRX, Ξ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Preprocessing Layer The preprocessing layer converts complex-valued features (if any) on TX-nodes, RX-nodes, and edges into the real-valued form that can be processed by neural networks, and then transforms the real-valued features into initial representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Specifically, the inputs of the preprocessing Preprocessing Layer Postprocessing Layer 1st Updating Layer Lth Updating Layer (0) (L) TX TX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (0) (L) RX RX RX RX (0) (L E15 layer (FTX, FRX, E) are converted into their corresponding real-valued forms � ˆFTX, ˆFRX, ˆE � , where the m-th row of FTX, the k-th row of FRX, and the (m, k)-th fiber of E are given by ˆfTX,m = � ℜ {fTX,m}T , ℑ {fTX,m}T�T , ∀m ∈ M, (19a) ˆfRX,k = � ℜ {fRX,k}T , ℑ {fRX,k}T�T , ∀k ∈ K, (19b) ˆ E(m,k,:) = � ℜ � E(m,k,:) �T , ℑ � E(m,k,:) �T�T , ∀(m, k) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (19c) Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' the initial representations of TX-nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' RX-nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' and edges are transformed by a one- layer MLP with rectified linear unit (ReLU) as the activation function: f (0) TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='m = ReLU � Wpre TXˆfTX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='m + bpre TX � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' ∀m ∈ M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (20a) f (0) RX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='k = ReLU � Wpre RXˆfRX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='k + bpre RX � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (20b) E(0) (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=':) = ReLU � Wpre ˆ E(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=':) + bpre� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' ∀(m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' k) ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (20c) where Wpre TX ∈ R ˘dTX×2dTX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' bpre TX ∈ R ˘dTX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Wpre RX ∈ R ˘dRX×2dRX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' bpre RX ∈ R ˘dRX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Wpre ∈ R ˘d E×2dE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' and bpre ∈ R ˘dE are trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Updating Layer The inputs and outputs of the updating layer l ∈ {1, · · · , L} are � F(l−1) TX , F(l−1) RX , E(l−1)� and � F(l) TX, F(l) RX, E(l)� , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' We next show the node- and edge-update mechanisms in the l-th updating layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 1) Node-Update Mechanism: The update of node representations in the l-th updating layer takes � F(l−1) TX , F(l−1) RX , E(l−1)� as the inputs, and then outputs the updated node representations � F(l) TX, F(l) RX � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In particular, when updating the representation of TXm, the inputs are composed of the previous layer’s representations of TXm, the neighboring RX-nodes RXk, and edges (m, k) for all k ∈ N TX m , where N TX m is the set of neighboring RX-nodes of TXm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' First, the input representations on RXk and edge (m, k) are concatenated and then processed by an MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Next, the processing results from all RXk with k ∈ N TX m are combined by an aggregation function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=', mean or max aggregators), which extracts information from all the neighboring RX-nodes regardless of their input order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Finally, the input representation of TXm and the aggregated result are concatenated and then processed by another MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The above procedure gives the following TX-update mechanism in the l-th updating layer: f (l) TX,m = MLP(l) 2 � f (l−1) TX,m, AGG(l) TX � MLP(l) 1 � f (l−1) RX,k , E(l−1) (m,k,:) �� k∈N TX m � , ∀m ∈ M, (21) 16 where f (l−1) TX,m is the m-th row of F(l−1) TX , f (l−1) RX,k is the k-th row of F(l−1) RX , E(l−1) (m,k,:) is the (m, k)- th fiber of E(l−1), MLP(l) 1 and MLP(l) 2 are two MLPs, and AGG(l) TX is an aggregation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Similarly, the RX-update mechanism in the l-th updating layer reverses the roles of TX-nodes and UE-nodes in (21): f (l) RX,k = MLP(l) 4 � f (l−1) RX,k , AGG(l) RX � MLP(l) 3 � f (l−1) TX,m, E(l−1) (m,k,:) �� m∈N RX k � , ∀k ∈ K, (22) where N RX k is the set of neighboring TX-nodes of RXk, MLP(l) 3 and MLP(l) 4 are two MLPs, and AGG(l) RX is an aggregation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Notice that the representations of TX-nodes and RX-nodes are updated differently in the proposed ENGNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' This is different from the previous work MPGNN [17] in which the node representations are updated homogeneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Moreover, in the proposed ENGNN, the input edge representations E(l−1) (m,k,:) in (21) and (22) are with superscript (l − 1) and hence are also updated (see the edge-update mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Taking a similar analysis in [17], we can show that (21) and (22) satisfy the following PE property: Property 1 (PE in Node-Update Mechanism): The node-update mechanism (21) and (22) are permutation equivariant with respect to TX-nodes and RX-nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Specifically, for any permutations πTX(·) and πRX(·), we have f (l) TX,πTX(m) = MLP(l) 2 � f (l−1) TX,πTX(m), AGG(l) TX � MLP(l) 1 � f (l−1) RX,k , E(l−1) (πTX(m),k,:) �� k∈N TX πTX(m) � , ∀m ∈ M, (23a) f (l) RX,πRX(k) = MLP(l) 4 � f (l−1) RX,πRX(k), AGG(l) RX � MLP(l) 3 � f (l−1) TX,m, E(l−1) (m,πRX(k),:) �� m∈N RX πRX(k) � , ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (23b) 2) Edge-Update Mechanism: The update of edge representations in the l-th updating layer takes � F(l−1) TX , F(l−1) RX , E(l−1)� as the inputs, and then outputs the updated edge representations E(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Different from the node-update mechanism, where the neighbors of a TX-node (or RX- node) are clearly defined as the connecting RX-nodes (or TX-nodes), it is more complicated to define the neighbors of an edge, let alone how to aggregate their representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Notice that an edge may connect with other edges through either a TX-node or an RX-node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In particular, for the edge (m, k) ∈ E, its neighboring edges through TXm are (m, k1), ∀k1 ∈ N TX m \\ {k}, while the neighboring edges through RXk are (m1, k), ∀m1 ∈ N RX k \\ {m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' For example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 4, the 17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The neighbors that share the connection with edge (1, 1) through TX1 are edge (1, 2) and edge (1, 3), which are denoted by dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The neighbor that shares the connection with edge (1, 1) through RX1 is edge (2, 1), which is denoted by a dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' neighboring edges of edge (1, 1) through TX1 are edge (1, 2) and edge (1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' On the other hand, the neighboring edge of edge (1, 1) through RX1 is edge (2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' This causes the neighbors of an edge to be innately divided into two categories based on the connecting node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Consequently, different from the node-update mechanism (21) and (22), the edge-update mechanism should provide two different aggregations for the two types of neighboring edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Specifically, when updating the representation of edge (m, k), the inputs are composed of the previous representations of edge (m, k), TXm, RXk, the neighboring edges (m, k1), ∀k1 ∈ N TX m \\ {k}, and neighboring edges (m1, k), ∀m1 ∈ N RX k \\ {m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The input representations of neighboring edges (m, k1), ∀k1 ∈ N TX m \\{k} and the connecting node TXm are concatenated and then processed by an MLP, while the input representations of neighboring edges (m1, k), ∀m1 ∈ N RX k \\{m} and the connecting node RXk are concatenated and then processed by another MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The processing results of all the neighboring edges are aggregated and then concatenated with the input representation of edge (m, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' An MLP is finally applied to produce the updated representation of edge (m, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' We can express the above edge-update procedure in the l-th (1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=':) RX1 (1,1,) (1,3,) (1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=') RX2 E(2,1,;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=') Edge (1,1) Edge (1,1) TX2 E (2,2,) and its and its E(2,3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=') neighbours neighbours RX3 through TXi through RXi18 updating layer as E(l) (m,k,:) = MLP(l) 7 � E(l−1) (m,k,:), AGG(l) E � MLP(l) 5 � E(l−1) (m,k1,:), f (l−1) TX,m � , MLP(l) 6 � E(l−1) (m1,k,:), f (l−1) RX,k �� k1∈N TX m \\{k},m1∈N RX k \\{m} � , ∀(m, k) ∈ E, (24) where E(l−1) (m,k,:) is the (m, k)-th fiber of E(l−1), MLP(l) 5 , MLP(l) 6 , and MLP(l) 7 are three MLPs, and AGG(l) E is an aggregation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Compared with the node-update mechanism (21) and (22), the edge-update mechanism (24) is more complicated, since the definition of neighbors in edge-update mechanism is more complex than that in the node-update one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In particular, the edge-update mechanism faces a more complicated situation where the neighboring edges are innately divided into two categories according to the two possible connected nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Consequently, different from the node-update mechanism (21) and (22), where the information from neighboring nodes are gathered by one MLP, the proposed edge-update mechanism applies two different transformations to extract the information from two different types of neighboring edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' We next show that (24) enjoys the following PE property: Property 2 (PE in Edge-Update Mechanism): The edge-update mechanism (24) is permu- tation equivariant with respect to TX-nodes and RX-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Specifically, for any permutations πTX(·) and πRX(·), we have E(l) (πTX(m),πRX(k),:) = MLP(l) 7 � E(l−1) (πTX(m),πRX(k),:), AGG(l) E � MLP(l) 5 � E(l−1) (πTX(m),k1,:), f (l−1) TX,πTX(m) � , MLP(l) 6 � E(l−1) (m1,πRX(k),:), f (l−1) RX,πRX(k) �� k1∈N TX πTX(m)\\{πRX(k)},m1∈N RX πRX(k)\\{πTX(m)} � , ∀(m, k) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (25) Proof: See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Postprocessing Layer The postprocessing layer converts the graph representations � F(L) TX , F(L) RX, E(L)� into the final output (STX, SRX, Ξ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' First, if the variables are complex, � F(L) TX , F(L) RX, E(L)� are transformed into 19 the complex form � ˜STX, ˜SRX, ˜Ξ � by � ℜ {˜sTX,m}T , ℑ {˜sTX,m}T�T = Wpost TX f (L) TX,m + bpost TX , ∀m ∈ M, (26a) � ℜ {˜sRX,k}T , ℑ {˜sRX,k}T�T = Wpost RX f (L) RX,k + bpost RX , ∀k ∈ K, (26b) � ℜ � ˜Ξ (m,k,:) �T , ℑ � ˜Ξ (m,k,:) �T�T = WpostE(L) (m,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' :) + bpost, ∀(m, k) ∈ E, (26c) where Wpost TX ∈ R2d′ TX× ˘dTX, bpost TX ∈ R2d′ TX, Wpost RX ∈ R2d′ RX× ˘dRX, bpost RX ∈ R2d′ RX, Wpost ∈ R2d′ E× ˘dE, and bpost ∈ R2d′ E are trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Next, � ˜STX, ˜SRX, ˜Ξ � are normalized to satisfy the constraints (if any), obtaining the final output (STX, SRX, Ξ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Key Insights The proposed ENGNN for representing φ(·, ·, ·) has been specified as a preprocessing layer, L updating layers, and a postprocessing layer, where the preprocessing and postprocessing layers utilize edge/node-wise MLPs, and the L updating layers are built on node- and edge-update mechanisms (21), (22), and (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Next, we provide some key insights of the proposed ENGNN for learning the beamforming design and power allocation as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 1) Permutation Equivariant with Respect to TX-nodes and RX-nodes: Proposition 1 (PE in ENGNN): The proposed ENGNN is permutation equivariant with respect to TX-nodes and RX-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Specifically, for any permutations πTX(·) and πRX(·), denote a permuted problem instance of (FTX, FRX, E) as � ˙FTX, ˙FRX, ˙E � , whose entries satisfy (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The corresponding outputs of the proposed ENGNN, � ˙STX, ˙SRX, ˙Ξ � = φ � ˙FTX, ˙FRX, ˙E � and (STX, SRX, Ξ ) = φ (FTX, FRX, E), always satisfy (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Proof: See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Proposition 1 implies that the proposed ENGNN is inherently incorporated with the PE property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' This is in sharp contrast to the generic MLPs, which require all permutations of each training sample to approximate this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Thus, the proposed ENGNN can reduce the sample complexity and training difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 2) Generalization on Different Numbers of TX-nodes and RX-nodes: In all the layers of the proposed ENGNN, the representations on different edges/TX-nodes/RX-nodes are transformed by the same architecture using the same trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Therefore, the dimensions of the trainable parameters are independent of the numbers of TX-nodes and RX-nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' This scale 20 adaptability empowers ENGNN to be trained in a setup with a small graph size, while being deployed to a much larger wireless network for the inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 3) Tackling Edge Variables: The proposed ENGNN is equipped with an edge-update mech- anism, which facilitates the update of the variables on graph edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' This allows ENGNN to be applied in a wider range of scenarios, where variables are defined between a pair of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' SIMULATION RESULTS In this section, we demonstrate the superiority of the proposed ENGNN on the three examples introduced in Section II-B via simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' We consider a downlink wireless network in a 2 × 2 km2 area, where the BSs and UEs are uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Each BS has a maximum transmit power of 33 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The path loss is 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 + 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='7 log10 d in dB, where d is the distance in meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The small scale channels follow Rayleigh fading and the noise power is −99 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' For the proposed ENGNN, each aggregation function in (21), (22), and (24) is implemented by a max aggregator, which returns the element-wise maximum value of the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' All the MLPs in (21), (22), and (24) are implemented by 3 linear layers, each followed by a ReLU activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In the training procedure, the number of epochs is set to 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Each epoch consists of 100 mini-batches of training samples with a batch size of 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' For each training sample, the BSs’ and UEs’ locations, and the small scale channels are randomly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' A learning rate γ = 10−4 is adopted to update the trainable parameters of ENGNN by maximizing (1a) using RMSProp [35] in an unsupervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' After training, we test the average performance of 100 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' All the experiments are implemented using Pytorch on one NVIDIA V100 GPU (32 GB, SMX2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Beamforming Design for Interference Channels First, we demonstrate the performance of the proposed ENGNN on the problem of beam- forming design for interference channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' During the training procedure, we set the wireless network with 20 BS-UE pairs, where each BS is equipped with 2 antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' An ENGNN with 1 updating layer is adopted, and the dimension of edge features ˘dE is set to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' As explained before (6), the beamforming variables in this scenario can be defined on either the edges or the nodes, which only affects the postprocessing layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The corresponding simulation results are termed as ENGNN-E and ENGNN-N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' For performance comparison, we include two state-of-the-art methods: 21 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Generalization on number of BS-UE pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (a) WMMSE: a widely used benchmark algorithm for sum rate maximization [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (b) MPGNN: the latest learning based method for sum rate maximization in interference channels [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 1) Generalization on Number of BS-UE Pairs: We first compare the performance of different approaches as the number of BS-UE pairs increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In particular, ENGNN-E, ENGNN-N, MPGNN are trained on 20 BS-UE pairs, while we test their performance in terms of sum rate on larger problem scales from 20 to 100 BS-UE pairs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' It can be seen that ENGNN-E, ENGNN-N, and MPGNN generalize well as the number of BS-UE pairs increases from 20 to 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' However, as the number of BS-UE pairs increases from 40 to 100, both ENGNN-E and ENGNN-N can still generalize very well, while the performance of MPGNN becomes worse than that of WMMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The superiority of ENGNN-E and ENGNN-N is owing to the proposed edge-update mechanism, which better extracts the features from channel states and hence further empowers the original node-update mechanism in MPGNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Since ENGNN-E and ENGNN-N result in similar performance, we only show ENGNN-E in the rest of simulations and term it as ENGNN for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 2) Generalization on Noise Power: To demonstrate the generalization performance on noise power, we set the noise power during the training procedure as −99 dBm, while we test the generalization performance under different noise powers from −99 dBm to −89 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 6 that ENGNN consistently achieves higher sum rate than those of MPGNN and WMMSE, which demonstrates the superiority of ENGNN in generalizing to different noise ENGNN-E 220 MPGNN WMMSE 200 ENGNN-N Sum rate (bps/Hz) 180 160 140 120 100 80 20 40 60 80 100 Number of BS-UE pairs22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Generalization on noise power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Generalization on different levels of interference among BS-UE pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 3) Generalization on Different Levels of Interference: In this experiment, we fix the field size during the training procedure as 2 × 2 km2, while we test the performance by varying the field size from 2 × 2 km2 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 km2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' By keeping the distance between each BS and its serving UE within 50-250 meters, the interference among different BS-UE pairs becomes weaker as the field size increases, and hence the sum rates of different approaches become higher in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Moreover, ENGNN consistently achieves higher sum rate than those of MPGNN and WMMSE as the field size increases, which demonstrates that ENGNN generalizes well on different levels of interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 150 ENGNN MPGNN 140 WMMSE 130 120 110 100 90 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 Field side length (km)ENGNN 95 MPGNN WMMSE Sum rate (bps/Hz) 90 85 80 75 99 97 95 93 91 89 Noise power (dBm)23 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Sample complexity comparison between ENGNN and MPGNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 4) Sample Complexity Comparison: We further compare the performance of ENGNN and MPGNN when trained on different numbers of training samples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' It can be seen that ENGNN outperforms MPGNN especially when the number of training samples is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In particular, when the number of training samples decreases to 10-100, the sum rate of ENGNN only decreases to 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='39-89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='68 bps/Hz, while that of MPGNN decreases sharply to 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='60-82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='10 bps/Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The required number of training samples of ENGNN is less than 10% of that of MPGNN when they achieve the same sum rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' This demonstrates the advantage of the proposed edge- update mechanism in sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Power Allocation for Interference Broadcast Channels Next, we demonstrate the performance of ENGNN on the problem of power allocation for interference broadcast channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In the training procedure, we set the wireless network with 5 BSs, with a minimum distance of 500 meters between BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Each BS is equipped with 16 antennas and serves 2 UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' We use zero-forcing beamforming to avoid multi-user interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The ENGNN is set with 1 updating layer and a dimension of ˘dE = 32 for the edge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' For performance comparison, we provide the simulation results of WMMSE and PGNN [30], which is the latest learning-based method for power allocation in interference broadcast channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 1) Generalization on Number of UEs: We first compare the performance of different approaches as the number of UEs increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' In particular, both ENGNN and PGNN are trained under 10 UEs, while we test their performance on larger problem scales from 10 to 50 UEs by varying the number of UEs in each cell from 2 to 10 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' It can be seen that both ENGNN 95 MPGNN 90 Sum rate (bps/Hz) 85 80 75 70 101 102 103 104 Number of training samples24 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Generalization on number of UEs for power allocation in interference broadcast channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Generalization on different levels of inter-cell interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' ENGNN and PGNN generalize well as the number of UEs increases from 10 to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' However, ENGNN always outperforms PGNN, and achieves competitive performance compared to that of WMMSE under different numbers of UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 2) Generalization on Different Levels of Interference: To demonstrate the generalization ability on different levels of interference, during the training procedure, the field size is fixed as 2 × 2 km2, while we test the performance by varying the field size from 2 × 2 km2 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 km2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' By keeping the distance between each BS and its serving UE within 50-250 meters, the inter-cell interference becomes weaker as the field size increases, and hence the sum rate becomes higher in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' It can be observed that the advantage of ENGNN over PGNN is stable as the field size increases, which demonstrates its superior generalization capability on 220 ENGNN PGNN 200 WMMSE Sum rate (bps/Hz) 180 160 140 120 100 80 10 20 30 40 50 Number of UEs85 ENGNN PGNN 84 WMMSE 83 82 81 80 79 78 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 Field side length (km)25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Generalization on power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Sample complexity comparison between ENGNN and PGNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' different levels of interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 3) Generalization on Power Budget: To demonstrate the generalization capability of ENGNN on different power budgets, we set the power budget at each BS as 33 dBm during the training procedure, while we test the generalization performance under different power budgets from 21 dBm to 33 dBm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' It can be seen that both ENGNN and PGNN generalize well as the power budgets decreases from 33 dBm to 21 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' However, ENGNN always outperforms PGNN under different power budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 4) Sample Complexity Comparison: We further compare the performance of ENGNN and PGNN when trained on different numbers of samples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' It can be seen that ENGNN outperforms PGNN especially when the number of training samples is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Particularly, when ENGNN 75 PGNN WMMSE 70 Sum rate (bps/Hz) 65 60 55 50 45 21 23 25 27 29 31 33 Power budget (dBm)78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='0 Sum rate (bps/Hz) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5 ENGNN 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='0 PGNN 2 101 5 102 103 Number of training samples26 (a) Sum rate comparison (b) Computation time comparison Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Generalization on number of UEs for cooperative beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' the number of training samples decreases to 2-5, the sum rate of ENGNN only drops to 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='30- 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='52 bps/Hz, while that of PGNN drops to 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='89-75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='12 bps/Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The required number of training samples of ENGNN is about 1% of that of MPGNN when they achieve the same sum rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' This demonstrates the advantage of the proposed edge-update mechanism in sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Cooperative Beamforming Design Finally, we demonstrate the performance of ENGNN on the problem of cooperative beam- forming design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' During the training procedure, we set the wireless network with 5 BSs and 2 UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Each BS is equipped with 2 antennas and the minimum distance between BSs is 500 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' An ENGNN with 2 updating layers is adopted, and the dimension of edge features ˘dE is set to 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' For performance comparison, we include WMMSE and GP [7], the latter being a computationally efficient first-order algorithm for solving simply constrained optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 1) Generalization on Number of UEs: To demonstrate the generalization ability of ENGNN with respect to different numbers of UEs, during the training procedure, the number of UEs is fixed as 2, while we test the performance of the trained ENGNN by varying the number of UEs from 2 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The performance comparison in terms of sum rate and computation time is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' We observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 13(a) that as the number of UEs increases, ENGNN always outperforms GP and WMMSE in terms of sum rate, which demonstrates its generalization ability with respect to different numbers of UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' On the other hand, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 13(b) shows that ENGNN ENGNN GP 30 WMMSE n rate (bps/Hz) 25 Sum 20 15 2 4 6 8 Number of UEs101 (s) Average computation time 100 ENGNN GP 10-1 WMMSE 2 4 6 8 Number of UEs27 (a) Sum rate comparison (b) Computation time comparison Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Generalization on numbers of BSs for cooperative beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' achieves a remarkable running speed, with over 100 times faster than that of GP and over 1000 times faster than that of WMMSE due to the computationally efficient feed forward computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 2) Generalization on Number of BSs: We further demonstrate the generalization ability of ENGNN with respect to different numbers of BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Specifically, the number of BSs is fixed as 5 during the training procedure, while we test the performance of the trained ENGNN by varying the number of BSs from 5 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' The performance comparison is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' We observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 14(a) that ENGNN achieves higher sum rate than those of GP and WMMSE under different numbers of BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Moreover, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 14(b) shows that ENGNN achieves a much faster running speed than that of GP and WMMSE under different numbers of BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' CONCLUSION This paper proposed a general problem formulation on the heterogeneous graph for the radio resource management problems, where the unknown variables to be designed can be defined on both the graph nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' A novel edge-update mechanism with desired PE property was incorporated, and a general neural network architecture was designed based on it, which can represent the mapping function from the node/edge features to variables for the radio resource management problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Simulation results demonstrated the superiority of the proposed architecture on three typical problems, with higher sum rate and much shorter computation time compared with state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Moreover, the proposed architecture generalizes well on 17 16 (zZH/sdq) 15 rate Sum 14 13 ENGNN GP WMMSE 12 5 6 7 8 Number of BSsAverage computation time (s) 100 ENGNN 10-1 GP WMMSE 10-2 5 6 7 8 NumberofBSs28 different numbers of BSs and UEs, different noise variances, interference levels, and transmit power budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' APPENDIX A PROOF OF PROPERTY 2 Let m′ = πTX(m) and k′ = πRX(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' Substituting these two equations into (24), we have E(l) (m′,k′,:) = MLP(l) 7 � E(l−1) (m′,k′,:), AGG(l) E � MLP(l) 5 � E(l−1) (m′,k′ 1,:), f (l−1) TX,m′ � , MLP(l) 6 � E(l−1) (m′ 1,k′,:), f (l−1) RX,k′ �� k′ 1∈N TX m′\\{k′},m′ 1∈N RX k′ \\{m′} � , ∀(m′, k′) ∈ E, (27) which implies that for any πTX(·) and πRX(·), we always have (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' APPENDIX B PROOF OF PROPOSITION 1 Substituting (17) into (19) and (20) we have ˙f (0) TX,πTX(m) = f (0) TX,m, ∀m ∈ M, (28a) ˙f (0) RX,πRX(k) = f (0) RX,k, ∀k ∈ K, (28b) ˙E (0) (πTX(m),πRX(k),:) = E(0) (m,k,:), ∀(m, k) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (28c) Next, we substitute (28) into (21), (22), and (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' According to Property 1 and Property 2, we have ˙f (l) TX,πTX(m) = f (l) TX,m, ∀m ∈ M, ∀l = 1, · · · , L, (29a) ˙f (l) RX,πRX(k) = f (l) RX,k, ∀k ∈ K, ∀l = 1, · · · , L, (29b) ˙E (l) (πTX(m),πRX(k),:) = E(L) (m,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' :), ∀(m, k) ∈ E, ∀l = 1, · · · , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' (29c) Substituting (29) into (26), we obtain ˙˜sTX,πTX(m) = ˜sTX,m, ∀m ∈ M, (30a) ˙˜sRX,πRX(k) = ˜sRX,k, ∀k ∈ K, (30b) ˙˜Ξ (πTX(m),πRX(k),:) = ˜Ξ 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=', “Lecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content='5-RMSProp: Divide the gradient by a running average of its recent magnitude,” COURSERA: Neural networks for machine learning, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} +page_content=' 26–31, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQf2vm0/content/2301.00757v1.pdf'} diff --git a/8NE1T4oBgHgl3EQfTwPq/content/tmp_files/2301.03083v1.pdf.txt b/8NE1T4oBgHgl3EQfTwPq/content/tmp_files/2301.03083v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f49811542f6ff764c1478e630680bcc288aa74a6 --- /dev/null +++ b/8NE1T4oBgHgl3EQfTwPq/content/tmp_files/2301.03083v1.pdf.txt @@ -0,0 +1,6509 @@ +arXiv:2301.03083v1 [math.OA] 8 Jan 2023 +Relative Cuntz–Pimsner algebras: +Classification of gauge-invariant ideals: +a simple and complete picture +Alexander Frei +January 10, 2023 +We give a simple and complete picture on the classification of relative +Cuntz–Pimsner algebras (and so also of gauge–equivariant representations) us- +ing their intuitive parametrisation by kernel–covariance pairs. +For this we first present a classification of kernel and cokernel mor- +phisms (in the general category of correspondences) which builds on the con- +cept of invariant ideals as realised and coined by Kajiwara–Pinzari–Watatani +(and implicitely by Pimsner). +The existence of all such kernel and cokernel +morphisms then enable us to reduce the general classification problem to the +faithful case of correspondences within ambient operator algebras. +The second component arises from an observation made by Katsura: +We unravel Katsura’s observation as an obstruction on the range of covari- +ance ideals for correspondences embedded in ambient operator algebras, which +comprises the second component of kernel–covariance pairs: As such our para- +metrisation runs over invariant ideals (as a discrete range of kernel ideals) +and on the other hand over bounded ideals below some maximal covariance +(as an upper bound on the range of covariance ideals). +We then illustrate the lattice of relative Cuntz–Pimsner algebras +(and so also of every gauge–equivariant representation) along the range of +kernel–covariance pairs. +Following, we provide the general version of the +gauge-invariant uniqueness theorem by its reduction to the faithful case, +for which we further recall a simplified proof by Evgenios Kakariadis. +This establishes the first half in our classification: Every gauge–equivariant +representation arises as a relative Cuntz–Pimsner algebra (for its own kernel– +covariance pair) and whence the class of gauge–equivariant representations co- +incides with the class of relative Cuntz–Pimsner algebras. As such the kernel– +covariance pairs exhaust the gauge–equivariant representations. +1 + +1 +Correspondences +Alexander Frei +For the second half in our classification we aim to uniquely determine the +relative Cuntz–Pimsner algebras by their parametrising kernel–covariance pairs: +More precisely, we will recover every abstract kernel–covariance pair as the +actual kernel and covariance from its relative Cuntz–Pimsner algebra, and as +such our kernel–covariance pairs are also classifying. +With our found classification by kernel–covariance pairs we then fur- +ther investigate the lattice structure of gauge–equivariant representations. +In particular, we elaborate the existence of connecting morphisms between +cokernel strands (given by the kernel component of kernel–covariance pairs) and +illustrate our results on examples of graph algebras. +Along this discussion we further clarify Katsura’s description (using T-pairs) +as a simple translation of kernel–covariance pairs (which had been already cov- +ered by Katsura himself) with the second component however not taken and +further pursued as describing the range of covariance ideals. +Altogether our classification is a simple reduction of the gauge– +invariant uniqueness theorem (along cokernel morphisms) together with the +identification of kernel and covariance for relative Cuntz–Pimsner algebras. +Finally we provide a realisation of relative Cuntz–Pimsner algebras as +absolute Cuntz–Pimsner algebra by maximal dilation and reveal Katsura’s +construction as the canonical dilation given by the maximal covariance. +We further reveal the canonical dilation as a familiar construction from graph +algebras and provide an example to illustrate the lack of minimal dilations. +As an application we provide a systematic approach for an earlier pullback +result by Robertson–Szymanski which we extend to the general context in up- +coming work with Mariusz Tobolski and Piotr M. Hajac. +1 +Correspondences +We begin with an introduction to correspondences and their representations. +In particular, we provide a less formal and more abstract angle. This seeks +to help to understand their gauge-equivariant representations from an abstract +perpsective — and so also to classify the entire lattice of gauge-invariant ideals. +Further, this allows one to better understand dilations and in further work +the shift equivalence problem from an abstract angle. A Hilbert module is a +right module over an operator algebra (seen as coefficient algebra) that comes +equipped with a pairing (compatible with the coefficient algebra) which renders +the right module complete with respect to the induced norm: +⟨−|−⟩ : X × X → A : +∥x∥2 := ∥ ⟨x|x⟩ ∥. +2 + +1 +Correspondences +Alexander Frei +Given a pair of Hilbert modules over a common coefficient algebra one may +introduce the notion of adjointable operators as those which admit an adjoint: +T : X → Y, +T ∗ : Y → X : +⟨T −|−⟩ = ⟨−|T ∗−⟩ . +We note that such adjointable operators are automatically continuous which +may be seen most easily via the closed graph theorem. Moreover the class of +adjointable operators over pairs of Hilbert modules defines a “consistent system” +of Banach spaces with composition and involution +L(Y |Z) ◦ L(X|Y ) ⊆ L(X|Z), +L(X|Y )∗ = L(Y |X) +satisfying the generalized C∗-identity +T ∈ L(X|Y ) : +∥T ∥2 = ∥T ∗T ∥. +The vertical separators hereby seek to convey the “Dirac bra–ket notation”, +that is we have the following identification which we term Dirac calculus: +Proposition 1.1. The identification (and its conjugate) +X = |X⟩ ⊆ L(A|X), +X∗ = ⟨X| ⊆ L(X|A) : +|x⟩ a := |xa⟩ , +⟨x| y := ⟨x|y⟩ : +⟨x|∗ = |x⟩ +define an isometric embedding (and its conjugate), +which renders the pairing as +⟨x|y⟩ = ⟨x| ◦ |y⟩ ∈ L(X|A) ◦ L(A|X). +This further renders the notion of compact operators as +K(X, Y ) := span |Y ⟩ ⟨X| ⊆ L(X|Y ) +and the notion of adjointable operators as +T x = T ◦ |x⟩ , +⟨T x|y⟩ = ⟨x| ◦ T ∗ ◦ |y⟩ = ⟨x|T ∗y⟩ . +All of the above enables one to split expressions as composition of operators. +Proof. We first note that the coefficient algebra itself defines a Hilbert module: +⟨−|−⟩ : A × A → A : +⟨x|y⟩ = x∗y. +3 + +1 +Correspondences +Alexander Frei +One may now verify that the assignments define mutual adjoints +⟨x|∗ = |x⟩ ∈ L(A|X) : +� +⟨x|y⟩ +��a +� += . . . = +� +y +��xa +� += +� +y +�� |x⟩ a +� +and that the identification defines an isometric embedding +X = |X⟩ ⊆ L(A|X) : +∥ |x⟩ ∥2 = ∥ ⟨x| ◦ |x⟩ ∥ = . . . = ∥ ⟨x|x⟩ ∥. +We leave these as an instructive exercise for the reader. +With this identification at hand, a Hilbert module conveniently reads as +nothing but a right module together with an abstract pairing given by involution +(X ↶ A) +XA ⊆ X, +(X × X → A) +X∗X ⊆ A +which we from now on simply indicate as such formal inclusions. +We may now turn our attention to the notion of C∗-correspondence: Formally +these are given as Hilbert modules together with a representation of the co- +efficient algebra as adjointable operators. With the viewpoint from above we +however obtain the alternative description as a bimodule over the coefficient +algebra together with some compatible inner product pairing +AX ⊆ X, +XA ⊆ X, +X∗X ⊆ A +(1) +satisfying some relations such as (now evident in Dirac formalism) +(ax)∗y = x∗(a∗y), +x∗(ya) = (x∗y)a, +etc. +We meanwhile note that the notion of correspondences has an intrinsic asym- +metry by the pairing. More precisely, the more traditional notion of Hilbert +modules comes equipped with a dual pairing which renders the notion symmet- +ric: +X∗X ⊆ A, +XA ⊆ X, +XX∗ ⊆ A, +AX ⊆ X. +In fact, this covers the main objective of covariant representations. We return +to this aspect in the following section. We continue with a swift introduction +to the (internal) tensor product of correspondences. Using our Dirac formalism +from above we may now simply introduce those as formal powers such as +A(XY . . . Z) ⊆ (AX)Y . . . Z, +(XY . . . Z)A ⊆ XY . . . (ZA) +4 + +1 +Correspondences +Alexander Frei +and where the inner product pairing now simply reads +(XY . . . Z)∗(XY . . . Z) = Z∗ . . . Y ∗� +X∗X ⊆ A +� +Y . . . Z +⊆ Z∗ . . . +� +Y ∗Y ⊆ B +� +. . . Z ⊆ . . . ⊆ +� +Z∗Z ⊆ A +� +. +Moreover, this automatically entails the balanced relation as for example +� +(xa)y − x(ay) +�∗� +(xa)y − x(ay) +� += += y∗(a∗x∗)(xa)y − . . . + (y∗a∗)(x∗x)(ay) = 0. +Let us give an illustrative example for such tensor products: +Example 1.2 (Graph correspondences: Tensor powers): +Consider a directed graph and regard its graph correspondence +X = ℓ2� +E = edges +� +, +A = c0 +� +V = vertices +� +with action and pairing given by range and source say +a +b +z +w +: +z∗z = a, +za = z = bz, +z∗z = b, +wb = w = bw. +On the other hand we know by [KPQ12] that every nondegenerate correspon- +dence over a “direct sum over a discrete set as vertices” arises as a graph +correspondence: +A = c0 +� +vertices = some set +� +=⇒ X = ℓ2(edges). +Indeed this follows by some simple counting argument: +a, b ∈ vertices : +bXa = ℓ2� +edges : b ← a +� += ℓ2(bEa). +From this argument it furthermore follows that degenerate correspondences arise +the same way when allowing for edges with heads pointing into the void such as +a +b +z +w +and furthermore also any power of a graph correspondence arises as a graph +5 + +1 +Correspondences +Alexander Frei +correspondence itself and indeed is given by its paths of according length: +XX . . . X = ℓ2(EE . . . E), +A = c0(vertices). +In other words, that is by concatenation of edges. +As such the tensor product may be seen as nothing but a formal power in +words and one may equivalently consider also mixed powers with the dual space +and its dual pairing (though formally only in the context of operator spaces). +We finish this section with an intrinsic characterization of Hilbert bimodules. +For this we first require the following well-known observation: +Proposition 1.3. Consider a correspondence (in Dirac braket notation) +⟨X|X⟩ ⊆ A, +XA ⊆ X, +AX ⊆ A +and regard the orthogonal complement for the kernel (by left action), +ker(A ↷ X) = {aX = 0} ⊆ A : +ker(A ↷ X)⊥ = {a ker(A ↷ X) = 0} ⊆ A. +This defines the largest ideal that renders the left action faithful. +In particular one may simultaneously identify the orthogonal complement with +its isometric image (by left action) within the space of adjointable operators. +Proof. Before we begin with the actual argument let us make the useful obser- +vation that the kernel defines an ideal (closed and two-sided) since +A ker(A ↷ X)X = 0, +ker(A ↷ X)AX = 0, +ker(A ↷ X)X ⊆ ker(A ↷ X)X = 0 +and so also a selfadjoint one (which one may also easily observe by hand) +� +X +��� ker(A ↷ X)∗X +� += +� +ker(A ↷ X)X +���X +� += 0 =⇒ ker(A ↷ X)∗X = 0. +This kernel determines now precisely those ideals that render the action faitfhul: +K ker(A ↷ X) = K ∩ ker(A ↷ X) = 0 ⇐⇒ K ↾ X = 0. +As such its orthogonal complement defines the largest such ideal: +K ker(A ↷ X) = 0 ⇐⇒ K ⊆ ker(A ↷ X)⊥. +6 + +1 +Correspondences +Alexander Frei +Note that the orthogonal complement (as we have defined from the left only) +defines itself an ideal (closed and two-sided) +A ker(. . .)⊥ ker(. . .) = 0, +ker(. . .)⊥A ker(. . .) = 0, +ker(. . .)⊥ ker(. . .) ⊆ ker(. . .)⊥ ker(. . .)0 +and so also a selfadjoint one (which this time is not too obvious). +Building on this, we may now further give an intrinsic characterization of +Hilbert bimodules (which we promote as theorem since it’s widely non-standard +even until now) and the author would like to thank Adam Skalski and Bar- +tosz Kwasniewski for helping with the key detail for this idea: +Theorem 1.4. Consider a correspondence as seen in (1) +and regard the orthogonal complement from Proposition 1.3 +ker(A ↷ X)⊥ = {a ker(A ↷ X) = 0} ⊆ A +and simultaneously identified as in that proposition by their action +A ⊇ ker(A ↷ X)⊥ = +� +ker(A ↷ X)⊥ ↷ X +� +⊆ L(X|X). +Then the correspondence defines a Hilbert bimodule if and only if the compact +operators (see Proposition 1.1) all lie within the orthogonal complement +|X⟩⟨X| ⊆ ker(A ↷ X)⊥ ⊆ L(X|X) +while the dual pairing is always given by +|−⟩⟨−| : X × X → A : +|X⟩⟨X| ⊆ ker(A ↷ X)⊥ ⊆ A +whence there is no ambiguity left anymore. As such the notion of Hilbert bi- +modules defines a pure property without any additional structure. +Before we begin with the proof, we note that the situation above may be +easiest pictured in mind (and remembered) by the following illustration: +7 + +1 +Correspondences +Alexander Frei +A +ker(A ↷ X)⊥ +L(X|X) +|X⟩ ⟨X| +As such a correspondence in general may be seen as a Hilbert bimodule with a +partial dual pairing. With this picture in mind let us get to the proof. +Proof. Instead of Hilbert bimodules, it suffices to consider the case of Hilbert +modules (meaning the case of a single action and pairing) say given by +⟨X|X⟩ ⊆ A, +XA ⊆ X. +The kernel for the single action (in this case from the right) agrees with the +orthgonal complement for the pairing (and so its support ideal): +ker(X ↶ A) = ⟨X|X⟩⊥ � += supp(X)⊥� +. +Indeed using Blanchard factorization [Bla96, Lemma 1.3] this immediately fol- +lows from +⟨X|X⟩ a = 0 ⇐⇒ +� +|X⟩ = |X⟩ ⟨X|X⟩ +� +a = 0. +In turn this observation tells us where to search for the pairing: +⟨X|X⟩ ⊆ ⟨X|X⟩⊥⊥ = ker(X ↶ A)⊥ +Replacing the right action from our consideration above by the left action from +the proposition we thus just revealed the condition from the proposition. Con- +versely, given the condition from the proposition we may simply retrieve the +dual pairing using the isometric image of the orthogonal complement. Using +Dirac calculus from Proposition 1.1 we finally obtain (simply as composition) +|x⟩ ◦ +� +⟨y| ◦ |z⟩ +� += +� +|x⟩ ◦ ⟨y| +� +◦ |z⟩ +and so the traditional compatibility (between pairings) holds trivially. +8 + +2 +Representations +Alexander Frei +2 +Representations +Consider an abstract correspondence as introduced in the first section: +X∗X ⊆ A, +XA ⊆ X, +AX ⊆ X. +This structure is in some sense freely floating, and so we wish to embed this +structure as a whole into an ambient operator algebra as illustrated: +X +A +X∗ +B +X +A +X∗ +A good analogy here is the embedding of Fell bundles into any crossed prod- +uct. This is what we understand as a representation. More precisely, that is a +representation of both the correspondence and the coefficient algebra into some +ambient operator algebra say +(X, A) +B : +τ : X → B, +ϕ : A → B +so the former being a morphism of vector spaces and the latter a morphism +of operator algebras and such that the pair is coherent with the structure in +between, which now reads in Dirac formalism (see Proposition 1.1): +ϕ(x∗y) = τ(x)∗τ(y), +τ(xa) = τ(x)ϕ(a), +τ(ax) = ϕ(a)τ(x). +It is well-known that the latter follows automatically from the former two: In- +deed using the C∗-identity we find (written in Dirac formalism) +� +ϕ(a)τ(x) − τ(ax) +�∗� +ϕ(a)τ(x) − τ(ax) +� += += ϕ(a)∗τ(x)∗τ(x)ϕ(a) − . . . + τ(ax)∗τ(ax) += ϕ(x∗a∗ax) − . . . + ϕ(x∗a∗ax) = 0. +Consider now the structure as embedded within the ambient operator algebra +X = τ(X) ⊆ B, +A = ϕ(A) ⊆ B +9 + +2 +Representations +Alexander Frei +then we may also view those as subspace and subalgebra which renders the +abstract inclusions from the previous section into actual inclusions +X∗X ⊆ A ⊆ B, +XA ⊆ X ⊆ B, +AX ⊆ X ⊆ B. +We note that while these representations are evidently not faitful in general, one +may always pass to its quotient correspondence which renders the representation +faithful. This will define the first parameter for the classification. +We now wish to extend a representation to higher and mixed tensor powers: +For this we recall the following result by Kajiwara–Pinzari–Watatani: +Proposition 2.1 ([KPW98, Lemma 2.1]). Representations canonically extend +to the tensor product and compact operators (denoted in Dirac formalism): +� +τ : X → Y +ϕ : A → B +� +=⇒ +� +XX → B : +XX∗ → B : +τ(xy) := τ(x)τ(y) +τ(xy∗) := τ(x)τ(y)∗ +� +The latter further satisfies the relations +KL ∈ (XX∗)(XX∗) ⊆ (XX∗) : +aKb ∈ A(XX∗)A ⊆ (XX∗) : +τ(K)τ(L) = τ(KL) +ϕ(a)τ(K)ϕ(b) = τ(aKb) +and so defines in particular a morphism of operator algebras. +Furthermore, suppose the morphism is isometric on the coefficient algebra +then so also on the correspondence and on compact operators: +∥(A → B)−∥ = ∥−∥ +=⇒ +∥(X → B)−∥ = ∥−∥, +∥(A → B)−∥ = ∥−∥ +=⇒ +∥(XX∗ → B)−∥ = ∥−∥. +By iteration the proposition includes all higher and any other mixed powers. +Proof. We provide the extension to compact operators since it demonstrates the +use of Dirac calculus with a neat trick by Kajiwara–Pinzari–Watatani: +We need to verify that the formal linear assignment on elementary compact +operators remains bounded +�����τ +�� +n +xny∗ +n +������ ≤ +����� +� +n +xny∗ +n +�����, +∀xn, yn ∈ X, +whence the assignment allows an (a posteriori well-defined) extension to the +completion of all compact operators. +For this we invoke matrix calculus to +10 + +2 +Representations +Alexander Frei +reformulate the linear sum as a product of matrices +(x)(y∗) := +� +n +xny∗ +n = +� +x1 +· · · +xN +� + + + + +y∗ +1 +... +y∗ +N + + + +. +Reformulated, the quite clever trick by Kajiwara Pinzari and Watatani is now +to use the C∗-identity (generalized to matrices of adjointable operators): +∥xy∗∥2 = +���(xy∗)(yx∗) = x(y∗y)x +��� = +���x√y∗y +��� = . . . = +��� +√ +x∗x√y∗y +���. +Note that this automatically invokes matrix inflations since for example +x∗x = + + + + +⟨x1| +... +⟨xN| + + + + +� +|x1⟩ +· · · +|xN⟩ +� += + + + + +⟨x1|x1⟩ +· · · +⟨x1|xN⟩ +... +... +... +⟨xN|x1⟩ +· · · +⟨xN|xN⟩ + + + +. +On the other hand, this equally applies when invoking the representation and +so we obtain the desired bound +∥τ(xy∗)∥ = +���ϕ +�√ +x∗x√y∗y +���� ≤ +��� +√ +x∗x√y∗y +��� = ∥xy∗∥. +In turn this trick also applies for the correspondence and tensor product +X → B : +∥τ(x)∥ = ∥ϕ(x∗x)∥ ≤ ∥x∗x∥ = ∥x∥, +XX → B : +∥τ(xy)∥ = ∥ϕ(y∗x∗xy)∥ ≤ ∥y∗x∗xy∥ = ∥xy∗∥ +as well as on every other type of tensor product such as of operator algebras. +Meanwhile, this moreover infers that isometricity passes from the coefficient +algebra to the correspondence (and tensor product) and to compact operators. +Moreover, the morphism satisfies the relations on compact operators since one +easily verifies on elementary compact operators: +ϕ(a)τ(xy∗)ϕ(b) = ϕ(a)τ(x)τ(y∗)ϕ(b) = τ(axy∗b), +τ(xy∗)τ(zw∗) = τ(x)ϕ(y∗z)τ(w∗) = τ(xy∗zw∗). +We therefore found the desired relations and the proof is complete. +With this at hand we may now introduce the notion of covariances. We begin +for this with the observation that (as in the previous proposition) we could have +11 + +2 +Representations +Alexander Frei +also defined the morphism (somewhat senseless) +A ⊇ ⟨X|X⟩ → B : +τ(x∗y) := τ(x)∗τ(y) +which agrees with the oringal one on the coefficient algebra (somewhat trivially): +⟨X|X⟩ +B +A +B. +τ(−) = ϕ(−) +Suppose on the other hand that some elements also act as compact operators: +A ∩ |X⟩⟨X| := +� +a ∈ A +��� a ∈ |X⟩⟨X| +� +⊆ A. +Then there is no evidence to believe that the induced morphism from the pre- +vious proposition would agree with the morphism for the algebra: +A ∩ |X⟩⟨X| +B +|X⟩⟨X| +B. +τ(−) ̸= ϕ(−) +Nevertheless the difference of morphisms surprisingly defines a morphism of +operator algebras since (using the property from Proposition 2.1) +(ϕ − τ)a(ϕ − τ)b = ϕ(a)ϕ(b) − ϕ(a)τ(b) − τ(a)ϕ(b) + τ(a)τ(b) += ϕ(ab) − τ(ab) − τ(ab) + τ(ab) = ϕ(ab) − τ(ab). +and so our representation decomposes into the sum of morphisms +� +A ∩ |X⟩⟨X| +B +� += +(2) += + + + + +A ∩ |X⟩⟨X| +|X⟩⟨X| +B + + + + + + + + + +A ∩ |X⟩⟨X| +B +B +− +A ∩ |X⟩⟨X| +|X⟩⟨X| +B + + + +. +As such we may capture the domain of equality by the covariance ideal: +cov +� +(X, A) → B +� +:= ker +� +′′ +� +⊴ A. +(3) +12 + +3 +Kernel and Covariance +Alexander Frei +We will often drop the dependency on the coefficient algebra for simplicity. +The covariance and kernel will classify the gauge-equivariant representations. +Let us thus take a closer look at kernel morphisms and possible covariances: +3 +Kernel and Covariance +We begin this section with a characterization of kernel morphisms in the cat- +egory of correspondences. For this let us introduce the general notion of mor- +phisms between correspondences. As for representations, these are given by a +pair of morphisms on the correspondence and the coefficient algebra +(X, A) +(Y, B) : +τ : X → Y, +ϕ : A → B +where the former defines a linear morphism and the latter a morphism of oper- +ator algebras and such that the pair is coherent with the structure in between, +which conveniently reads in Dirac formalism: +ϕ(x∗y) = τ(x)∗τ(y), +τ(ax) = ϕ(a)τ(x), +τ(xa) = τ(x)ϕ(a). +While the resulting category fails to be abelian (basically due to the algebraic +morphism on the coefficient algebra) it still possesses all kernels and cokernels. +As the notion of kernels and cokernels is nonstandard however in categories +beyond abelian ones, let us give a quick introduction. Our catgory of corres- +pondences and their morphisms has zero morphisms in the sense: +(X, A) +(Y, B) +(Z, C) +0 += (X, A) +(Z, C), +0 +(X, A) +(Y, B) +(Z, C) +0 += (X, A) +(Z, C). +0 +A kernel for a morphism is the universal annihilating morphism: +(?, ?) +(X, A) +(Y, B) +ker += 0 +That is any other annihilating morphism factors uniquely over the kernel: +(X′, A′) +(X, A) +(Y, B) = 0 +=⇒ +(X′, A′) +(?, ?) +(X, A). +∃! +ker +13 + +3 +Kernel and Covariance +Alexander Frei +Dually one may define the cokernel of morphisms. Moreover, we define a short +exact sequence denoted by +0 +(?, ?) +(X, A) +(?, ?) +0 +whenever each side is the kernel respectively cokernel of the other: +(?, ?) +(X, A) = ker +� +(X, A) +(?, ?) +� +(X, A) +(?, ?) = coker +� +(?, ?) +(X, A) +� +We will fill the question marks in the proposition below. +But before we note the following equivalent notions of invariant and hereditary +ideals which date all the way back to Pimsner and as explicitely coined by +Kajiwara–Pinzari–Watatani (we refrain from the notion of negatively invariant +ideals as we will find those from a different perspective later on): +Lemma 3.1 ([Pim97, Lemma 3.5 following] and [KPW98, section 4]; +see further also [FMR03, section 2] and similarly also [Kat07, section 1]): +The notion of invariant and hereditary ideals coincide. More precisely, one has +the characterization for ideals in the coefficient algebra K ⊴ A, +XK = {x ∈ X | X∗x ∈ K} = {x ∈ X | x∗x ∈ K} +(4) +and so it furthermore holds the equivalence +X∗KX ⊆ K ⇐⇒ KX ⊆ XK. +(5) +The former is generally refered to as invariant and the latter as hereditary. +Proof. Katsura gives a neat proof based on some factorization result by Lance, +more precisely [Lan95, Lemma 4.4], which itself however still requires some +rather technical approximation. Instead we may verify the equivalence with +the following fairly elementary observations: Let us first note that both of the +right-hand spaces are automatically linear and closed by Cohen–Hewitt: +KX = span KX ⊆ X, +XK = span XK ⊆ X. +Now we clearly have the forward inclusions since +X∗(XK) = (X∗X)K ⊆ AK ⊆ K. +14 + +3 +Kernel and Covariance +Alexander Frei +Conversely, we have using any approximate identity for the ideal: +x∗x ∈ K =⇒ (1 − e)x∗x(1 − e) → 0 =⇒ x = lim +e (xe) ∈ XK. +With the characterization at hand we further obtain the forward direction +X∗KX ⊆ K =⇒ KX ⊆ XK : +(X∗K∗)(KX) = X∗KX ⊆ K. +Alternatively, one may verify the forward direction using Blanchard factoriza- +tion (whose original proof is very neat and elementary, see [Bla96, Lemma 1.3]): +KX = (KX)(KX)∗(KX) = (KX)(X∗KX) ⊆ (KX)K ⊆ XK +This however implicitely invokes the yet technical Cohen–Hewitt (see above). +If one wishes to refrain from using Cohen–Hewitt altogether, one may also argue +in the following elementwise way (formulated in Dirac notation): +K |X⟩ ∋ k |x⟩ = |y⟩ ⟨y|y⟩ =⇒ ⟨y|y⟩3 = ⟨x| k∗k |x⟩ ∈ K +=⇒ ⟨y|y⟩ = +3� +⟨x| k∗k |x⟩ ∈ K =⇒ k |x⟩ = |y⟩ ⟨y|y⟩ ∈ XK. +All of these variants for the forward direction have their advantage. +For the converse direction we may simply argue +KX ⊆ XK =⇒ X∗KX ⊆ X∗XK ⊆ AK ⊆ K. +So the notion of invariant and hereditary ideals coincide. +Let us give an example to illuminate the notion of hereditary ideals: +Example 3.2 (Graph correspondences: hereditary ideals): +Consider a graph correspondence as in Example 1.2: +X = ℓ2� +E = edges +� +, +A = c0 +� +V = vertices +� +. +Then every ideal corresponds to some collection of vertices +K = c0 +� +S = some vertices +� +⊴ c0(V ) = A +and hereditary ideals become the hereditary collection of vertices +c0(S)ℓ2(E) ⊆ ℓ2(E)c0(S) +⇐⇒ +SE ⊆ ES +15 + +3 +Kernel and Covariance +Alexander Frei +which reads written out in words +range(some edge) ∈ S =⇒ source(some edge) ∈ S +and whence their name: hereditary ideals. +In order to understand kernel morphisms (and so in turn also cokernel mor- +phisms) we make the following observation: Suppose a morphism vanishes on +the coefficient algebra, then it does so also on the entire correspondence: +( A +B ) = 0 +=⇒ +( X +Y ) = 0 +(6) +Indeed one easily verifies (in a way using the C∗-identity) +τ(x) = 0 =⇒ ϕ(x∗x) = τ(x)∗τ(x) = 0 =⇒ x∗x = 0 =⇒ x = 0 +which is equivalently the commutative diagram (see the previous section) +⟨X|X⟩ +0 +A +0 ⊆ B. +τ(−) = ϕ(−) +As such one may already expect the kernel of morphisms to involve the kernel +on the coefficient algebra in some crucial way. With this in mind, we may now +give a new intrinsic characterization of kernel and cokernel morphism. +Meanwhile the author would like to note that the idea to consider concrete +kernel correspondences (by invariant ideals) dates back to [Pim97] and their quo- +tient correspondence to [KPW98] with their representations already appearing +in the proof of [KPW98, theorem 4.3] and as explicitely in [FMR03]. +The author identified those as partial results on the intrinsic characterisation +of categorical kernel and cokernel morphisms, which further expanded and +completed provide the following entire classification of kernel and cokernel +morphisms in the category of correspondences, which we promote as theorem +due to its usefulness (also in upcoming work). +Theorem 3.3 (Classification: kernel and cokernel morphisms): +The category of correspondences has all kernel and all cokernel morphisms. +More precisely, they are all of the special form +0 +(XK, K) +(X, A) +� X +XK , A +K +� +0 +16 + +3 +Kernel and Covariance +Alexander Frei +precisely for the invariant (and equivalently hereditary) ideals +K ⊴ A : +X∗KX ⊆ K +( ⇐⇒ KX ⊆ XK). +Conversely, given a morphism its kernel is given by +(XK, K) +(X, A) +(Y, B) +ker +: +K = ker(A → B) ⊴ A. +As a consequence, its cokernel is given as (within the image) +(X, A) +(Y, B) +� Y +Y L, B +L +� +coker +: +L = +� +n≥0 ⟨Y n| BAB |Y n⟩ ⊴ B. +Furthermore, the cokernel morphisms are precisely the surjective mor- +phisms. +Before we begin with the proof for the above proposition let us give a quick +clarification on the quotient norm (and a rather quite simplified proof thereof): +Lemma 3.4 (compare with [FMR03, Lemma 2.1] and [Kat07, Lemma 1.5]): +It holds for quotient correspondences: The norm for Hilbert modules agrees with +the norm for quotient Banach spaces, +��� x0 +XK +��� +2 +Hilbert = inf ∥ ⟨x0|x0⟩ + K∥ = inf ∥x0 + XK∥2 = +��� x0 +XK +��� +2 +Banach. +In particular, the quotient correspondence is complete (by Cohen–Hewitt). +Proof. As compared to both references, we give a simplified proof: +On the one hand we have the obvious inclusion +∥x0 + XK∥2 ⊆ ∥ ⟨x0 + XK|x0 + XK⟩ ∥ ⊆ ∥ ⟨x0|x0⟩ + K∥ +and as such the bound (when applying infima) +��� x0 +XK +��� +2 +Hilbert = inf ∥ ⟨x0|x0⟩ + K∥ ≤ inf ∥x0 + XK∥2 = +��� x0 +XK +��� +2 +Banach. +For the converse implication we may invoke the well-known formula for the +quotient norm on operator algebras (the simplifying trick) +���� +⟨x0|x0⟩ +K +���� = inf ∥ ⟨x0|x0⟩ + K∥ = lim +e→1 ∥(1 − e) ⟨x0|x0⟩ (1 − e)∥ +where the limit runs over some or any approximate identity for the ideal. +As such we obtain the converse containment +∥ ⟨x0 − x0e|x0 − x0e⟩ ∥ = ∥x0 − x0e∥2 ∈ ∥x0 + XK∥2 +17 + +3 +Kernel and Covariance +Alexander Frei +and so also the desired converse bound: +��� x0 +XK +��� +2 +Banach = inf ∥x0 + XK∥2 ≤ lim +e→1 ∥(1 − e) ⟨x0|x0⟩ (1 − e)∥ = +��� x0 +XK +��� +2 +Hilbert +So the desired Hilbert module norm and Banach space norm coincide. +Having clarified the completeness for the quotient correspondence, we may +now savely get to the desired kernel and cokernel morphisms. +Proof of theorem 3.3. We begin with the statements about kernel morphisms. +We first verify that each morphism admits a (categorical) kernel given by +(XK, K) = ker( (X, A) +(Y, B) ) : +K = ker(A → B). +Clearly the kernel annihilates the morphism due to our observation (6): +� +ker(A → B) → A → B +� += 0 +=⇒ +� +X ker(A → B) → X → Y +� += 0 +Meanwhile, let us also note that any such ideal is invariant: +⟨X| ker(A → B) |X⟩ ⊆ ker(A → B) : +(A → B) +� +⟨X| ker(A → B) |X⟩ +� += += (Y ← X) ⟨X| · (A → B) ker(A → B) · (Y ← X) |X⟩ = 0. +Conversely, suppose another morphism annihilates from the left +(W, D) +(X, A) : +(W → X → Y ) = 0, +(D → A → B) = 0. +(The annihilation suffices on coefficient algebras due to our observation above.) +Trivially, we have the desired inclusion at the level of coefficient algebras: +(D → A → B) = 0 +=⇒ +D → ker(A → B) +On the other hand, recall that the induced morphism on the support ideal +coincides with the morphism at the level of coefficient algebras. As such we +necessarily also have an inclusion into the kernel, +⟨W|W⟩ +ker(A → B) +A +D +ker(A → B) +A. +18 + +3 +Kernel and Covariance +Alexander Frei +We however have the equivalance (using the characterization (4)): +W → X ker(A → B) ⇐⇒ ⟨W|W⟩ → ker(A → B). +As such we obtain also the desired inclusion at the level of correspondences, +D → ker(A → B) +=⇒ +W → X ker(A → B). +So there exists also a unique factorization over the kernel correspondence above. +Every morphism thus admits a kernel given by the above kernel morphism. +We next verify that each such morphism is indeed some kernel, namely the +kernel from the short exact sequence in the proposition: +0 +(XK, K) +(X, A) +� X +XK , A +K +� +0 : +ker +� +(X, A) +� X +XK , A +K +� � += (XK, K) +(X, A). +For this let us first verify that the quotient gives a well-defined correspondence +� X +XK +�∗� X +XK +� +⊆ +� A +K +� +, +� X +XK +�� A +K +� +⊆ +� X +XK +� +, +� A +K +�� X +XK +� +⊆ +� X +XK +� +. +In other words we need to verify the relations +(XK)∗X ⊆ K, +X∗(XK) ⊆ K, +. . . , +(K ⊆ A)X ⊆ (XK). +The only nontrivial one here (which is not guaranteed automatically) is the last +one which precisely calls for hereditary ideals (equivalently invariant ideals) +(K ⊆ A)X = KX ⊆ XK +( ⇐⇒ KX ⊆ XK) +and so the quotient gives a well-defined correspondence. +Let us now get to its kernel. We already found the unique (categorical) kernel +of morphisms above. As such the desired equality now easily follows from +ker +� +(X, A) +� X +XK , A +K +� � += +� +X ker +� +A → A +K +� +, ker +� +A → A +K +�� += (XK, K). +So far about kernel morphisms for correspondences. Let us now get to cokernel +morphisms. We first verify that each cokernel is also of the special form as above +0 +(XK, K) +(X, A) +� X +XK , A +K +� +0 : +coker +� +(XK, K) +(X, A) +� += (X, A) +� X +XK , A +K +� +. +19 + +3 +Kernel and Covariance +Alexander Frei +Clearly the quotient annihilates the kernel (using our obersation (6)): +� +K → A → A +K +� += 0 +=⇒ +(XK → X → +X +XK ) = 0 +Conversely, given any other annihilating morphism say +(X, A) +(Y, B) +(τ,ϕ) +: +(XK → X → Y ) = 0, +(K → A → B) = 0. +Then both morphisms factor uniquely over the quotient (as linear maps): +0 +(XK, K) +(X, A) +� X +XK , A +K +� +0 +0 +0 +(Y, B) +(Y, B) +0. +(τ,ϕ) +So it remains to verify that the factorization defines a morphism of corres- +pondences. That however basically follows from the original morphism (using +congruence classes): +ϕ +� +(x + XK)∗(y + XK) +� += ϕ(x∗y) = τ(x)∗τ(y) = τ(x + XK)∗τ(y + XK), +τ +� +(x + XK)(a + K) +� += τ(xa) = τ(x)ϕ(a) = τ(x + XK)ϕ(a + K). +Recall here that the coherence with the left action follows automatically, +=⇒ τ +� +(a + K)(x + XK) +� += ϕ(a + K)τ(x + XK). +So we have found the desired cokernel: +coker +� +(XK, K) +(X, A) +� += +� X +XK , A +K +� +. +We now wish to find the cokernel for general morphisms: +coker( (X, A) +(Y, B) ) = (?, ?) +For this we may now make use of the following relation to our advantage: +The kernel and cokernel operator satisfy the Galois connection (verbatim from +[Lan78, section VIII.1] and confer further [Fre64, chapter 1 and 2]): +ker coker ker = ker, +ker ker = 0 = coker coker, +coker ker coker = coker . +20 + +3 +Kernel and Covariance +Alexander Frei +On the other hand, we have already found the form of kernel morphisms: +ker +� +(Y, B) → (?, ?) +� += (Y L, L) → (Y, B). +as well as the cokernel for kernel morphisms +coker( (Y L, L) +(Y, B) ) = (Y, B) +� Y +Y L, B +L +� +. +We may thus combine these with the Galois connection between the kernel and +cokernel operator to find the necessary shape of cokernel morphisms: +(?, ?) = coker( (X, A) +(Y, B) ) = += coker ker coker( (X, A) +(Y, B) ) = +� Y +Y L, B +L +� +. +So we are left with finding the invariant ideal which determines the quotient. +Consider for this the factorization over the image (kernel of cokernel) +0 +(X, A) +(X, A) +0 +0 +0 +(Y L, L) +(Y, B) +� Y +Y L, B +L +� +0. +As such the invariant ideal (which determines the kernel) necessarily contains +the image of the coefficient algebra +(X, A) +(Y L, L) +=⇒ +(A → L → B) +and is necessarily also the smallest such ideal (generated by the image) +L = BAB + ⟨Y | BAB |Y ⟩ + . . . = +� +n≥0 ⟨Y n| BAB |Y n⟩ ⊴ B. +Indeed any larger invariant ideal defines nothing but a quotient beyond: +0 +(Y L, L) +(Y, B) +� Y +Y L, B +L +� +0 +0 +(Y L′, L′) +(Y, B) +� Y +Y L′ , B +L′ +� +0 +So we have also found the cokernel for general morphisms. For the final assertion +21 + +3 +Kernel and Covariance +Alexander Frei +we note that every cokernel is onto as easily seen from their special form +� +X +X +XK +0 +� +and +� +A +A +K +0 +� +. +For the converse consider a surjective morphism (X, A) → (Y, B) : +� +X +Y +0 +� +and +� +A +B +0 +� +. +Recall from above that the kernel and cokernel operator define a Galois connec- +tion and so there is no choice for our morphism (to define a cokernel) than to +arise as its own coimage (cokernel of kernel) +(X, A) → (Y, B) = coker +� +some morphism +� += coker ker coker +� +some morphism +� += coker ker +� +(X, A) → (Y, B) +� +. +Recall for this the factorisation over its coimage (as outlined before): +0 +(XK, K) +(X, A) +� X +XK , A +K +� +0 +0 +0 +(Y, B) +(Y, B) +0 +ker +coker ker +So we need to verify that the factorisation defines an isomorphism. +For this we obtain as our morphism is onto (on the coefficient algebra): +0 +K +A +A/K +0 +0 +K +A +B +0 +So the factorisation defines an isomorphism on the level of coefficient algebras. +As a consequence, the factorisation is also faithful on the correspondence (recall +for instance proposition 2.1): +ker +� +A +K +B +� += 0 +=⇒ +ker +� +X +XK +Y +� += 0. +On the other hand, the factorisation is also onto (for the correspondence) +im +� +X +XK +Y +� += im +� +X +X +XK +Y +� += im +� +X +Y +� += Y +and as such the factorisation defines an isomrphism as desired. +That is every surjective morphism defines a cokernel morphism as well. +22 + +3 +Kernel and Covariance +Alexander Frei +Let us identify such quotients in the context of directed graphs: +Example 3.5 (Graph correspondences: quotient graphs): +Consider a graph correspondence as in Example 1.2 +X = ℓ2� +E = edges +� +, +A = c0 +� +ver = vertices +� +and its hereditary ideals given by hereditary collections (as in Example 3.2) +K = c0 +� +S ⊆ vertices +� +⊴ A : +SE ⊆ ES. +It is already clear that its quotients themselves arise as a graph, simply since +their coefficient algebras define discrete direct sums of vertices as in 1.2: +0 +K = c0(S) +A = c0(V ) +A +K = c0 +� +T := V \ S +� +0. +And indeed we may now simply reveal the quotient as +0 +XK = ℓ2(ES) +X = ℓ2(E) +X +XK = ℓ2(T E) +0. +Thus its quotients simply arise as the complementary graphs. +We have now found everything about kernel and cokernel morphisms. +With this in our toolbox, we now proceed to covariances. For this we note that +we may render any representation faithful: Indeed we may simply factor any +representation over the quotient correspondence +0 +(XK, K) +(X, A) +� X +XK , A +K +� +0 +0 +0 +(B, B) +(B, B) +0. +ker +The resulting factorization is faithful (isometric) on the coefficient algebra and +as such also on the entire correspondence (see proposition 2.1): +K = ker(A → B) : +ker +� A +K → B +� += 0 =⇒ ker +� X +XK → Y +� += 0. +So we may always first pass to the (unique) quotient correspondence to ren- +der any representation faithful. +This is the first step in the classification of +gauge-equivariant representations. The second step is to understand the possi- +bly occuring covariances in nature. This is captured as an important observation +23 + +3 +Kernel and Covariance +Alexander Frei +by Katsura below. For this let us recall Katsura’s ideal +max(X, A) = ker(A ↷ X)⊥ ∩ XX∗ ⊴ A +(7) +which we denoted as maximal ideal for reasons which will become clear in the +result below, and we will often drop the dependence on the coefficient algebra +for simplicity. Note also that we have already encountered this ideal in different +context on Hilbert bimodules (see Proposition 1.3 and 1.4). With this ideal in +mind let us get to Katsura’s observation (which we split as a result on faithful +representations and further as one on faithful morphisms in general): +Proposition 3.6 (First part of [Kat04, Proposition 3.3]): +Consider an embedding into some ambient operator algebra +(X, A) ⊆ B : +A ⊆ B +( =⇒ X ⊆ B). +Then its covariance (3) lies perpendicular to the kernel +cov( (X, A) ⊆ B ) ⊥ ker(A ↷ X). +In particular it holds for general representations (and factored as above) +(X, A) +� X +XK , A +K +� +B : +0 ⊆ cov +� � X +XK , A +K +� +B +� +⊆ max +� X +XK , A +K +� +. +So the range of possible covariances is bounded from above by Katsura’s ideal, +whence its name and notation as maximal ideal in (7). +Proof. We recast the arguments from Katsura in our language. For this let us +first recall the commutative diagram for the covariance ideal, +cov(X → B) ⊆ A +B +|X⟩⟨X| +B. +In our case the representation is also faithful (all the horizontal paths): +A ⊆ B, +X ⊆ B, +XX∗ ⊆ B, +. . . +On the other hand, the covariance ideal defines an ideal in the coefficient algebra +24 + +3 +Kernel and Covariance +Alexander Frei +(as we have observed in the previous section). As such we also have +ker(A ↷ X) cov(X → B) ⊆ cov(X → B). +So one may place this expression in the commutative diagram above to obtain +ker(A ↷ X) cov(X → B) +B +ker(A ↷ X)|X⟩⟨X| = 0 +B +and simply trace back the desired orthogonality (using the top path). +The remaining points follow from the discussion preceeding the proposition. +Let us reveal the maximal covariance in the case of graph algebras: +Example 3.7 (Graph correspondences: maximal covariance): +Consider a graph correspondence as in Example 1.2: +X = ℓ2� +E = edges +� +, +A = c0(vertices). +For these the trivially acting portion (the kernel for the left action) and the +compactly acting portion correspond to sources and finite receivers: +ker(A ↷ X) = c0 +� +a : |a edges | = 0 +� += c0(sources), +A ∩ XX∗ = c0 +� +a : |a edges | < ∞ +� += c0(fin receivers). +As such the orthogonal complement reads together +max(X, A) = ker(A ↷ X)⊥ ∩ XX∗ = += c0 +� +a : 0 < |a edges| < ∞ +� += c0(regular). +That is the maximal covariance corresponds to the regular vertices and as +such any other covariance ideal corresponds to simply some collection of such. +We finish this section with an investigation of covariances for morphisms be- +tween correspondences (as opposed to representations into operator algebras). +We begin with faithful morphisms between correspondences. As for represent- +ations we have (compare Proposition 2.1): Being faithful passes from the coef- +ficient algebra to the correspondence (and to any other power): +(X, A) → (Y, B) : +A ⊆ B +=⇒ +X ⊆ Y. +25 + +3 +Kernel and Covariance +Alexander Frei +So the faithful morphisms may be seen as nothing but a subcorrespondence: +That is those whose inner product already lies in the subalgebra and similar for +the action from either side, +⟨Y |Y ⟩ ⊆ B, +BY ⊆ Y, +Y B ⊆ Y : +X ⊆ Y, +A ⊆ B : +⟨X|X⟩ ⊆ A, +AX ⊆ X, +XA ⊆ X +(8) +while for comparison mixed expressions only satisfy +⟨X|Y ⟩ ⊆ B, +BX ⊆ Y, +XB ⊆ Y. +Schematically the inclusions for subcorrespondences may look like +⟨X|X⟩ +A +B +AX X = XA +Y +which compare to mixed expressions as possibly only +⟨X|Y ⟩ +⟨Y |X⟩ +A +B +BX +X ̸= +XB +Y +. +So one may think of a subcorrespondence as some sort of coherent restriction: +Think of global actions and their restrictions to possibly partial actions only. +The action of compact operators on the ambient correspondence further reads +(XX∗)Y ⊆ X(Y ∗Y ) ⊆ XB ⊆ Y +Altogher, we may thus really think of a subcorrespondence simply as a +subspace and subalgebra, which will be quite convenient also further on. +In particular one has (for rather trivial reason) +(T − T ′)Y = 0 +=⇒ +(T − T ′)X = 0 +and as such also for a ∈ A and k ∈ XX∗, +(a − k)Y = 0 +=⇒ +(a − k)X = 0. +(9) +26 + +3 +Kernel and Covariance +Alexander Frei +This basically trivial implication already verifies Katsura’s second observation, +and note how thinking in terms of subspaces and subalgebras paid off: +Proposition 3.8 (Second part of [Kat04, Proposition 3.3]): +For a subcorrespondence as in (8) the covariance arises as pullback +cov +� +X ⊆ Y +� += +� +a ∈ A +��� im(a ↷ Y ) ∈ im(XX∗ ↷ Y ) +� += (A ↷ Y )−1� +im(A ↷ Y ) ∩ im(XX∗ ↷ Y ) +� +which we usually abbreviate simply as their common intersection. +As a consequence it follows the characterization for trivial covariance +cov +� +(X, A) ⊆ B +� += 0 +⇐⇒ +im(A → B) ∩ im +� +XX∗ → B +� += 0 +which reveals the familiar slogan for the Toeplitz representation: +Those whose coefficient algebra has trivial intersection with compact operators. +Proof. This is nothing but the trivial implication (9). +We continue with the covariance for kernel and cokernel morphisms. +We begin for this with the following fairly standard result due to Kajiwara– +Pinzari–Watatani which will become particularly useful in later context: +Proposition 3.9 (see [KPW98, Proposition 4.2]): +A short exact sequence of correspondences (see theorem 3.3) +0 +(XK, K) +(X, A) +� X +XK , A +K +� +0 +induces a morphism (of operator algebras) +L(X) +L +� X +XK +� +: +T (x + XK) := T x + XK +whose kernel admits the equivalent characterization +X−1(K) := +� +X∗T X ⊆ K +� += +� +T X ⊆ XK +� += ker +� +L(X) → L +� X +XK +�� +. +(10) +Further the morphism commutes with the left action by the coefficient algebra, +0 +K +A +A +AK +0 +L(X) +L +� X +XK +� +. +27 + +3 +Kernel and Covariance +Alexander Frei +Proof. In order to verify the induced morphism let us first note the following: +Adjointable operators restrict to kernel correspondences: +T ∈ L(X) =⇒ T ∈ L(XK) : +T (XK) ⊆ (T X)K ⊆ XK. +As such we obtain a commutative diagram and whence the adjointable operator +descends also to the quotient, +0 +XK +X +X +XK +0 +0 +XK +X +X +XK +0. +T +The resulting operator is adjointable as one easily verifies +⟨T (x + XK) | y + XK⟩ = ⟨T X | y⟩ = ⟨x | T ∗y⟩ = ⟨x + XK | T ∗(y + XK)⟩ . +As such we got the desired morphism (of operator algebras) +L(X) +L +� X +XK , A +K +� +: +T (x + XK) := T x + XK. +We meanwhile note that the morphism is generally not onto! +Moreover, the morphism clearly commutes with the left action by the coefficient +algebra as one easily verifies (think in terms of subspaces) +(a + K)(x + XK) = ax + XK = a(x + XK). +The remaining claim about the kernel immediately follows from (4). +We continue with the analogous result for compact operators as for example +in Fowler–Muhly–Raeburn (for which we provide a slightly simplified proof): +Proposition 3.10 (see [FMR03, Lemma 2.6]): +A short exact sequence of correspondences (see theorem 3.3) +0 +(XK, K) +(X, A) +� X +XK , A +K +� +0 +28 + +3 +Kernel and Covariance +Alexander Frei +induces a short exact sequence at the level of compact operators +0 +(XK)(XK)∗ = XKX∗ +XX∗ +� X +XK +�� X +XK +�∗ +0 +L(X) +L +� X +XK +� +which restricts from the morphism of adjointable operators as indicated. +Proof. Clearly, the induced morphism on compact operators commutes with the +morphism of adjointable operators since (think in terms of subspaces) +(x + XK)(y + XK)∗(z + XK) = xy∗z + XK = xy∗(z + XK). +Next we already know that the kernel correspondence (as a subcorrespondence) +defines an embedding at the level of compact operators: +(XK, K) ⊆ (X, A) +=⇒ +(XK)(XK)∗ ⊆ XX∗. +For kernel correspondences these now further define an ideal as for example +� +(XK)(XK)∗ = XKX∗� +XX∗ = XK(X∗XX∗) = XKX∗. +On the other hand, the right-hand morphism is also clearly onto since +X +X +XK +0 +=⇒ +XX∗ +� X +XK +�� X +XK +�∗ +0. +Regarding exactness in the middle, we may now invoke the characterization of +the kernel from the previous proposition, which reads for compact operators +X−1(K) ∩ XX∗ = ker +� +XX∗ +� X +XK +�� X +XK +�∗ � +. +While the ideal clearly lies inside the kernel (left as an exercise for the reader) +the converse inclusion now easily follows from the above description: +X−1(K) ∩ XX∗ = XX∗� +X−1(K) ∩ XX∗� +XX∗ +⊆ X +� +X∗X−1(K)X +� +X∗ ⊆ XKX∗. +So we have found the desired exactness for compact operators. +We may now easily derive the desired covariance for kernel and cokernel +morphisms (with the previous results in mind) +29 + +3 +Kernel and Covariance +Alexander Frei +Proposition 3.11. Kernel and cokernel morphisms have full covariance, +0 +(XK, K) +(X, A) +� X +XK , A +K +� +0 : +cov +� +(XK, K) → (X, A) +� += K ∩ XKX∗, +cov +� +(X, A) → +� X +XK , A +K +� � += A ∩ XX∗. +This stands in contrast to the covariance for representations: +The covariance for representations is bounded by Katsura’s ideal (7). +Proof. Note that kernel morphisms define a particular type of subcorres- +pondences and so we find ourselves in the situation of proposition 3.8: +The nontrivial converse of implication (9) holds for kernel correspondences, +a ∈ K, +k ∈ XKX∗ : +(a − k)XKX∗ = 0 +=⇒ +(a − k)X = 0. +Note that we may equivalently verify the implication +KX∗(a − k)X∗K = 0 +=⇒ +X∗(a − k)X = 0. +For this we note the inclusion (due to invariance): +X∗(a − k)X ⊆ X∗(K + XKX∗)X = (X∗KX) + (X∗X)K(X∗X) ⊆ K. +As such the above implication holds true since for +K +� +X∗(a − k)X ⊆ K +� +K = 0 +=⇒ +X∗(a − k)X = 0 +so kernel morphisms have full covariance as desired. +Regarding the cokernel morphism, we may combine proposition 3.9 and 3.10 to +obtain the desired full covariance +A ∩ XX∗ +A +K +L(X) +L +� X +XK +� +XX∗ +� X +XK +�� X +XK +�∗. +30 + +3 +Kernel and Covariance +Alexander Frei +Concluding that kernel and cokernel morphisms have full covariance. +Recall next that we may always render representations faithful by passing +to the induced representation on the quotient +0 +(XK, K) +(X, A) +� X +XK , A +K +� +0 +0 +0 +(B, B) +(B, B) +0. +ker +The previous result allows us now to further clarify the relation between the +covariance on the quotient correspondence (the relevant one) and the covariance +for the original representation (the author would like to note that he found this +special instance as an observation made by Katsura in [Kat07]): +Proposition 3.12. Consider a cokernel morphism (simply some onto mor- +phism as observed in proposition 3.3) followed by an arbitrary morphism +(X, A) +(Y, B) +0 +and +(Y, B) +(Z, C). +Then the covariance for the composition arises as pullback +cov(X → Y → Z) = (A → B)−1 cov(Y → Z) ∩ XX∗. +In particular the covariance for a representation may be recovered from the in- +duced representation on the quotient correspondence, +(X, A) +� X +XK , A +K +� +B : +cov +� +X → B +� += +� +A → A +K +�−1 cov +� X +XK → B +� +∩ XX∗ +which resambles Katsura’s observation [Kat07, lemma 5.10 statement (v)]. +Proof. The result now easily follows from our previous proposition: Indeed as +cokernel morphisms have full covariance we obtain a covariance diagram for our +cokernel and one for the arbitrary morphism + + + + + +A ∩ XX∗ +B ∩ Y Y ∗ +XX∗ +Y Y ∗ + + + + + +and + + + + + +cov(Y → Z) +C ∩ ZZ∗ +Y Y ∗ +ZZ∗ + + + + + +31 + +3 +Kernel and Covariance +Alexander Frei +and so also a commuting diagram for their composition +(A → B)−1� +. . . +� +∩ XX∗ +cov(Y → Z) +C ∩ Y Y ∗ +XX∗ +Y Y ∗ +ZZ∗. +That is the pullback lies within the covariance +(A → B)−1 cov(Y → Z) ∩ XX∗ ⊆ cov(X → Z). +For the converse inclusion it suffices to establish +cov(X → Z) ⊆ (A → B)−1 cov(Y → Z) +⇐⇒ +(A → B) cov(X → Z) ⊆ cov(Y → Z) +which is to verify the covariance diagram +cov(X → Z) +(A → B) cov(X → Z) +C ∩ ZZ∗ +Y Y ∗ +ZZ∗ ? +Indeed this may be easily seen by following the covariance diagram +cov(X → Z) +. . . +C ∩ ZZ∗ +XX∗ +Y Y ∗ +ZZ∗ +and the full covariance for our cokernel morphisms (once more) +A ∩ XX∗ +B ∩ Y Y ∗ +XX∗ +Y Y ∗ +ZZ∗. +So the covariance for the composition arises from the pullback as desired. +The remaining statement arises now as special instance from above. +We finish this section with the following negative result about the covari- +ance for subcorrespondences: While we have found that kernel and cokernel +morphisms have full covariance, this is not the case for subcorrespondences in +32 + +3 +Kernel and Covariance +Alexander Frei +general. That is simply the converse of implication (9) fails in general: +a ∈ A, +k ∈ XX∗ : +(a − k)X = 0 +̸=⇒ +(a − k)Y = 0. +For this we consider the following somewhat minimal example: +Example 3.13 (Subcorrespondence with zero covariance): +Consider the direct sum of an operator algebra as both the coefficient algebra +and the Hilbert module (which we depict as diagonal operators) +Y = +� D +D +� += B +=⇒ +Y ∗Y ⊆ B, +Y B ⊆ Y +✓ +but with left action given by the flip automorphism on B = D ⊕ D: +� d1 +d2 +� +↷ +� y1 y2 +� += +�� +1 +1 +�� d1 +d2 +�� +1 +1 +��� y1 y2 +� +. +Regard the subcorrespondence given by the subalgebra (noninvariant ideal!) +� +X = +� D +0 +� += A +� +⊆ +� +Y = +� D +D +� += B +� +. +Then its left action vanishes identically (and in particular by compact operators): +A ↷ X = +�� +1 +1 +�� D +0 +�� +1 +1 +��� D +0 +� += +� 0 +D +�� D +0 +� += 0 +On the other hand it never vanishes on the ambient correspondence +A ↷ Y = +�� +1 +1 +�� D +0 +�� +1 +1 +��� D +D +� += +� 0 +D +�� D +D +� += +� 0 +D +� +. +As such the covariance diagram is maximally noncommuting: +A = A ∩ XX∗ +B +XX∗ +Y Y ∗ +̸= +: +ker +� +A ⊆ B ⊆ Y Y ∗ � += 0, +ker +� +A → XX∗ ⊆ Y Y ∗ � += A. +So one needs to stay cautious about the covariance for subcorrespondences: +In worst case one needs to verify a particular covariance by hand. +This finishes our section on kernel and cokernel morphisms on one hand, +and the possible covariances on the other hand. With this at hand we may +now proceed to the gauge-equivariant representations and their classification +— in other words the classification of relative Cuntz–Pimsner algebras. +33 + +4 +Relative Cuntz–Pimsner algebras +Alexander Frei +4 +Relative Cuntz–Pimsner algebras +We introduce in this section the relative Cuntz–Pimsner algebras and elaborate +why and how these serve to classify the gauge-equivariant representations. +We begin with the Toeplitz algebra: That is the universal representation +(more precisely the initial representation) as such that any other representation +uniquely factors via the universal one, +(X, A) +(X, A) +T (X, A) +(B, B). +Next recall from the previous section that we may render any representation +faithful by passing to the quotient correspondence +0 +(XK, K) +(X, A) +� X +XK , A +K +� +0 +0 +0 +(B, B) +(B, B) +0 +ker +and recall from theorem 3.3 that any such quotient arises precisely for some +invariant ideal (equivalently hereditary ideal) +K = ker(A → B) ⊴ A : +X∗KX ⊆ K +( ⇐⇒ KX ⊆ XK). +On the other hand, we found in proposition 3.6 that faithful representations +have their covariance bounded from above by the maximal covariance, +0 ⊆ cov +� � X +XK , A +K +� +B +� +⊆ max +� X +XK , A +K +� +. +As such we may aim to classify representations by pairs of some invariant ideal +(as possible covariance) and another bounded ideal (as possible covariance): +� +K ⊴ A : X∗KX ⊆ K +��� I ⊴ A +K : I ⊆ max +� X +XK +� � +(11) +To handle this task, we may consider the class of representations with kernel +and covariance at least a given pair of invariant and bounded ideal, +(X, A) +(B, B) : +K ⊆ ker(A → B), +I ⊆ cov +� X +XK → B +� +. +Note that any such class contains at least the trivial representations and +34 + +4 +Relative Cuntz–Pimsner algebras +Alexander Frei +furthermore that representations may well have larger kernel and covariance +only (which we cannot exclude at this moment as we will explain below). +The universal such representation defines now the relative Cuntz–Pimsner alge- +bra for a given kernel–covariance pair as in (11): +(X, A) +� X +XK , A +K +� +� X +XK , A +K +� +O(K, I) +(B, B). +K⊆ker(A→B) +I⊆cov +� +X +XK →B +� +As such we obtain an entire “2-dimensional lattice” of relative Cuntz–Pimsner +algebras (as class of universal representations) by first following along the lattice +of cokernel morphisms (given by invariant ideals) +(X, A) +. . . +. . . +. . . +. . . +� +X +X(K∩L), +A +K∩L +� +� X +XK , A +K +� +. . . +. . . +� X +XL, A +L +� +� +X +X(K+L), +A +K+L +� +. . . +. . . +. . . +. . . +(0, 0) +35 + +4 +Relative Cuntz–Pimsner algebras +Alexander Frei +followed by the lattice of covariance ideals (on the chosen cokernel) +. . . +� X +XK , A +K +� +. . . +T +� X +XK , A +K +� += O(K, 0) +. . . +. . . +. . . +. . . +O(K, I ∩ J) +O(K, I) +. . . +. . . +O(K, J) +O(K, I + J) +. . . +. . . +. . . +. . . +O(K, max) = O +� X +XK , A +K +� +with Toeplitz algebras and absolute Cuntz–Pimsner algebras as extreme points. +We now explain the precise goals for the classification of relative Cuntz– +Pimsner algebras. +We will first note that that relative Cuntz–Pimsner al- +gebra come equipped with gauge-actions rendering the representation gauge- +equivariant (we +explain +this +in +more detail +in +the +following +section). +As such the possible range of relative Cuntz–Pimsner algebras are the gauge- +equivariant representations at most. The surprising first goal of their classifica- +tion now states that every gauge-equivariant representation indeed arises itself +as a relative Cuntz-Pimsner algebra (and in particular every gauge-equivariant +representation is itself a universal representation). More precisely, consider first +the induced representation rendering the representation faithful +(X, A) +� X +XK , A +K +� +(B, B) : +K = ker(A → B) +followed by the covariance for the resulting representation +� X +XK , A +K +� +(B, B) : +I := cov +� � X +XK , A +K +� +⊆ B +� +. +To better illuminate the problem let us restrict the representation to its range, +that is the operator algebra generated by (the image of) the correspondence, +(X, A) +� X +XK , A +K +� +C∗(X ∪ A) ⊆ B. +36 + +4 +Relative Cuntz–Pimsner algebras +Alexander Frei +With this description the first problem states that +(X, A) +� X +XK , A +K +� +O(K, I) = C∗(X ∪ A). +Or put in other words, the pairs of invariant ideals as kernel and bounded ideals +covariance exhaust the gauge-equivariant representations. We will solve this +problem via the familiar gauge-invariant uniqueness theorem. +The second goal is to determine that in fact every possible kernel and covari- +ance arises itself as an actual kernel and covariance. More precisely, consider +any pair of invariant ideal as kernel and bounded ideal as covariance: +� +K ⊴ A : +I ⊴ A/K : +X∗KX ⊆ K +I ⊆ max(X/XK) +� +Then there simply may be no such representation with precisely the given kernel +and covariance ideal (so that not every such pair would arise in nature): +(X, A) +� X +XK , A +K +� +(B, B) : +� +ker(A → B), cov +� X +XK → B +� � += (K, I) ? +Or put in other words, two possibly different pairs of kernel and covariance ideal +could in principle lead to one and the same relative Cuntz–Pimsner algebra: +O(K, I) = O(K′, I′) +=⇒ +(K, I) = (K′, I′) ? +We will however find the following relations (both nontrivial!): +(X, A) +� X +XK , A +K +� +O(K, I) : +� +ker +� A +K → O(K, I) +� += 0 +��� cov +� X +XK → O(K, I) +� += I +� +and as such also the desired kernel–covariance pair +� +ker(A → O(K, I)) = K +��� cov +� X +XK → O(K, I) +� += I +� +. +(12) +For this the Fock representation will come into play: Its concrete represent- +ation allows us to actually compute its kernel and covariance and so to verify +the desired relations. As such each possible pair of invariant ideal (as kernel) +and bounded ideal (as covariance) arises itself as actual kernel and covariance. +Summarizing these goals, the kernel–covariance pairs completely parametrize +the entire lattice of gauge-equivariant representations. In other words, the lat- +37 + +4 +Relative Cuntz–Pimsner algebras +Alexander Frei +tice of relative Cuntz–Pimsner algebras (as schematically given above) classifies +the entire lattice of gauge-equivariant representations. +With this at hand, we then further investigate the connecting morphisms +between relative Cuntz–Pimsner algebras. More precisely, we will find that our +parametrisation given by kernel–covariance pairs defines a lattice isomorphism +in the sense that +( K ⊆ L | “I ⊆ J” ) +⇐⇒ +O(K, I) ≤ O(L, J) +where the latter denotes the factorization (as common notation): +(X, A) +(X, A) +O(K, I) +O(L, J). +This has been observed already by Katsura in his 2007 article on gauge- +invariant ideals using however instead so-called T-pairs (and further O-pairs): +We will unravel these as nothing but transformed versions of our kernel– +covariance pairs and as such give a natural interpretation for such pairs. +Following we further give precise descriptions for when connecting morphisms +exist between cokernel strands (as from where and to where) +O(K, I = ??) +O(L, J) , +O(K, I) +O(L, max) ?? +which generalize results from Katsura. Together we thus obtain a plethora of +connecting morphisms between cokernel strands such as +� X +XK , A +K +� +� X +XL, A +L +� +T +� X +XK , A +K +� +T +� X +XL, A +L +� +O(K, I) +O(L, J) +O +� X +XK , A +K +� +O +� X +XL, A +L +� +and note that plenty of connecting morphisms also will be missing. +For this we will give some further examples from graph algebras to illuminate +38 + +5 +Gauge actions: Fourier spaces +Alexander Frei +the lack of connecting morphism. With this in mind and without further ado, +we now get to the class of gauge-equivariant representations: +5 +Gauge actions: Fourier spaces +We begin with the following observation to motivate gauge-equivariant +representations: For every fixed complex number on the torus we may consider +the automorphism which rotates the correspondence +z ∈ T : +(X, A) +(X, A) : +z ↷ x = zx, +z ↷ a = a +which together define a circle action T ↷ (X, A). +Clearly these do not affect the kernel of representations as for +(X, A) +B : +ker( A +A +B +1 +) = ker( A +B ). +Similarly they do not affect the covariance as they act trivially on compacts, +zz∗ = 1 : XX∗ +XX∗ : +x(zz∗)y∗ = xy∗ : +A ∩ XX∗ +A ∩ XX∗ ⊆ A +B +XX∗ +XX∗ +B. += +=? +As such every relative Cuntz–Pimsner algebra (as universal representation for +some relations) admits a unique gauge action which renders its representation +gauge-equivariant (think in terms of generators and relations): +∃T ↷ O(K, J) : +� +T ↷ (X, A) +� +� +T ↷ O(K, I)) +� +. +As such at best we may hope to classify the relative Cuntz–Pimsner algebras +amongst those representations which come along with a gauge-action rendering +the representation equivariant. More precisely, that is written out +(ϕ, τ) : +� +T ↷ (X, A) +� +� +T ↷ B +� +: +z ↷ τ(x) = τ(z ↷ x) = τ(zx), +z ↷ ϕ(a) = ϕ(z ↷ a) = ϕ(a). +39 + +5 +Gauge actions: Fourier spaces +Alexander Frei +Equivalently these are the conventional gauge-equivariant representations of op- +erator algebras (from any of the preceeding relative Cuntz–Pimsner algebras) +π : +� +T ↷ O(K, I) +� +� +T ↷ B +� +: +z ↷ π(−) = π(z ↷ −). +Consider now the relative Cuntz–Pimsner algebras (as universal representations) +which allow factorizations for our gauge-equivariant representation: +(X, A) +O(K, I) +B +⇐⇒ +� +K ⊆ ker(A → B) +��� I ⊆ cov +� X +XK → B +� � +As such — if the gauge-equivariant representation has a chance of being a uni- +versal representation for some kernel–covariance pair — then certainly the best +chance is given by the kernel–covariance pair for the representation itself: +(K, I) = +� +ker(A → B) +��� cov +� X +XK → B +� � +: +(X, A) +O(K, I) +C∗(X ∪ A) ⊆ B ? +This is what the gauge-equivariant uniqueness theorem will establish in the next +section and so also the first goal in the classification of relative Cuntz–Pimsner +algebras. So let us prepare ourselves a bit more for this by taking a closer look +at gauge-equivariant representations and their Fourier spaces. +It is well known that every operator algebra equipped with a circle action +(such as our gauge-equivariant representations) comes along with Fourier spaces +B(n ∈ Z) = +� +b +��� +� +b(z) := z ↷ b +� += znb +� +⊆ B +and in particular its fixed point algebra +B(n = 0) = +� +b +��� +� +b(z) := z ↷ b +� +≡ b +� +⊆ B +which define a Fell bundle over the integers +B(m)B(n) ⊆ B(m + n), +B(n)∗ = B(−n) +together with a conditional expectation onto its fixed point algebra +E : B +B(0) : +E(b) = +� +T +� +b(z) = z ↷ b +� +dz +40 + +5 +Gauge actions: Fourier spaces +Alexander Frei +and more generally with projections onto any of its Fourier spaces +En : B +B(n) : +En(b) = +� +T +z−nb(z) dz. +Recall that conditional expectations given by averaging are automatically faith- +ful since the state space separates the positive elements (and since averaging +runs as Bochner integral): +b ≥ 0 =⇒ +� +b(z) = z ↷ b +� +≥ 0 =⇒ SB +� +b(z) = z ↷ b +� +≥ 0 : +E(b ≥ 0) = 0 =⇒ SB(E(b)) = SB +�� +b(z) dz +� += +� +SB(b(z)) dz = 0 +=⇒ SB +� +b(z) = z ↷ b +� +≡ 0 =⇒ SB +� +b = 1 ↷ b +� += 0 =⇒ b = 0. +We meanwhile note that the Fourier spaces densely span the operator algebra by +[Exe94, proposition 2.5] (which seems a rather intuitive yet nontrivial relation) +B = +� +. . . + B(−1) + B(0) + B(1) + . . . +� += +� +n +B(n) +and we encourage the reader to have a look into the beautiful proof by Exel: +It does not require any Cesaro approximations and instead only invokes the +elementary isomorphism (as the basic version of Coburn) +C∗(Z) = C∗(u∗u = 1 = uu∗) = C( σu | u∗u = 1 = uu∗) = C(T) +=⇒ C∗(Z) = +� +. . . + Cu∗ + Ce + Cu + . . . +� += C(T). +A short digression: One may meanwhile wonder how it may be possible that +any element may be approximated by Fourier sums while the series of its Fourier +coefficients only converges in Cesaro mean (and generally diverges in norm). The +answer to this question becomes evident when replacing for instance the torus +with an open disk and the trigonometric with polynomial sums +{. . . , 1/z, 1, z, z2, . . .} ⊆ C(T) +⇝ +{1, z, z2, . . .} ⊆ C(D) +It is well-known here that any continuous function may be approximated in +norm by polymoials (by Stone–Weierstrass). On the other hand, the convergent +power series correspond to the class of Taylor analytic functions +f(z) = +� +n +anzn ⇐⇒ f ∈ C∞(D) +41 + +5 +Gauge actions: Fourier spaces +Alexander Frei +The difference lies in the fact that a polynomial approximation of continuous +functions generally changes each of the coefficient in the sequence — also the +previously already set coefficients! As such one may think of the convergent +Fourier series as a generalization of analytic functions. +We now further restrict our attention on the range of the representation (also +since the ambient rest does not reflect the correspondence): More precisely, that +is the operator algebra generated by (the image of) the correspondence, +(X, A) +C∗(X ∪ A) ⊆ B +⇝ +B = C∗(X ∪ A). +Recall that a correspondence (and so its image under the representation) satisfies +its defining inclusions as explained in more detail in section 1: +X∗X ⊆ A, +XA ⊆ X, +AX ⊆ X. +As such the the range (as generated operator algebra) admits “a fine structure” +C∗(X ∪ A) = span +� +. . . + (X∗ + XX∗X∗ + . . .)+ ++(A + XX∗ + XXX∗X∗ + . . .) + (X + XXX∗ + . . .) + . . . +� +which we organised in groups according to the Fourier spaces they belong to. +As such we also obtain “a fine structure” for the fixed point algebra +B(n = 0) = E +� +B = C∗(X ∪ A) +� += span +� +A + XX∗ + XXX∗X∗ + . . . +� +which allows us to deduce the gauge-invariant uniqueness theorem by induction. +Note further that for any correspondence (including possibly degenerate ones!) +X∗X ⊆ A, +XA ⊆ X, +AX ⊆ X +=⇒ XA = X, +XXA = XX, +. . . +As such the Fourier spaces further satisfy +B(n ≥ 1) = En +� +B = C∗(X ∪ A) +� += span +� +Xn + XnXX∗ + . . . +� += Xn� +A + XX∗ + XXX∗X∗ + . . . +� += XnB(0) +where we left the linear spans and closures implicit for more pointy statements, +and by involution also for negative Fourier spaces and as such altogether +B(n ≥ 1) = XnB(0), +B(n ≤ −1) = B(0)X−n. +42 + +5 +Gauge actions: Fourier spaces +Alexander Frei +As a consequence, the Fell bundle (restricted on the range) further satisfies +B(n ≥ 1) = B(1)n : +B(n) = XnB(0) ⊆ B(1)nB(0) ⊆ B(1)n ⊆ B(n) +and similarly for its negative Fourier spaces, and as such defines altogether a +semi-saturated Fell bundle (see for instance [Exe94, proposition 4.8]): +B(m ≥ 0)B(n ≥ 0) = B(m + n), +B(m ≤ 0)B(n ≤ 0) = B(m + n). +These relations are quite useful for bipartite inflations as introduced in [MPT08]. +With the knowledge about Fourier spaces at hand, we may now get back to the +classification problem from above: +(K, I) = +� +ker(A → B) +��� cov +� X +XK → B +� � +: +(X, A) +O(K, I) +C∗(X ∪ A) ⊆ B ? +At first we may pass to the quotient correspondence +(X, A) +� X +XK , A +K +� +O(K, I) +B +to obtain an honest embedding since +K = ker(A → B) +=⇒ +A/K ⊆ B +=⇒ +X/XK ⊆ B. +As such we may assume that our correspondence lies faithfully in the ambient +operator algebra (upon replacing the original with the quotient correspondence) +� X +XK , A +K +� +⇝ (X, A) +=⇒ +O(K, I) ⇝ O(0, I) +=⇒ +X ⊆ B, A ⊆ B +(13) +and at the same time restrict our attention to its range: +C∗(X ∪ A) ⊆ B +⇝ +C∗(X ∪ A) = B. +We may thus think of both ambient algebras as faithful completions for the +correspondence — but under a priori possibly different norm topologies: +O(X; I) = C∗(X ∪ A) +X ∪ A +C∗(X ∪ A) = B +As another valuable perspective, one may also think of the above prob- +lem as gauge-equivariant envelopes (comparable to the maximal and minimal +43 + +5 +Gauge actions: Fourier spaces +Alexander Frei +C∗-envelope of non-selfadjoint operator algebras): +T (X) +. . . +O(X; I) +Q +. . . : +(X, A) ⊆ (T ↷ Q). +The author would like to thank Elias Katsoulis for bringing closer this idea. +Next as our representation is gauge-equivariant it restricts in particular to fixed +point algebras and further commutes with conditional expectations: +O(X; I) +B +O(X; I)(0) +B(0) +O(X; I) +B +O(X; I)(0) +B(0) +As such it suffices to verify the above equality on fixed point algebras: +π : O(X; I)(0) +B(0) +=⇒ +π : O(X; I) +B +(14) +Indeed this follows as standard argument from the faithful conditional expecta- +tion (sufficiently for the relative Cuntz–Pimsner algebra): +π(a) = 0 =⇒ π(a∗a) = 0 =⇒ π(E(a∗a)) = E(π(a∗a)) = 0 +=⇒ E(a∗a) = 0 =⇒ a∗a = 0 =⇒ a = 0. +Another interesting less commonly known argument due to Exel basically +exploits that continuous functions, whose Fourier coefficients all vanish, already +vanish identically themselves (see [Exe94, proposition 2.5 and 2.9]): +En(a)En(a)∗ ∈ O(X; I) +� +n − n = 0 +� +: +π(a) = 0 =⇒ En(πa)En(πa)∗ = π +� +En(a)En(a)∗� +≡ 0 +=⇒ En(a)En(a)∗ ≡ 0 =⇒ En(a) ≡ 0 =⇒ a = 0. +For the fixed point algebra we however found the “fine structure” +O(X; I)(n = 0) = span +� +A + XX∗ + XXX∗X∗ + . . . +� +so the fixed point algebras arises as an increasing union (inductive limit) +O(X; I)(n = 0) = closure +�� +N +span +� +A + XX∗ + . . . + XNX−N�� +. +44 + +6 +Uniqueness theorem +Alexander Frei +As such we may verify the faithfulness by induction along n ∈ N: +O(X; I) ⊇ span(A + XX∗ + . . . + XnX−n) +B ? +(15) +The point here is not that each summand embeds separately (which is trivial) +A +B, +XX∗ +B, +XXX∗X∗ +B, +. . . +but instead that the summands will have plenty of correlation among each other +(namely precisely the amount of covariance) which causes lots of sums to collapse +within the representation such as the covariance itself, +cov(X → B) = ker + + + + + + +cov(X → B) +B +B + + + + − + + + + +cov(X → B) +XX∗ +B + + + + + + +=⇒ +� +(a) + (k = −a) ∈ A + XX∗ +���� a ∈ cov(X → B) +� +⊆ ker +� +A + XX∗ +B +� +. +In fact, proposition 3.8 tells us that the covariance precisely captures this kernel. +As an interesting question to the encouraged reader: can you figure out why? +Before we proceed, we would like to note that the idea for the above in- +duction and its proof are due to Evgenios Kakariadis as in [Kak16] and note +that this simplifies the original proof by Katsura substantially! We will not +go further into how these approaches compare (since this would not benefit +our current work) but we would like to note that Katsuras approach analysing +cores may be quite valuable after all in the more general context of product +systems and so we refer the curious reader to [Kat04, section 5] for comparison. +With this we may now proceed to the gauge-invariant uniqueness theorem. +6 +Uniqueness theorem +We may now state and proof our version of the gauge-invariant uniqueness +theorem for arbitrary gauge-equivariant representations which arise as relative +Cuntz–Pimsner algebra along kernel–covariance pairs: +Theorem 6.1 (Gauge-invariant uniqueness theorem: The general version). +Consider a gauge-equivariant representation (as in the previous section) +� +T ↷ (X, A) +� +� +T ↷ B +� +45 + +6 +Uniqueness theorem +Alexander Frei +and choose the kernel–covariance pair for the representation +(K, I) = +� +ker(A → B) +��� cov +� X +XK → B +� � +. +Then the quotient from the relative Cuntz–Pimsner algebra is faithful: +(X, A) +O(K, I) +C∗(X ∪ A) ⊆ B. +Thus each gauge-equivariant defines a relative Cuntz–Pimsner algebra. +On the other hand, each relative Cuntz–Pimsner algebra defines itself a gauge- +equivariant representations. As such the gauge-equivariant representations agree +with relative Cuntz–Pimsner algebras (the range of kernel–covariance pairs). +We have basically already proven the theorem: +Indeed we have al- +ready reduced the general version to the case of faithful representations +(see the previous section) and the faithful case is a classical result due to Kat- +sura from [Kat04] resp. the simplified version due to Kakariadis in [Kak16]. +For convenience of the reader we review the simplified version: +We begin for this with a couple of observations and useful relations. +At first consider the increasing subalgebras from the induction problem (15) +(which exhaust the fixed point algebra): +span(A + XX∗ + . . . + XnX−n) ⊆ O(X; I) +In order to simplify their induction we need to address a technical detail: +Note that while the sum of ideals is already closed, this fails in general for the +sum of subalgebras, +A ⊆ B, +A′ ⊆ B : +(A + A′) ⊆ (A + A′). +However the sum of an ideal and a subalgebra is closed nevertheless: +A ⊆ B, +J ⊴ B : +(A + J) = (A + J). +Quick proof for an algebra and ideal (as for pairs of ideals): +Consider the short exact sequences of possibly incomplete algebras, +0 +J +(A + J) +(A + J)/J +0 +0 +J +(A + J) +(A + J)/J +0. +π +46 + +6 +Uniqueness theorem +Alexander Frei +Note that the morphism between quotients defines an embedding since +J ∩ (A + J) = J +=⇒ +(A + J)/J ⊆ (A + J)/J. +Since the range of ∗-homomorphisms is always closed we obtain also +π(A + J) = π(A) = π(A) = π(A + J) = π +� +A + J +� +. +As such the quotients agree and so (by basic homological algebra) +(A + J)/J = (A + J)/J +=⇒ +(A + J) = (A + J). +(16) +So the sum of an algebra and ideal defines a closed subalgebra. +With this at hand we find in our case (using the obvious inclusion) +AXX∗ ⊆ XX∗ +=⇒ +span(A + XX∗) = A + span(XX∗) +and similarly on larger sums +span +� +A + XX∗ + . . . + XnX−n� += A + span +� +XX∗ + . . . + XnX−n� +. +This finishes our discussion on the technical detail. +With this the induction problem asks for the kernel +ker +� +O(X; I) ⊇ A + span +� +XX∗ + . . . + XnX−n� +B +� +and so for the coefficient algebra that intersects compact operators +A ∩ +� +XX∗ + . . . + XnX−n� +⊆ B +where we drop from now on the closed linear span for more pointy statements. +We would like to better understand this intersection. For this we first discuss +the following result, which gives an interesting algebraic description for non- +commutative Cartan subalgebras from [Exe11] as the nondegenerate ones: +Lemma 6.2. A subalgebra contains some approximate identity for the ambient +algebra if and only if the subalgebra is nondegenerate: +A ⊆ B : +AB = B +⇐⇒ +(1 − e)B → 0 for A ∋ e → 1. +On the other hand, the nondegeneracy holds equivalently if the subalgebra reaches +47 + +6 +Uniqueness theorem +Alexander Frei +the entire ambient algebra hereditarily (with implicit Cohen–Hewitt) +AB = B +⇐⇒ +her(A) = ABA = B. +Further, the subalgebra remains nondegenerate on intermediate subalgebras: +A ⊆ B0 ⊆ B : +AB = B +=⇒ +AB0 = B0. +The analogous statements also hold when replacing the subalgebra and the am- +bient algebra by any pair of operator algebras (after suitable modifications). +Proof. The forward direction is obvious since for any approximate unit +A ∋ e → 1 : +(1 − e)B = (1 − e)AB → 0 +where we implicitly use Cohen–Hewitt as usual. +The converse is also obvious as evidentily +(1 − e)B → 0 +=⇒ +B ⊆ AB. +Regarding the hereditary subalgebra we have: +AB = B +( =⇒ B = B∗ = (AB)∗ = B∗A∗ = BA) +=⇒ +B = (AB) = A(BA) = ABA +=⇒ +B = ABA ⊆ AB +Regarding intermediate algebras we have: +AB = B +(1 − e)B → 0 for A ∋ e → 1 +AB0 = B0 +(1 − e)B0 → 0 for A ∋ e → 1 +For arbitrary pairs of operator algebras (instead of a subalgebra): +One needs to replace the equality by an inclusion and an additional closure. +With the nondegeneracy in mind we may now unravel the intersection above: +For this we first note that the compact operators are nondegenerate in the sum +of higher order compact operators, +� +XX∗ + . . . + XnX−n� += XX∗� +XX∗ + . . . + XnX−n� +. +Indeed we may simply pull out a rabbit from each summand (using Blanchard): +XX∗ = (XX∗)XX∗, +XXX∗X∗ = (XX∗)XXX∗X∗, +. . . +48 + +6 +Uniqueness theorem +Alexander Frei +As a consequence we obtain the inclusion (similarly as in lemma 6.2): +A ∩ +� +XX∗ + . . . + XnX−n� +⊆ +� +A ∩ (XX∗ + . . . + XnX−n) +� +XX∗. +The latter however lies inside the algebra of compact operators since +� +A ∩ (XX∗ + . . . + XnX−n) +� +XX∗ ⊆ AXX∗ ⊆ XX∗ +and as such inside the covariance for the representation (see proposition 3.8). +As the covariance cannot decrease we obtain the combined relation: +A ∩ +� +XX∗ + . . . + XnX−n� += cov(X → B) = cov +� +X → O(X; I) +� +. +In particular the kernel arises as sum of compact operators: +ker +� +A + +� +XX∗ + . . . + XXnX−nX∗� +B +� +⊆ XX∗ + +� +XX∗ + . . . + XXnX−nX∗� +⊆ O(X; I). +So far our introductory observations and relations. With these in mind we may +now get to the proof of the gauge-equivariant uniqueness theorem. +Proof of theorem 6.1 (Gauge-invariant uniqueness theorem): We have already +reduced the general version to the case of faithful representations in (13): +� X +XK , A +K +� +⇝ (X, A) +=⇒ +O(K, I) ⇝ O(0, I) +=⇒ +X ⊆ B, +A ⊆ B +Furthermore the problem reduces to fixed point algebras as in (14) +O(X; I)(0) +B(0) +=⇒ +O(X; I) +B +which may be solved as an induction along its fine structure as in (15): +O(X; I) ⊇ A + +� +XX∗ + . . . + XnX−n� +B ? +With this at hand we may now begin the proof by induction from [Kak16]: +The base case simply states the inclusion that we are already well aware of, +O(X; I) = C∗(X ∪ A) +A +C∗(X ∪ A) = B. +Before we begin with the induction combine the observations from above Con- +49 + +6 +Uniqueness theorem +Alexander Frei +sider now the induction step (while assuming the induction hypothesis): +ker +� +A + +� +XX∗ + . . . + Xn+1X−n−1� +B +� += 0 ? +We may reduce this problem to the induction hypothesis via compression +X∗ ker +� +A + +� +XX∗ + . . . + Xn+1X−n−1� +B +� +X +⊆ ker +� +X∗X + X∗� +XX∗ + . . . + Xn+1X−n−1� +X +B +� +⊆ ker +� +A + +� +XX∗ + . . . + XnX−n� +B +� += 0 +from which we infer that the kernel vanishes for the compression +XX∗ ker +� +A + +� +XX∗ + . . . + X(XnX−n)X∗� +B +� +XX∗ = 0. +We however found that the kernel lies inside the sum of compact operators: +ker +� +A + +� +XX∗ + . . . + XXnX−nX∗� +B +� +⊆ XX∗ + +� +XX∗ + . . . + XXnX−nX∗� +⊆ O(X; I). +At the same time, the compact operators define a nondegenerate subalgebra +within higher order compact operators and as such have trivial annihilator: +� +XX∗ + . . . + XXnX−nX∗� += XX∗� +XX∗ + . . . + XXnX−nX∗� +=⇒ +(XX∗)⊥ ∩ +� +XX∗ + . . . + XXnX−nX∗� += 0. +So the kernel above necessarily also vanishes without compression. +Concluding the current and previous section, we have achieved the first +half in the classification of relative Cuntz–Pimsner algebras: that is we have +found that every gauge-equivariant representation arises itself as relative Cuntz– +Pimsner algebra for some kernel–covariance pair. In short, that is the kernel– +covariance pairs exhaust the gauge-equivariant representations. +We may now proceed with the second half on the classification of rela- +tive Cuntz–Pimsner algebras: The kernel–covariance pairs classify the rela- +tive Cuntz–Pimsner algebra and equivalently every possible kernel–covariance +pair arises itself in nature. +As such we need to construct sufficiently many +representations (as well as any nontrivial one whatsoever). This is where the +50 + +7 +Fock representation +Alexander Frei +Fock representation and the quotients thereof will come into play: +7 +Fock representation +We begin with the following problem: Note that up until now we have not come +across any representation whatsoever (put aside sufficiently many) except the +trivial representation +B = 0 : +X +0, +A +0 +and as such it could be in principle that the relative Cuntz–Pimsner algebras +(for some correspondences) all coincide as the only trivial representation +(X, A) +T (X, A) = . . . = O(K, I) = O(K′, I′) = . . . = 0 ?? +As such the question arises whether a correspondence admits always a +nontrivial representation (and further also sufficiently many faithful ones). +For this we consider a failing attempt which leads us in turn to the Fock +representation: As a first guess we may consider the linking algebra associated +to our correspondence (seen as Hilbert module only) +B = +� +A +X +�� +A∗ +X∗ � += +� +A +X∗ +X +XX∗ +� +⊆ L +�� +A +X +�� +together with the representation given by the canonical embedding +� +A +0 +� +⊆ +� +A +X∗ +X +XX∗ +� += B, +� +0 +X +� +⊆ +� +A +X∗ +X +XX∗ +� += B. +Now while this representation respects the Hilbert module structure (basically +by construction) it does not respect the left module structure: +� +x∗ +0 +�� +0 +y +� += +� x∗y +0 +� +, +� +0 +x +�� a +0 +� += +� +0 +xa +� +, +� a +0 +�� +0 +x +� += +� +0 +0 +� +̸= +� +0 +ax +� +. +The solution is to simply extend these down the diagonal to infinity which brings +us straight to the Fock space representation: More precisely, the Fock space is +51 + +7 +Fock representation +Alexander Frei +the infinite direct sum of increasing tensor powers (as in section 1) +F(X) := A ⊕ X ⊕ XX ⊕ XXX ⊕ . . . = + + + + + + +A +X +XX +... + + + + + + +and the representation as diagonal action and as right shift respectively: +A → L(FX) : +a +� +A ⊕ X ⊕ XX ⊕ . . . +� +:= aA ⊕ aX ⊕ aXX ⊕ . . . +X → L(FX) : +x +� +A ⊕ X ⊕ XX ⊕ . . . +� +:= 0 ⊕ xA ⊕ xX ⊕ xXX ⊕ . . . +We visualize them in matrix notation (using the formal left and right shift): +X⊗R = X + + +0 +1 0 +1 0 +. . . + +, +A⊗1 = A + + +1 +1 +1 +. . . + +, +X∗⊗L = X∗ + + +0 1 +0 1 +0 +. . . + +. +Further the Fock representation comes with the (inner) circle action +T ↷ L(FX) : +(z ↷ −) := + + +1 +z +z2 +. . . + + − + + +1 +z +z2 +. . . + + +∗ +which renders the representation gauge-equivariant: + + +1 +z +z2 +. . . + +A + + +1 +1 +1 +. . . + + + + +1 +z +z2 +. . . + + +∗ += 1A + + +1 +1 +1 +. . . + +, + + +1 +z +z2 +. . . + +X + + +0 +1 0 +1 0 +. . . + + + + +1 +z +z2 +. . . + + +∗ += zX + + +0 +1 0 +1 0 +. . . + +. +Furthermore the representation of compact operators reads +XX∗ �→ X + + +0 +1 0 +1 0 +. . . + + + + +0 1 +0 1 +0 +. . . + +X∗ = XX∗ + + +0 +1 +1 +. . . + + +With these preliminary computations ready we may now reveal the Fock +representation as the universal representation (i.e. the Toeplitz algebra): +Proposition 7.1 (Fock representation = Toeplitz algebra): +52 + +7 +Fock representation +Alexander Frei +The Fock representation has trivial kernel and covariance +ker +� +A +L(FX) +� += 0, +cov +� +X +L(FX) +� += 0 +and as such defines the universal representation (by theorem 6.1). +Proof. Clearly the representation is faithful (and so has trivial kernel) as +� a +a +. . . +�� A +. . . +. . . +� += +� aA +. . . +. . . +� += +� 0 +0 +. . . +� +=⇒ +a = 0. +As a consequence, we may further compute the covariance as the intersection +within the representation (as the special instance from proposition 3.8) +cov +� +X +L(FX) +� += A +� 1 +1 +. . . +� +∩ XX∗ +� 0 +1 +. . . +� += 0. +So the covariance is trivial as well and the proposition follows. +Note that more importantly we established the existence of any non- +trivial representation whatsoever (equivalently the universal representation is +different from the trivial representation) and further also the existence of +faithful representations (equivalently the universal representation is faithful). +Put in other words, we may now distinguish Toeplitz algebras along the lattice +of quotient correspondences (the first dimension of kernel–covariance pairs): +T +� X +XK +� += O(K, 0) = O(K′, 0) = T +� +X +XK′ +� +=⇒ +K = K′ +Indeed we have even found the stronger kernel relation from (12): +ker +� +A +T (X) +� += 0 +=⇒ +ker +� +A +A +K +T +� X +XK +� � += K +We are still left with the question (which we get to next) +O(K, I) = O(K′, I′) +=⇒ +( K = K′ | I = I′ ) +? +For this we may now consider the Fock representation as a concrete realization +for the Toeplitz algebra and as such further construct every relative Cuntz– +Pimsner algebra (along kernel–covariance pairs) as a concrete quotient thereof: +For example given first a quotient correspondence (for some invariant ideal) +K ⊴ A : +X∗KX ⊆ K : +(X, A) +� X +XK , A +K +� +we may construct the corresponding Toeplitz algebra as quotient by the kernel +53 + +7 +Fock representation +Alexander Frei +(more precisely its ideal generated within the original Toeplitz representation) +0 +(XK, K) +(X, A) +� X +XK , A +K +� +0 +0 +T X(K ⊆ A)T X +T X +T +� X +XK +� +0 +where we omit as usual the closed linear span for more pointy statements. +Before continuing we replace for convenience the original correspondence by the +quotient correspondence, +� X +XK , A +K +� +⇝ (X, A) +=⇒ +T +� X +XK +� +⇝ T (X). +Consider next an ideal (as possible covariance) bounded from above by the +maximal covariance (as in proposition 3.6), +I ⊴ A : +I ⊆ max(X, A). +The bound from above is necessary as larger covariances force an additional +kernel and as such would factor over some further quotient correspondence. +Similarly as for relative Toeplitz algebras, the relative Cuntz–Pimsner algebra +arises now as coequalizer for the chosen covariance +O(K, I) = coeq + + + + +I ⊆ A ∩ XX∗ +T X +XX∗ +T X + + + + +and as such also as quotient by their difference as in (2): +0 +T X +� +(ϕ − τ)I +� +T X +T X +O(X; I) +0 : +(ϕ − τ) = + + + +A ∩ XX∗ +T X +T X + + + − + + + +A ∩ XX∗ +XX∗ +T X + + +. +We now invoke the Fock representation as Toeplitz algebra: Recall that we have +already found here the concrete embedding for the coefficient algebra as well as +the concrete embedding of compact operators (see above) +T X ⊆ L +�� A +X +. . . +�� +: +A �→ A +� 1 +1 +. . . +� +, +XX∗ �→ XX∗ +� 0 +1 +. . . +� +54 + +7 +Fock representation +Alexander Frei +and as such their difference reads +A ∩ XX∗ �→ A ∩ XX∗ +� 1 +0 +. . . +� +. +We therefore found the relative Cuntz–Pimsner algebra as quotient +0 +T X +� I +0 +. . . +� +T X +T X +O(X; I) +0. +(17) +This was originally established by Muhly and Solel in [MS98, theorem 2.19]. +We now wish to verify that the induced quotient representation remains faithful: +That is we note that the quotient could in principle introduce new kernel, +I ⊆ A ∩ XX∗ : +ker +� +A +T X +O(X; I) +� += 0 ? +Indeed we note that the quotient does introduce new kernel as soon as the +covariance exceeds the maximal covariance (as we have noted also above). +As such we have to make sure this does not happen as long as the covariance lies +below the maximal covariance from proposition 3.6. For this we note that the +trivial kernel above may be equivalently verified now as the trivial intersection +I ⊆ max(X, A) : +A +� 1 +1 +. . . +� +∩ T X +� I +0 +. . . +� +T X = 0 ? +(18) +On the other hand, we wish to also verify the covariance relation from (12): +cov +� +A +T X +O(X; I) +� += I ? +The problem here is that the covariance could in principle increase as well. +Indeed the construction (and even our very definition) of relative Cuntz–Pimsner +algebras guarantees just a least covariance for the provided covariance ideal. +This becomes more evident as follows: Consider for this the difference morphism +which factors by the universal property via the Toeplitz algebra: +A ∩ XX∗ +A ∩ XX∗ +0 +T X +� I +0 +. . . +� +T X +T X +O(X; I) +0. +ϕ − τ +55 + +7 +Fock representation +Alexander Frei +As such the covariance may be read off from the common intersection as +� A ∩ XX∗ +0 +. . . +� +∩ T X +� I +0 +. . . +� +T X = +� I +0 +. . . +� +? +(19) +Note this meanwhile also highlights how the covariance could increase: how? +Let us begin to verify that the representation remains faithful along the quotient. +For this we begin with the following well-known relation (which goes all the way +back to an observation by Joachim Cuntz made in [Cun77]): +Proposition 7.2. The ideal generated by the covariance as in (17) coincides +with the ideal of compact operators of the form +T X +� I +0 +. . . +� +T X = +� A +X +. . . +� +I( A X∗ . . . ) = K +�� A +X +. . . +� +I +� +with implicit closed linear spans for more pointy statements. +Sketch of proof: The result follows most easily using the formal right and left +shift operators (as further above) from which the covariance ideal reads +� I +0 +. . . +� += I +�� 1 +1 +. . . +� +− +� 0 +1 +. . . +�� += I ⊗ (1 − RL). +Recall however that these satisfy the well-known relation +LR = 1 +=⇒ +L(1 − RL) = 0 = (1 − RL)R. +As such we obtain as the only contributions for the ideal +T X +� I +0 +. . . +� +T X = +� +mn +� +X +� 0 +1 0 +. . . +��m� I +0 +. . . +�� +X∗ +� 0 1 +0 +. . . +��n +. +On the other hand they generate the system of matrix units such as +� 0 +1 0 +. . . +�m� 1 +0 +. . . +�� 0 1 +0 +. . . +�n += +� +0 +0 1 0 +0 +� += +� 0 +1 +0 +� +( 0 1 0 ). +As such we obtain for the above ideal +T X +� I +0 +. . . +� +T X = . . . = +� +mn +� 0 +Xn +0 +� +I( 0 X−n 0 ) = F(X)F(X)∗ +which is the desired relation for the covariance ideal. +In order to handle the kernel for the induced representation on the quotient +56 + +7 +Fock representation +Alexander Frei +we need to take a closer look into the ideal of compact operators from 7.2: +We begin for this with the well-known approximation by “finite rank matrices”. +More precisely, one has for compact operators on Fock space and S ⊆ N: +� 0 +K[. . .] +0 +� +∋ +� 0 +k(S) +0 +� +� k00 +k01 +k10 +k11 +. . . +� +∈ K +�� A +X +. . . +�� +. +S→N +Indeed one easily verifies this (using Dirac calculus from 1.1): +� . . . +0 +1 +�� A +X +. . . +� +(A∗ X∗ . . .) +0 +� A +X +. . . +� +(A∗ X∗ . . .) +� . . . +0 +1 +� +In particular we obtain for the diagonal compact operators as in 7.2: +K +�� A +X +. . . +� +I +� +∩ +� L(A) +L(X) +. . . +� += +� I +XIX∗ +→ 0 +� +. +Meanwhile the author would like to take a moment to thank his previous tutor +Dominic Enders for highlighting this perspective during personal discussions. +The idea is now to use the previous relation in contrast to the following +observation by Katsura which we reformulate in our language: Consider for +this the diagonal operators on Fock space +� L(A) +L(X) +. . . +� +⊆ L +� +FX = +� A +X +. . . +�� +and the representation between such diagonal operators: + + +0 +L(Xn) +0 +0 + + + + +0 +0 +L(Xn) ⊗ 1 +0 + + ⊆ + + +0 +0 +L(Xn ⊗ X) +0 + + +This generally fails to define a faithful representation: one may for instance +consider the graph correspondence for any finite acyclic graph such as +X = ℓ2� +E = • +• +� +=⇒ +XX = ℓ2� +EE = ∅ +� += 0. +Katsura’s crucial obervation tells us now that this becomes faithful when re- +stricted to the subspace of compact operators by our covariance ideal: +Proposition 7.3 ([Kat04, Lemma 4.7]). The representation above defines an +57 + +7 +Fock representation +Alexander Frei +embedding when restricted to the maximal covariance as in proposition 7.2, + + +0 +|Xn⟩ max(X, A) ⟨Xn| +0 +0 + + ⊆ + + +0 +0 +|Xn⟩ max(X, A) ⟨Xn| ⊗ 1 +0 + +. +As such the representation defines also an embedding for any covariance ideal +below the maximal covariance. +Proof from [Kat04]: We note for the kernel (in Dirac braket notation) +⟨Xn| ker +� +|Xn⟩ max(X, A) ⟨Xn| ↷ |Xn⟩ ⊗ |X⟩ +� +|Xn⟩ +⊆ ker +� +⟨Xn|Xn⟩ max(X, A) ⟨Xn|Xn⟩ ↷ |X⟩ +� +⊆ ker(A ↷ X) ∩ ker(A ↷ X)⊥ = 0 +where we have used the obvious inclusion +⟨Xn|Xn⟩ max(X, A) ⟨Xn|Xn⟩ ⊆ max(X, A) ⊆ ker(A ↷ X)⊥. +(Note the intersection reflects also the first level as in proposition 1.3.) +For an ideal such as the kernel above it holds however +⟨Xn| +� +ker[. . .] = ker[. . .]∗ ker[. . .] +� +|Xn⟩ = 0 +=⇒ +ker[. . .] |Xn⟩ = 0. +As such we found the inclusion +ker +� +|Xn⟩ max(X, A) ⟨Xn| ↷ |Xn⟩ ⊗ |X⟩ +� +⊆ ker +� +|Xn⟩⟨Xn| ↷ |Xn⟩ +� += 0. +In particular, the same holds true for any covariance below the maximal. +With Katsura’s observation at hand we may now verify the desired kernel +and covariance relation as in (12) which will resolve the second half of our +classification of relative Cuntz–Pimsner algebras. We note here that the kernel +relation is already due to Katsura as established in [Kat04]: +Theorem 7.4 (Relative Cuntz–Pimsner algebras: Kernel and Covariance): +For the relative Cuntz–Pimsner algebra as above it holds +ker +� +A → T → O(X; I) +� += 0 +cov +� +X → T → O(X; I) +� += I +As a consequence, the kernel–covariance pairs are also classifying. +58 + +7 +Fock representation +Alexander Frei +Proof. We begin with the kernel relation due to Katsura from [Kat04]: +As in the beginning discussion we need to verify the relation (18): +I ⊆ max(X, A) : +A +� 1 +1 +. . . +� +∩ T X +� I +0 +. . . +� +T X = 0 ? +For this we first revealed in proposition 7.2 that the ideal generated by +our covariance (within the Toeplitz algebra) agrees with compact operators. +As such the intersection with the coefficient algebra reads +A +� 1 +1 +. . . +� +∩ T X +� I +0 +. . . +� +T X = A +� 1 +1 +. . . +� +∩ +� I +|X⟩ I ⟨X| +→ 0 +� +. +In contrast, Katsura’s observation from proposition 7.3 tells us that these embed +along the diagonal and as such the norm remains also constant throughout, +� a +a +. . . +� +∈ +� I +|X⟩ I ⟨X| +. . . +� +: +∥a∥ = ∥a ↷ X∥ = . . . = ∥a ↷ Xn∥. +Both the vanishing of compact operators along the diagonal and the constant +norm are only possible for the trivial intersection and as such the trivial kernel. +We continue with the covariance relation from (12): For this we may now +simply verify the common intersection as in (19) also using proposition 7.2: +� A ∩ XX∗ +0 +. . . +� +∩ T X +� I +0 +. . . +� +T X = += +� A ∩ XX∗ +0 +. . . +� +∩ +� I +IX∗ +XI XIX∗ +. . . +� += +� I +0 +. . . +� +. +As such the covariance does not increase and the theorem is proven. +We have thus established also the second half in our classification: +More precisely, we have first found that the class of relative Cuntz–Pimsner +algebras exhausts the gauge-equivariant representations (which was the content +of the gauge-invariant uniqueness theorem). On the other hand we now found +that the parametrisation via kernel–covariance pairs is also classifying: +O(K, I) = O(K′, I′) +=⇒ +(K, I) = (K′, I′) +✓ +Altogether we have thus found: +the lattice of kernel–covariance pairs +parametrises the entire lattice of gauge-equivariant representations (as points). +59 + +8 +Lattice structure +Alexander Frei +8 +Lattice structure +While our discussion (so far) captured the lattice of gauge-equivariant represent- +ations as individual points along the lattice, this still leaves open how the +lattice structure of kernel–covariance pairs reflects the lattice structure of +gauge-equivariant representations (among each other) to which we now get: +Recall for this that our kernel–covariance pairs encode the covariance for the +quotient correspondence (which rendered the representation faithful) +(X, A) +B : +K = ker +� +A → B +� +=⇒ +I = cov +� +X +XK → B +� +and its intrinsic characterisation on the quotient (as bounded ideal) +� +K ⊴ A +��� X∗KX ⊆ K +� +=⇒ +� +I ⊴ A/K +��� I ⊆ max +� X +XK +� � +. +We therefore begin with a translation of our kernel–covariance pairs which +lives on the original correspondence. +This allows us to give an intrinsic or- +der on kernel–covariance pairs reflecting the lattice structure of representations. +Along this translation we further reveal Katsura’ mysterious T-pairs as nothing +but our original kernel–covariance pairs (with maximal covariance in disguise). +For our translation we first recall that the covariance for an embedding +(faithful representation) may be simply read off as common intersection within +the ambient algebra (as in proposition 3.8) +(X, A) ⊆ B : +cov(X → B) = im(A → B) ∩ im(XX∗ → B). +More precisely, one may take the portion of the coefficient algebra +cov(X → B) = +� +a ∈ A +��� a ∈ im(XX∗ → B) +� += A ∩ im(XX∗). +That is however the same amount of information as the actual intersection, +as long as one keeps track of the embedding for the coefficient algebra: +60 + +8 +Lattice structure +Alexander Frei +B +XX∗ +A +im(XX∗) +im(XX∗) +im(A) +With this picture in mind, we continue on some general representation +(X, A) +� X +XK , A +K +� +B : +K = ker(A → B). +We first note that for a quotient (i.e. surjective mapping) there is absolutely no +loss of generality when pulling back any ideal along the quotient since for +I ⊴ A/K +⇝ +� +A → A +K +�−1I ⊴ A : +I = +� +A → A +K +�� +A → A +K +�−1I +and so we may use the equivalent intrinsic definition of kernel–covariance pairs +as those with covariance below the maximal covariance within the quotient: +(I + K) ⊴ A : +I/K ⊆ max +� X +XK +� +!! +(20) +We wrote the covariance ideal (here and below) as sum with the kernel ideal +simply to guarantee that the ideal arises indeed as pullback from the quotient. +On the other hand, note that the amount of covariance (as described above) +does not change either in the sense of how much common intersection the coef- +ficient algebra has with compact operators since (see also proposition 3.10): +im +� +A +A +K +B +� += im +� +A +K +B +� +, +im +� +XX∗ +� X +XK +�� X +XK +�∗ +B +� += im +� � X +XK +�� X +XK +�∗ +B +� +. +So there is really no loss of generality from this perspective either. +As such we obtain another equivalent extrinsic definition of kernel–covariance +pairs as those with covariance ideal describing the amount of common intersec- +tion between the coefficient algebra and compact operators: +(X, A) +B : +(I + K) = im(A → B) ∩ im(XX∗ → B). +61 + +8 +Lattice structure +Alexander Frei +For comparison between kernel–covariance pairs we however take from now on +the portion within the coefficient algebra as above, that is +(I + K) = +� +a ∈ A +��� a ∈ im(XX∗ → B) +� += A ∩ im(XX∗ → B) +and we note this agrees with our intrinsic definition (somewhat obvious now). +Meanwhile we also keep in mind the viewpoint on the covariance as amount of +common intersection as it provides an interesting perspective on representations. +With both these definitions at hand (the intrinsic and the extrinsic) we may +now get to the lattice of gauge-equivariant representations. +Given a pair of +representations we define the usual order of representations as +� +(X, A) → B +� +≤ +� +(X, A) → B′ � +: +(X, A) +B +B′. +More precisely, that is the representation factors over the other and note that the +sole existence of such a factorisation entails a unique such as the representations +are all generated as an operator algebra by (the image of) the correspondence: +B = C∗(A ∪ X) +=⇒ +B +B′ +uniquely +✓ +Given a factorisation we now easily infer for their kernel and covariance +ker(A → B) ⊆ ker(A → B → B′), +A ∩ im(XX∗ → B) ⊆ A ∩ im(XX∗ → B → B′). +Indeed the latter may be easily seen as (somewhat trivially) +im(a → B) ∈ im(XX∗ → B) +=⇒ +im(a → B → B′) ∈ im(XX∗ → B → B′). +Schematically the amount of intersection could look something like: +B +B′ +So we have found the following converse direction (using theorem 7.4): +� +K ⊆ L +��� I + K ⊆ J + L +� +⇐= +O(K, I) ≤ O(L, J) +✓ +What about the forward direction? That is assume we have an inclusion of +62 + +8 +Lattice structure +Alexander Frei +kernel–covariance pairs as above. As we have an inclusion of kernel ideals we +obtain in particular for their quotient correspondence +(X, A) +� X +XK , A +K +� +� X +XL, A +L +� +O(K, I) +O(L, J) +and so we may replace our correspondence as usual by the quotient +� X +XK , A +K +� +⇝ (X, A) +=⇒ +O(K, I) ⇝ O(0, I). +Recall that the relative Cuntz–Pimsner algebra satisfies (by definition) +(X, A) +O(X; I) +B +⇐⇒ +I ⊆ cov( X → B ) +and as such we need to verify the least amount of covariance +I ⊆ cov +� +(X, A) +� X +XL, A +L +� +O(L, J) +� +? +This basically follows now from our study of kernel morphisms and covariance +ideals from section 3, which we recall now for more clarity in our context: +At first we found that kernel and cokernel morphisms have full covariance. That +is in our context the fully commutative diagram for the quotient morphism and +on the other hand the covariance diagram for the quotient representation: +A ∩ XX∗ +A/L +XX∗ +� X +XL +�� X +XL +�∗ +and +J/L +O(L, J) +� X +XL +�� X +XL +�∗ +O(L, J). +Combing these with our assumption on covariance ideals we obtain +(I + 0) +J/L +O(L, J) +XX∗ +� X +XL +�� X +XL +�∗ +O(L, J) +63 + +8 +Lattice structure +Alexander Frei +and as such the desired amount of covariance, +(I + 0) ⊆ (J + L) +=⇒ +I ⊆ cov +� +(X, A) → O(L, J) +� +. +As such we also found the forward direction and so the order isomorphism, +which is the main conclusion of this article: +Theorem 8.1 (Kernel–covariance pairs: Order isomorphism): +The kernel–covariance pairs as in (20) define the order isomorphism +� +K ⊆ L +��� I + K ⊆ J + L +� +⇐⇒ +O(K, I) ≤ O(L, J) +(21) +and as such the lattice of kernel–covariance pairs with its natural order by +inclusion describes the entire lattice of gauge-equivariant representations, +equivalently the entire lattice of gauge-invariant ideals. +Let us give an example of our result for some graph algebra, +also to illustrate the ease of working with such kernel–covariance pairs: +Example 8.2 (Graph correspondences: gauge-invariant ideals): +Consider a graph correspondence as in example 1.2, +X = ℓ2� +E = edges +� +, +A = c0(vertices) +and recall its quotient graphs as in 3.5 (given by hereditary ideals as in 3.2) +as well as their covariance ideals given by their sets of regular vertices as in 3.7. +As an example consider the following graph and its quotient graphs +/2 +K = 0 : + + + + +a +b + + + + +K = (a) : + + + + +b + + + + +K = (a ∪ b) : +� +∅ +� +. +64 + +8 +Lattice structure +Alexander Frei +Then its lattice of gauge-equivariant representations reads (as Hasse diagram) +K = 0 : +K = (a) : +K = (a ∪ b) : +I = 0 +I = (a) +I = (a) +I = (b) +I = (a ∪ b) +I = (a ∪ b) +I = (a ∪ b) +and so equivalently also the lattice of gauge-invariant ideals. +Note how we now easily read off the order via kernel–covariance pairs. +Before we continue let us make a few remarks on the order isomorphism: +In particular the following discussion on connecting morphisms will basically +cover the notion of suprema and infima as addressed in the following remark: +Remark 8.3 (Lattice isomorphism: suprema and infima): +Note that as both lattices are order isomorphic they will be also lattice isomor- +phic as unions and intersections (finite or arbitrary) as well as top and bottom +elements are determined as suprema and infima respectively +order iso +� +sup +s as +� += sup +s +� +order isoas +� +which aside their existence depend only on the given order. +Put in other words the notion of a lattice is really just a pure property and +defines no additional structure so there is really no difference between the +lattice of gauge-equivariant representations and the lattice of kernel–covariance +pairs. +We will however later discover that arbitrary suprema and infima of +kernel–covariance pairs do not necessarily always arise as intersections and +sums of their kernel and covariance ideals, that is we only have +inf +s (Ks|Is) ≤ +� � +s +Ks +���� +� +s +Is +� +and +sup +s (Ks|Is) ≥ +� � +s +Ks +���� +� +s +Is +� +. +Indeed while the intersection und sum of invariant ideals remain invariant +⟨X| +�� +Ks +� +|X⟩ ⊆ +� +Ks +and +⟨X| +�� +Ks +� +|X⟩ ⊆ +� +Ks +65 + +8 +Lattice structure +Alexander Frei +the intersection and sum of covariance ideals may not always end up below the +maximal covariance, +� +Is ⊆ max +� +X +X(� Ks) +� +and +� +Is ⊆ max +� +X +X(� Ks) +� +? +Schematically that is the next possible kernel–covariance pair may lie just further +beyond (as seen within the lattice of any pairs of ideals) such as +� +K : . . . +��� I ⊆ max( X +XK ) ✓ +� +� +(K1 + K2) : . . . +��� (I1 + I2) ⊈ max +� +X +X(K1+K2) +� � +� +Ks : . . . +��� Is ⊆ max +� +X +XKs +� +✓ +� +. +For example the requirement to have at least as much covariance as all the given +covariance ideals can force larger kernel than just the sum of given kernel +ideals, or put more drastically there may exist no connecting morphism from +each relative Cuntz–Pimsner algebra to the cokernel strand over the sum of +kernel ideals. We will see such an example in 8.5 below involving already even +just a pair of kernel–covariance pairs: +(X, A) +� +X +XK1 , A +K1 +� +� +X +XK2 , A +K2 +� +� X +XK , A +K +� +. . . +O(K1, I1) +O(K2, I2) +O(K, I) +. . . +. . . +. . . +. . . +× +× +Furthermore we note that from either description the order defines a partial +order (as opposed to just a preorder) as easily seen from +� +(X, A) +C∗(X ∪ A) = B +� +and +� +(X, A) +C∗(X ∪ A) = B′ � +: + + + + +(X, A) +(X, A) +(X, A) +B +B′ +B + + + + = + + + + +(X, A) +(X, A) +B +B + + + + +so one is a retract of the other and similarly the other way around, or one may +66 + +8 +Lattice structure +Alexander Frei +equivalently also argue using their kernel from the Toeplitz algebra +� +T X +B +B′ � +=⇒ +ker +� +T X → B +� +⊆ ker +� +T X → B′� +from which they had been already the same quotient +ker +� +T X → B +� += ker +� +T X → B′� +=⇒ +� +T X +B = B′ � +. +Or one may now also argue using kernel–covariance pairs, +� +K ⊆ L ⊆ K +��� I ⊆ J ⊆ I +� +=⇒ +� +K = L +��� I = J +� +. +Altogether we remark that the lattice of gauge-equivariant representations co- +incides entirely with the lattice of kernel–covariance pairs, while suprema and +infima of kernel–covariance pairs may lay only beyond of just the intersection +and sum of their kernel and covariance ideals. +With this in mind we continue on the remaining questions from section 4: +More precisely, we wish to find suitable characterisations when morphisms exists +between different quotient strands (based on kernel–covariance pairs) +� X +XK , A +K +� +O(K, 0) +O(K, I) +O(K, max) +� X +XL, A +L +� +O(L, 0) +O(L, J) +O(L, max) +and we warn ahead that these won’t always exist. +Now at first we note +that clearly the existence infers the inclusion of kernel ideals and we may for +simplicity replace the original correspondence with the quotient +(X, A) := +� X +XK , A +K +� +O(K = 0, 0) +O(K = 0, I) +. . . +In particular there always exist connecting morphism from at least the Toeplitz +algebra and so also some further relative Cuntz-Pimsner algebras +T +� X +XK +� += O(K, 0) +. . . +O(K, I =?) +. . . +T +� X +XL +� += O(L, 0) +O(L, J) +O(L, max). +✓ +✓ +✓ +× +As such the first question is to find the smallest relative Cuntz–Pimsner algebra +67 + +8 +Lattice structure +Alexander Frei +from which connecting morphisms exist. This may be now easily solved as +O(K, I) +O(L, J) +⇐⇒ +(I + K) ⊆ (J + L) +and as such the largest covariance ideal (i.e. within the maximal covariance) +simply arises as intersection with the given covariance from the quotient, +I = max +� X +XK +� +∩ +� +A/K → A/L +�−1J = max +� X +XK +� +∩ (J + K). +Schematically the intersection can look something like this: +A/K +J ∩ max +max +� +X +XK +� +A/L +J +max +� X +XL +� +The reader may easily find some examples with (using graph algebras as above): +max +� X +XK +� +̸= 0 : +(J ∩max) = 0 / 0 ̸= (J ∩max) ̸= max / (J ∩max) = max ? +Moreover one may now easily guess the meet of kernel–covariance pairs: +� +s +( Ks | Is ) = +� +K = +� +Ks +���� I = +� +Is ∩ max +� X +XK +� � +So far about connecting morphisms from preceeding cokernel strands. +The other direction however is more interesting: That is a relative Cuntz– +Pimsner does not necessarily connect to every following quotient correspondence +and in there not even beginning at every relative Cuntz-Pimsner algebra either, +O(K, 0) +O(K, I) +O(K, max) +O(L =?, 0) +O(L =?, J =?) +O(L =?, max). +Note this also describes the lattice of gauge-invariant ideals for the given +relative Cuntz–Pimsner algebra (simply as each such defines a quotient). +We note however that as soon as it connects to another relative Cuntz–Pimsner +algebra (in some following quotient correspondence) then it certainly also does +so to the absolute Cuntz–Pimsner algebra for that quotient correspondence. +68 + +8 +Lattice structure +Alexander Frei +As such this introduces an obstruction which may be now handle using (21): +O(K, I) ≤ O(L, max) +⇐⇒ +Jmin := +� +A/K → A/L +� +I ⊆ max +� X +XL +� +This condition fails from time to time (we provide a simple example below). +In case this condition is met we obtain as smallest solution +O(K, 0) +O(K, I) +. . . +O(L, 0) +. . . +O(L, Jmin) +. . . +O(L, max) +× +✓ +✓ +✓ +while in case the condition fails then there simply is no connecting morphism. +As such we also obtain the lattice for any relative Cuntz-Pimsner algebra which +is really just our main result restated (while this also generalizes O-pairs): +Corollary 8.4 (Relative Cuntz-Pimsner algebra: gauge-invariant ideals): +Consider a relative Cuntz-Pimsner algebra (as described in section 4) +(X, A) +� X +XK , A +K +� +O(K, 0) +O(K, I) +for some kernel–covariance pair as in (20) above. +Then its lattice of gauge-invariant ideals simply runs over pairs as in (21) +� +K ⊆ L +��� I + K ⊆ J + L +� +⇐⇒ +� +K ⊆ L +��� Jmin ⊆ J ⊆ max +� X +XL +� � +or in words simply over all larger kernel–covariance pairs. +Let us give an example for when there is no connecting morphism: +Example 8.5 (Graph correspondences: no connecting morphism): +Consider as an example the following graph and its quotient graphs +/2 +K = 0 : +� +a → b +� +K = (a) : +� +b +� +K = (a ∪ b) : +� +∅ +� +. +Then there is no connecting morphism for the absolute Cuntz-Pimsner algebras +69 + +8 +Lattice structure +Alexander Frei +between the first and second quotient (by simply reading off covariance ideals): +T +� +X = ℓ2(a → b) +� +T +� +X = ℓ2(b) +� +T (X = 0) +O +� +X = ℓ2(a → b) +� +O +� +X = ℓ2(b) +� +O(X = 0). +× +Indeed the obstruction fails for the covariance ideal: +I1 = (b) = reg +� +a → b +� +and +K2 = (a) ⊆ her(a → b) : +(A → A/K2)I1 = (b) ⊈ reg +� +quotient graph = b +� +. +The issue here is that the hereditary ideal is simply not saturated. Alternatively +one may note that the first defines the simple algebra of 2×2 matrices. +In particular, we obtain for the join of kernel–covariance pairs +� +K1 = 0 +��� I1 = (b) +� +∨ +� +K2 = (a) +��� I2 = (a) +� += +� +K = (a ∪ b) +��� . . . +� +. +In other words, the join as next possible kernel–covariance pair lies only +beyond of just the sum of kernel and covariance ideals. So we found an example +for the issue (mentioned further above) that suprema and infima will be generally +beyond just intersections and sums of ideals. +We finish this section with a widely missed identification of Katsura’s work: +That is we clarify how Katsura’s T-pairs (and O-pairs) are nothing but the +pullback version of our kernel–covariance pairs from above. More precisely, we +elaborate Katsura’s cryptic requirement +J(K) := +� +a ∈ A +��� . . . and aX−1(K) ⊆ K +� +: +K ⊆ I ⊆ J(K) +and how this defines a translation of the constraint from proposition 3.6: +That is any covariance for an embedding into an operator algebra is necessarily +orthogonal to the kernel (for its left action) which read in our case +cov +� +X +XK → B +� +⊥ ker +� A +K ↷ +X +XK +� +and as such these cannot exceed the maximal covariance (a.k.a. Katsura’s ideal) +cov +� +X +XK → B +� +⊆ +�� X +XK +�� X +XK +�∗ ∩ ker +� A +K ↷ +X +XK +�⊥� += max +� X +XK +� +. +70 + +8 +Lattice structure +Alexander Frei +As such we found instead our kernel–covariance pairs as given by invariant +ideals as kernel (which defines some sort of discrete range for kernel ideals) +K ⊴ A : +X∗KX ⊆ K +together with ideals bounded from above as covariance (which defines an +upper bound on the range of covariance ideals) +I ⊴ A/K : +0 ⊆ I ⊆ max +� X +XK +� +while we found in the second +half +of +our +classification that each +such kernel–covariance pair indeed arises itself (more precisely theorem 7.4). +In order to establish Katsura’s requirement we first note the obvious +I = +� +A → A/K +�−1 +J +⇐⇒ +K ⊆ I +and one the other hand the inclusion (for quotient maps) +J ⊆ max +� X +XK +� +⇐⇒ +� +A → A/K +�−1 +J ⊆ +� +A → A/K +�−1 +max +� X +XK +� +. +As such Katsura’s condition simply states (see [Kat07, lemma 5.2]) +J(K) = (A → A/K)−1 max +� X +XK +� +for which it further suffices to verify (see also [Kat07, lemma 5.2]) +� +aX−1(K) ⊆ K +� += +� +A → A/K +�−1 +ker +� A +K ↷ +X +XK +�⊥ +since the dotted condition represents nothing but compactly acting coefficients. +This is however now easily verified: Consider for this the pullback (which con- +tains the same information) +ker +� +A ↷ +X +XK +� += +� +A → A/K +�−1 +ker +� A +K ↷ +X +XK +� +: +� +A → A/K +�� +A → A/K +�−1 +( . . . ) = ( . . . ) +and which further reads (as in proposition 3.9) +ker +� +A ↷ +X +XK +� += +� +a X +XK = 0 +� += {aX ⊆ XK} = {X∗aX ⊆ K} = X−1(K). +Put together we obtain the desired relation for the orthogonal complement. +We note that the relation has been worked out by Katsura in [Kat07, lemma 5.2] +71 + +9 +Pimsner dilations +Alexander Frei +which however has been not continued further on: Katsura chose to work with +the cryptic requirement instead of pursuing their kernel–covariance counterpart. +Possibly because they got only partially recognized as covariance ideals. +Finally the author notes that the results here arose from a more detailed +study of [Kat07] which builds on [FMR03] and further [KPW98] and [Pim97]. +More precisely, the author realized the relations drawn in [Kat07, lemma 5.10] +(which extend [FMR03, lemma 2.9]) as a partial result on categorical kernel +and cokernel morphisms which led to their intrinsic characterization (in +theorem 3.3) and so also on the range of possible kernel ideals. +On the other hand the author realized the first observation made in [Kat04, +proposition 3.3] as an intrinsic characterisation on the range of possi- +ble covariance +ideals for the induced representation on the quotient. +These let the author to systematically employ such kernel–covariance pairs, +which allowed on one hand to handle the general version of the gauge- +invariant uniqueness-theorem by reduction to the faithful case which follows +from the sleek and simplifying proof by Evgenios Kakariadis in [Kak16] +(which draws from the second observation made in [Kat04, proposition 3.3]) +and on the other hand the critical observation made by Takeshi Katsura in +[Kat04, lemma 4.7] in his seminal paper from 2004, which led the author to +retrieve kernel–covariance pairs from their relative Cuntz-Pimsner algebra +(in theorem 7.4). The main difference however is that we didn’t need to +build any ad-hoc semi-kind-of categorical pushout for correspondences as was +handled in [Kat07]. Instead it is all based on the simple idea of reduction +to faithful representations using kernel and cokernel morphisms. +9 +Pimsner dilations +This final section introduces the notion of dilations and verifies the existence +of the maximal dilation as Hilbert bimodule. We further reveal Katsura’s +construction as a particular nonmaximal dilation and illustrate the lack of +minimal dilations. Meanwhile, the author would like to take this opportunity +to thank Ralf Meyer for sharing his enlightening perspective on the Pimsner +dilation as maximal dilation. +We begin with the concept of dilations: More precisely that is any gauge- +equivariant factorisation over some intermediate correspondence such as +(Y, B) +(X, A) +O(K, I) +72 + +9 +Pimsner dilations +Alexander Frei +where the gauge-equivariance boils down to simply +Y +O(K, I)(1) +and +B +O(K, I)(0). +As the original correspondence generates the relative Cuntz–Pimsner algebra, +so does also the intermediate one +C∗(X ∪ A) = O(K, I) +=⇒ +C∗(Y ∪ B) = O(K, I) +whence the factorisation defines a relative Cuntz–Pimsner algebra itself: +(Y, B) +O(K, I) = O +� +Y, B +��� L =? J =? +� +As such the task is now to find dilations which generate the relative Cuntz– +Pimsner algebra as an absolute Cuntz–Pimsner algebra: +(X, A) +� +Y =?, B =? +� +O(K, I) = O +� +Y, B +��� L = 0, J = max +� +? +As the kernel should be trivial we have no choice than to look within the rel- +ative Cuntz–Pimsner algebra itself. +Also we may as usual assume that our +original correspondence embeds itself (simply by replacing our original corres- +pondence by its quotient correspondence). As such our intermediate correspon- +dence necessarily arises as an intermediate subspace +X ⊆ Y ⊆ O(K = 0, I)(1) +and +A ⊆ B ⊆ O(K = 0, I)(0). +On the other hand we found (as a well-known description) that the absolute +Cuntz–Pimsner algebra arises as the smallest gauge-equivariant quotient for +which the coefficient algebra faithfully embeds into (and so also the correspon- +dence) or in other words the coefficient algebra detects the gauge-invariant ideals +within the absolute Cuntz–Pimsner algebra. So we aim to find an intermediate +subspace which detects the remaining gauge-invariant ideals +J ⊴ O(K = 0, I) : +B ∩ J = 0 =⇒ J = 0 ? +By our main result (theorem 7.4) we found a parametrisation for the entire +73 + +9 +Pimsner dilations +Alexander Frei +lattice of gauge-equivariant representations +O(K = 0, I = 0) +O(K ̸= 0, I = 0) +. . . +O(K = 0, I) +. . . +. . . +O(K = 0, max) +. . . +. . . +given by kernel–covariance pairs +� +K ⊴ A : X∗KX ⊆ K +��� I ⊴ A +K : I ⊆ max +� X +XK +� � +. +and so also of gauge-invariant ideals (within the Toeplitz algebra) whence also +for the relative Cuntz-Pimsner algebra. Our original coefficient algebra however +already detects the kernel component: +A ∩ T (X, A)(K ⊆ A)T (X, A) = 0 +=⇒ +K = 0 +✓ +As such the only gauge-invariant ideals which our original coefficient algebra +cannot detect are precisely the covariance ideals (with trivial kernel component) +I ⊆ max(X, A) : +A ∩ T (X, A) +� I +0 +� +T (X, A) = 0. +Furthermore we have already taken a quotient for some covariance (which +brought us to our relative Cuntz–Pimsner algebra) and so the remaining gauge- +invariant ideals arise as remaining quotients +O(K = 0; I = 0) +O(K = 0, I) +. . . +O(K = 0, max). +As such we need to find an intermediate coefficient algebra which detects all of +the remaining covariance ideals beyond the given one I ⊆ J ⊆ max(X, A): +A ⊆ B ⊆ O(K, I)(0) : +B ∩ O(K, I) +� J +0 +� +O(K, I) ̸= 0 +(22) +with an intermediate subspace as correspondence (as described in section 1) +X ⊆ Y ⊆ O(K, I)(1) : +Y ∗Y ⊆ B, +BY ⊆ Y, +Y B ⊆ Y. +As such one may first choose an intermediate subalgebra which detects the re- +maining covariance ideals (and if desired also a chosen subspace) and from there +74 + +9 +Pimsner dilations +Alexander Frei +simply enlarge the chosen subalgebra to form a correspondence, for instance as +the smallest correspondence above: +�� � +Y0 ⊆ Y +��B0 ⊆ B +� ��� Y ∗Y ⊆ B, BY ⊆ Y, Y B ⊆ Y +� +. +Indeed the intersection of any class of correspondences forms a correspondence: +� +Y = +� +Yn +��� B = +� +Bn +� +: +Y ∗ +n Yn ⊆ Bn, . . . +=⇒ +Y ∗Y ⊆ B, . . . +We meanwhile need to verify that there always exists one above: For this we +simply consider the maximal dilation (also known as Pimsner dilation) +� +Y = O(K, I)(1) +��� B = O(K, I)(0) +� +. +Indeed the maximal dilation defines even a Hilbert bimodule and so also a +correspondence simply as Fourier spaces define Fell bundles (confer section 5): +O(K, I)(−1)O(K, I)(1) ⊆ O(K, I)(−1 + 1 = 0), +O(K, I)(0)O(K, I)(1) ⊆ O(K, I)(0 + 1 = 1), +. . . +On the other hand the maximal dilation is also easily seen to detect all of the +remaining covariance simply as each is generated from the fixed point algebra: +� J +0 +� +⊆ A +� 1 +1 +� ++ XX∗� 0 +1 +� +⊆ O(K, I)(0) +=⇒ +� J +0 +� +⊆ O(K, I)(0) ∩ +� +O(K, I) +� J +0 +� +O(K, I) +� +(23) +In fact one may argue more generally: Given any operator algebra with a +given circle action and consider its fixed point algebra (as in section 5) +T ↷ B : +B(n = 0) = +� +b +��� +� +b(z) := z ↷ b +� +≡ b +� +. +Then its fixed point algebra detects every gauge-invariant subalgebra: +� +A = T ↷ A +� +⊆ B : +A ∩ B(0) = 0 =⇒ A = 0. +Indeed this follows using the conditional expectation (confer section 5) +T ↷ B : +E(b) = +� +T +� +b(z) = z ↷ b +� +dz. +As the subalgebra is gauge-invariant the conditional expectation does not leave +75 + +9 +Pimsner dilations +Alexander Frei +the subalgebra (using its construction as Bochner integral) +� +A = T ↷ A +� +=⇒ +E(A) = +� +T +� +z ↷ A ⊆ A +� +dz ⊆ A. +On the other hand, every operator algebra is spanned by its positive portion, +A = pos(A) − pos(A) + i pos(A) − i pos(A), +pos(A) := {0 ≤ a ∈ A}. +The conditional expectation (given by averaging) is however faithful on the +positive portion and as such we have have found the detection +E(pos A) ⊆ A ∩ B(0) = 0 +=⇒ +pos(A) = 0 +=⇒ +A = 0. +This well-known technique is quite worthwhile in other context. +As such we have found the following familiar result with ease (note there was +basically nothing left to prove anymore): +Theorem 9.1 (Maximal dilation: absolute Cuntz–Pimsner algebra): +The maximal dilation realises relative Cuntz–Pimsner algebras as absolute one +O(K, I) = O +� +Y = O(K, I)(1) +��� B = O(K, I)(0) +� +. +and further defines the maximal Hilbert bimodule. +Proof. This is now an immediate consequence from (23) satisfying (22). +This dilation however is rather large in the sense that there is not much +control over its behavior (besides its universal description). For instance one +may think of the maximal dilation similar to maximal Furstenberg boundary. +Instead we therefore seek for some dilation small enough to be tractable com- +binatorially while large enough to detect covariance. As explained above, one +may for this simply begin with a small subalgebra which detects covariance and +simply enlarge the chosen subalgebra to form a correspondence. In practice one +may for instance attempt to run the algorithm +Y = BX + X + XB + BXB +=⇒ +B′ = B + Y ∗Y +=⇒ +Y ′ = B′Y + Y + Y B′ +=⇒ +. . . +(24) +with implicit closed linear span as usual. +While there always exist a smallest correspondence above (as we found above) +this process may never halt and whence leave us clueless about its combina- +torial behavior. In good cases however, the algorithm halts and thus allows +76 + +9 +Pimsner dilations +Alexander Frei +for its combinatorial description. This happens in particular for the canonical +subalgebra given by the maximal covariance itself: +A ⊆ +� +B = A + +� max(X, A) +0 +� � +⊆ O(K, I)(0) +(25) +whose sum defines a subalgebra since +� max(X, A) +0 +� +A +� 1 +1 +� +⊆ +� max(X, A)A +0 +� +⊆ +� max(X, A) +0 +� +which is nothin but the relation (from proposition 2.1) +(ϕ − τ) max(X, A) · ϕ(A) = (ϕ − τ) +� +max(X, A)A +� +⊆ (ϕ − τ) max(X, A). +On the other hand its left action keeps the space invariant +� max(X, A) +0 +� +X ⊆ (A + XX∗)X ⊆ X +(26) +and as such the algorithm halts right after the first round, +Y = X + X +� max(X, A) +0 +� += XB : +Y ∗Y = BX∗XB ⊆ B +and as such we got the canonical dilation as a combinatorial object. +Using the left and right shift (as in section 7) we further even note +� max(X, A) +0 +� +↷ X = max(X, A)(1 − RL) · XR = 0 +(27) +which is nothing but the obvious relation (see also proposition 2.1) +τ(XX∗)τ(X) = τ(XX∗X) +=⇒ +(ϕ − τ) +� +A ∩ XX∗� +τ(X) = 0. +This resambles Katsura’s construction from [Kat04] and so we refer to the canon- +ical dilation given by the maximal covariance as Katsura dilation (and note +also here that there was basically nothing left to prove anymore): +Theorem 9.2 (Katsura dilation: absolute Cuntz–Pimsner algebra): +The canonical dilation given by the maximal covariance realises a relative Cuntz– +Pimsner algebra as an absolute Cuntz–Pimsner algebra +O(K = 0, I) = O +� +Y = X + X +� max(X, A) +0 +� ��� B = A + +� max(X, A) +0 +� � +and the analogous dilation for kernel–covariance pairs with kernel ideal. +This dilation may well fail to define a minimal dilation (detecting covariance) +77 + +9 +Pimsner dilations +Alexander Frei +and even if minimal, it generally fails to be the only minimal dilation. +Proof. This is now an immediate consequence from (25) satisfying (22). +We further provide examples for the failure of minimality in example 9.8. +Corollary 9.3 (Katsura dilation: intrinsic description): +The canonical dilation given by the maximal covariance allows the intrinsic +description as the operator algebra freely generated by their abstract copies +� +A = A +� 1 +1 +� � +∪ +� +max(X, A)/I = +� max(X, A) +0 +� � +with multiplication given by +A +� 1 +1 +� +· +� max(X, A) +0 +� +⊆ +� A max(X, A) +0 +� +⊆ +� max(X, A) +0 +� +and similarly for the correspondence itself. +The analogous expression holds for kernel–covariance pairs with kernel ideal. +This further recovers the particular description from [Kat07, definition 6.1]. +Proof. By part of our main result (the nontrivial part of theorem 7.4) we found +that the relative Cuntz–Pimsner algebra does not introduce additional kernel +which is the faithful copy of the coefficient algebra (as a familiar result): +(X, A) +O(K = 0, I) : +A = A +� 1 +1 +� +. +On the other hand the relative Cuntz–Pimsner algebra also does not introduce +additional covariance (asides the already given covariance) which reads +� A ∩ XX∗ +0 +� +∩ T (X, A) +� I +0 +� +T (X, A) = +� I +0 +� +. +and as such the added maximal covariance defines a faithful copy up to +max(X, A)/I = +� max(X, A) +0 +� +⊆ O(K = 0, I). +Note that the maximal covariance absorbs the coefficient algebra and as such +their sum already defines a closed and thus complete operator algebra (see the +quick proof (16) on the sum of an algebra and ideal from section 6): +A +� 1 +1 +� ++ +� max(X, A) +0 +� += A +� 1 +1 +� ++ +� max(X, A) +0 +� +⊆ O(K = 0, I). +78 + +9 +Pimsner dilations +Alexander Frei +In fact this holds in any representation as also their universal: +C∗� +A ∪ max(X, A)/I +��� A max(X, A)/I ⊆ max(X, A)/I +� += A + max(X, A)/I = A + max(X, A)/I. +As such any concrete representation which provides faithful disjoint copies for +the coefficient algebra and the maximal covariance (mod covariance ideal) de- +fines a faithful representation for their freely generated operator algebra: +A ∩ max(X, A)/I = 0 ⊆ B +=⇒ +C∗(A ∪ max(X, A)/I) ⊆ B +Indeed this simply follows by some basic linear algebra. This holds in particular +for their copy in the relative Cuntz–Pimsner algebra +A +� 1 +1 +� +∩ +� max(X, A) +0 +� += +� ker(A ↷ X) +0 +� +∩ +� max(X, A) +0 +� += 0 +where we have used their trivial intersection +ker(A ↷ X) ∩ max(X, A) ⊆ ker(A ↷ X) ∩ ker(A ↷ X)⊥ = 0 +and as such the dilation arises as universal representation +C∗� +A ∪ max(X, A)/I +��� A max(X, A)/I ⊆ max(X, A)/I +� += A +� 1 +1 +� ++ +� max(X, A) +0 +� +⊆ O(K = 0, I). +Note this defines a quite general argument which applies also in other context. +Finally Katsura’s description is nothing but the isomorphism +C∗� +A ∪ M +��� AM ⊆ M +� += +� +a ⊕ (im a + m) ∈ A ⊕ (im A + M ⊆ B) +� +which simply enforces faithful disjoint copies for any A → B: +A = A(1 ⊕ 1), +A(1 ⊕ 1) ∩ (0 ⊕ M) = 0, +M = 0 ⊕ (M ⊆ B). +There is however nothing special about this choice of formal description. +Instead it is the universal description as freely generated copies with one ab- +79 + +9 +Pimsner dilations +Alexander Frei +sorbing which captures its properties: +C∗� +A ∪ max(X, A)/I +��� A max(X, A)/I ⊆ max(X, A)/I +� += A +� 1 +1 +� ++ +� max(X, A) +0 +� +⊆ O(K = 0, I). +The reader may now similarly argue for the correspondence. +We may now easily recover the classical result that any gauge-equivariant +quotient for some (possibly relative) graph algebra remains a graph algebra. +For this we first recall that any quotient correspondence (as kernel component) +arises as a quotient graph (confer example 3.5) +A/K = c0(V )/c0 +� +H = hereditary +� += c0 +� +W = V \ H +� +X/XK = ℓ2(E)/ℓ2(EH) = ℓ2� +F := WE +� +and as such we may replace the original graph by the quotient graph +X = ℓ2(E := F), +A = c0(V := W) +=⇒ +O(K = 0, I). +On the other hand recall that any covariance ideal for a graph (in our case the +quotient graph) arises simply as a regular set of vertices (confer example 3.7): +max(X, A) = c0 +� +regular +� +=⇒ +I = c0 +� +R ⊆ regular +� +. +With this notation in mind we may now find the canonical dilation as a graph. +We note for this that as it was given by the algorithm above we may re- +cover the canonical dilation as a combinatorial object from the original data, +which boils down in our case to the canonical dilation arising as a graph: +Corollary 9.4 (Katsura dilation: graph correspondences): +The canonical dilation given by the maximal covariance as in (25) +� +XB = X + X +� max(X, A) +0 +� ��� B = A + +� max(X, A) +0 +� � +arises as the following canonical graph (with notation from above): +Its coefficient algebra arises as the orthogonal sum of vertices +W = singular +� 1 +1 +� +∪ +� 0 +regular +� +∪ +� regular \ R +0 +� +80 + +9 +Pimsner dilations +Alexander Frei +together with the correspondence given by the graph +EW = E +� +singular +� 1 +1 +� +∪ +� 0 +regular +� +∪ +� regular \ R +0 +� � +and its left action given by (whence defining the range of edges) +� a +a +� +EW = (aE)W +and +� 0 +b +� +EW = +� b +b +� +EW = (bE)W +while trivially acting for the left over last summand. +As such the graph reads in more classical terms +W = +� +all vertices = singular ∪ regular +� +∐ +� +regular \ R +� +EW = E +� +all vertices +� +∐ E +� +regular \ R +� +with range and source map given by +s(− ∐ ∅) = s(−) ∐ ∅ +and +s(∅ ∐ −) = ∅ ∐ s(−), +r(− ∐ ∅) = r(−) ∐ ∅ = r(∅ ∐ −). +Note that the combined summands basically recover the original graph whereas +the last provides an additional copy to make up for the maximal covariance. +On the other hand the edges all point into the original copy of vertices. +As such this recovers the familiar construction for graph algebras: Any gauge- +equivariant quotient arises as a graph algebra itself. +Proof. In order to find the canonical dilation as a graph it suffices to recover +its coefficient algebra as an orthogonal sum of vertices (see example 1.2). +In our case the coefficient algebra already reads as a sum of vertices +A +� 1 +1 +� ++ +� max(X, A) +0 +� += c0 +� +vertices +�� 1 +1 +� ++ +� +c0 +� +regular +� +0 +� +which we may decompose now further into an orthogonal sum: +First the singular vertices (that is the nonregular ones) are trivially disjoint +from the regular ones and as such define an orthogonal summand, +vertices = singular ∪ regular : +singular +� 1 +1 +� +⊥ +� regular +regular +� +. +On the other hand the sum on regular vertices may be also taken as +regular +� 1 +1 +� ++ +� regular +0 +� += +� 0 +regular +� ++ +� regular +0 +� +81 + +9 +Pimsner dilations +Alexander Frei +whose summands belong to the relative Cuntz–Pimsner algebra since +τ(XX∗) = +� 0 +XX∗ +� +and +(ϕ − τ) +� +A ∩ XX∗� += +� A ∩ XX∗ +0 +� +which is available only for the compactly acting portion! The latter summand +vanishes precisely for the given covariance (see theorem 7.4 or corollary 9.3) +� regular \ R +0 +� +̸= 0 +and +� R +0 +� += 0 +and as such the sum reduces to the non-zero vertices +� 0 +regular +� ++ +� regular +0 +� += +� 0 +regular +� ++ +� regular \ R +0 +� +. +As such the coefficient algebra for our dilation decomposes into +vertices +� 1 +1 +� ++ +� regular +0 +� += singular +� 1 +1 +� ++ +� 0 +regular +� ++ +� regular \ R +0 +� +and so we have found the vertices for our graph correspondence. +We may now simply read off the edges from our correspondence as +X + X +� max(X, A) +0 +� += ℓ2(E) +� +singular +� 1 +1 +� ++ +� 0 +regular +� ++ +� regular \ R +0 +� � +with the source of edges as evident. On the other hand its left action reads +(using the induced morphism on compact operators from 2.1) +� a +a +� +EW = ϕ(a)τ(E)W = τ(aE)W = (aE)W +� 0 +b +� +EW = τ(b)τ(E)W = τ(bE)W = (bE)W +� c +0 +� +EW = (ϕ − τ)cτ(E)W = 0 +and so we have found the desired graph. +We illustrate the canonical dilation for the following prominent graph: +Example 9.5 (Katsura dilation: Toeplitz graph): +Consider the correspondence given by the single loop +� +X = ℓ2 +� +a +x � += Cx +����� A = c0 +� +vertices +� += Ca +� +. +82 + +9 +Pimsner dilations +Alexander Frei +Its Toeplitz algebra recovers the traditional Toeplitz algebra since +T (X, A) = C∗� +x ∪ a +��� x∗x = a, ax = x +� += C∗� +x∗x = 1 +� += T +and as such its suggestive name as Toeplitz graph. On the other hand its +absolute Cuntz–Pimsner algebra recovers the traditional circle algebra +O(X, A) = C∗� +x ∪ a +��� x∗x = a = xx∗ � += C∗� +x∗x = 1 = xx∗ � += C(T). +These already define all its gauge-equivariant representations: +Indeed there is only a single covariance ideal given by the single vertex +max(X, A) = c0 +� +regular = a = vertices +� += A +and no further quotient graph (except the trivial one). +As such we found the entire lattice of gauge-equivariant representations. +Further we may now compute the graph for the canonical dilation: +For this we may now simply read off the graph as (confer corollary 9.4) +� +W = (a) ∐ (a \ ∅) = a ∐ a +��� EW = (xa) ∐ (xa) = x ∐ x +� +: + + +s(x ∐ ∅) = a ∐ ∅, s(∅ ∐ x) = ∅ ∐ a +r(x ∐ ∅) = a ∐ ∅ = s(∅ ∐ x) + + =⇒ + + + + +a ∐ ∅ +x ∐ ∅ +∅ ∐ a +∅ ∐ x + + + +. +Put together we found the realisation for the Toeplitz algebra: +T + + +• + + = O + + +• +• + + +The latter appears sometimes as well under the name Toeplitz graph. +We now continue with the issue about the existence of minimal dilations. +For this we begin with the following positive result for relative graph algebras: +Corollary 9.6 (Katsura dilation: minimal dilation). For graph correspondences +the canonical dilation is minimal. That is roughly speaking, there is no smaller +graph which realises the relative graph algebra as an absolute one. +Proof. Consider a subalgebra detecting each covariance B ⊆ c0(regular): +B ∩ c0 +� +S ⊆ regular +� += 0 +=⇒ +c0 +� +S ⊆ regular +� += 0. +83 + +9 +Pimsner dilations +Alexander Frei +Then the subalgebra necessarily contains each summand and so also +c0 +� +S = {r} +� += C(r) ⊆ B +=⇒ +B = +� +rC(r) = c0 +� +regular +� +. +As such the maximal covariance defines a minimal dilation. +The previous observation suggests now the following result for correspon- +dences over spaces (as coefficient algebra) and in particular for integer actions: +Corollary 9.7 (Katsura dilation: correspondences over spaces): +It holds for correspondences with abelian maximal covariance (and so also for +correspondences over spaces): The canonical dilation given by the maximal co- +variance defines a minimal dilation if and only if the maximal covariance has +discrete spectrum (resp. the maximal covariance defines a discrete subspace). +Maximal covariances with totally disconnected spectra are not sufficient. +Proof. Note that the entire problem deals with an abelian operator algebra. +By Stone–Weierstrass it thus suffices to focus on any ideal over some point in +the spectrum and any subalgebra over some pair of points of the form +ker +� +ω : B → C +� += +� +b(ω) = 0 +� +and +eq +� +ωi : B ⇒ C +� += +� +b(ω1) = b(ω2) +� +. +For the ideal we first observe for a pair of closed subspaces E, F ⊆ X = ΓB: +ker(E) ∩ ker(F) = ker(E ∪ F) = 0 +⇐⇒ +E ∪ F = X. +As such the only ideals some non-isolated point cannot detect are +ker(ω) ∩ ker(F) = 0 : +ker +� +F = X − ω = X +� += 0 +whence the non-isolated point would already detect every ideal. +Suppose on the other hand that the spectrum defines a discrete space. +Then every subalgebra as above fails detect even two ideals: +eq +� +ω1, ω2 +� +∩ ker(X − ωi) = +� +b(X − ωi) = 0 = b(ωi) +� += 0 +The corollary now follows from combining these two observations. +We leave the final statement about totally disconnected spaces to the reader. +With the previous corollary in mind, we may now illustrate some negative +examples on the existence of minimal dilations altogether: +84 + +9 +Pimsner dilations +Alexander Frei +Example 9.8 (Minimal dilations: failures of existence). For this we may now +simply consider any integer action as our correspondence, +Z ↷ C0(space) +=⇒ +X = C0(space) = A. +Note that any integer action defines a regular correspondence and as such the +coefficient algebra defines the maximal covariance. Consider now some contin- +uous space such as for instance the real line for which we thus obtain: +a) The canonical dilation fails to be minimal as for instance the following +ideals already detect each covariance +C0(R − r) = ker(r) ⊆ C0(R) : +C0(R − r) ∩ I = 0 +=⇒ +I = 0. +b) On the other hand none of those is minimal either since furthermore detection +of covariance already happens for those with discrete complement such as for +C0 +� +R − {r1, . . . , rn} +� += ker +� +r1 ∪ . . . ∪ rn +� +⊆ C0(R). +Consider now some enumeration of rational numbers {q1, q2, . . .} = Q. +As such we obtain some decreasing sequence of ideals with detection: +ker(q1) ⊇ ker(q1 ∪ q2) ⊇ . . . ⊇ ker(q1 ∪ . . . ∪ qn) ⊇ . . . +Their intersection however fails to detect any covariance since +ker +� +Q = {q1, q2, . . .} +� += ker +� +{q1, q2, . . .} = R +� += 0. +As such the given sequence of ideals admits no minimal dilation. +In fact we have even found that the axiom of choice fails to apply! +Acknowledgements +The author would like to thank his supervisor Søren Eilers for his kind support +and encouragements, as well as Evgenios Kakariadis for pointing out to [Kat07] +as a valuable follow-up article by Takeshi Katsura, which inspired the author +on finding the missing components. +Moreover, the author acknowledges the +support under the Marie–Curie Doctoral Fellowship No. 801199. +85 + +References +Alexander Frei +References +[Bla96] +Etienne Blanchard. D´eformations de C∗-alg`ebres de Hopf. Bulletin +de la Soci´et´e Math´ematique de France, 124(1):141–215, 1996. +[Cun77] +Joachim Cuntz. Simple C∗-algebras generated by isometries. Com- +munications in Mathematical Physics, 57(2):173 – 185, 1977. +[Exe94] +Ruy Exel. Circle Actions on C∗-Algebras, Partial Automorphisms, +and a Generalized Pimsner-Voiculescu Exact Sequence. Journal of +Functional Analysis, 122(2):361–401, 1994. +[Exe11] +Ruy Exel. Noncommutative Cartan subalgebras of C∗-algebras. New +York Journal of Mathematics, 17:331–382, 2011. +[FMR03] Neal Fowler, Paul Muhly, and Iain Raeburn. +Representations of +Cuntz-Pimsner algebras. Indiana University Mathematics Journal, +52(3):569–605, 2003. +[Fre64] +Peter Freyd. Abelian Categories. Harper and Row and John Weath- +erhill, 1964. +[Kak16] +Evgenios Kakariadis. +A note on the gauge invariant uniqueness +theorem for C∗-correspondences. +Israel Journal of Mathematics, +215(2):513—-521, Sep 2016. +[Kat04] +Takeshi +Katsura. +On +C∗-algebras +associated +with +C∗- +correspondences. +Journal of Functional Analysis, 217(2):366–401, +2004. +[Kat07] +Takeshi Katsura. Ideal structure of C∗-algebras associated with C∗- +correspondences. +Pacific Journal of Mathematics, 230(1):107–145, +2007. +[KPQ12] Steven Kaliszewski, Nura Patani, and John Quigg. +Characteriz- +ing graph C∗-correspondences. +Houston Journal of Mathematics, +38(3):751–759, 2012. +[KPW98] Tsuyoshi Kajiwara, Claudia Pinzari, and Yasuo Watatani. +Ideal +Structure and Simplicity of the C∗-Algebras Generated by Hilbert +Bimodules. Journal of Functional Analysis, 159(2):295–322, 1998. +[Lan78] +Saunders Mac Lane. Categories for the Working Mathematician, vol- +ume 5 of Graduate Texts in Mathematics. Springer New York, second +edition edition, 1978. +86 + +References +Alexander Frei +[Lan95] +Christopher Lance. Hilbert C∗-Modules: A Toolkit for Operator Al- +gebraists, volume 210 of London Mathematical Society Lecture Note +Series. Cambridge University Press, 1995. +[MPT08] Paul Muhly, James Pask, and Mark Tomforde. Strong shift equiva- +lence of c*-correspondences. Israel Journal of Mathematics, 167:315– +346, 2008. +[MS98] +Paul +Muhly +and +Baruch +Solel. +Tensor +algebras +over +C∗- +correspondences: Representations, dilations, and C∗-envelopes. Jour- +nal of Functional Analysis, 158(2):389–457, 1998. +[Pim97] +Michael +Pimsner. +A +class +of +C∗-algebras generalizing +both +Cuntz–Krieger algebras and crossed products by Z. In Dan-Virgil +Voiculescu, editor, Free Probability Theory, volume 12 of Fields In- +stitute Communications, pages 189–212, 1997. +87 + diff --git a/8NE1T4oBgHgl3EQfTwPq/content/tmp_files/load_file.txt b/8NE1T4oBgHgl3EQfTwPq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae95692fc3b1fe56fd5159025faa63bd8488bb39 --- /dev/null +++ b/8NE1T4oBgHgl3EQfTwPq/content/tmp_files/load_file.txt @@ -0,0 +1,1705 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf,len=1704 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='03083v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='OA] 8 Jan 2023 Relative Cuntz–Pimsner algebras: Classification of gauge-invariant ideals: a simple and complete picture Alexander Frei January 10, 2023 We give a simple and complete picture on the classification of relative Cuntz–Pimsner algebras (and so also of gauge–equivariant representations) us- ing their intuitive parametrisation by kernel–covariance pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we first present a classification of kernel and cokernel mor- phisms (in the general category of correspondences) which builds on the con- cept of invariant ideals as realised and coined by Kajiwara–Pinzari–Watatani (and implicitely by Pimsner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The existence of all such kernel and cokernel morphisms then enable us to reduce the general classification problem to the faithful case of correspondences within ambient operator algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The second component arises from an observation made by Katsura: We unravel Katsura’s observation as an obstruction on the range of covari- ance ideals for correspondences embedded in ambient operator algebras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' which comprises the second component of kernel–covariance pairs: As such our para- metrisation runs over invariant ideals (as a discrete range of kernel ideals) and on the other hand over bounded ideals below some maximal covariance (as an upper bound on the range of covariance ideals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We then illustrate the lattice of relative Cuntz–Pimsner algebras (and so also of every gauge–equivariant representation) along the range of kernel–covariance pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Following, we provide the general version of the gauge-invariant uniqueness theorem by its reduction to the faithful case, for which we further recall a simplified proof by Evgenios Kakariadis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This establishes the first half in our classification: Every gauge–equivariant representation arises as a relative Cuntz–Pimsner algebra (for its own kernel– covariance pair) and whence the class of gauge–equivariant representations co- incides with the class of relative Cuntz–Pimsner algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the kernel– covariance pairs exhaust the gauge–equivariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 1 1 Correspondences Alexander Frei For the second half in our classification we aim to uniquely determine the relative Cuntz–Pimsner algebras by their parametrising kernel–covariance pairs: More precisely, we will recover every abstract kernel–covariance pair as the actual kernel and covariance from its relative Cuntz–Pimsner algebra, and as such our kernel–covariance pairs are also classifying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With our found classification by kernel–covariance pairs we then fur- ther investigate the lattice structure of gauge–equivariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In particular, we elaborate the existence of connecting morphisms between cokernel strands (given by the kernel component of kernel–covariance pairs) and illustrate our results on examples of graph algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Along this discussion we further clarify Katsura’s description (using T-pairs) as a simple translation of kernel–covariance pairs (which had been already cov- ered by Katsura himself) with the second component however not taken and further pursued as describing the range of covariance ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Altogether our classification is a simple reduction of the gauge– invariant uniqueness theorem (along cokernel morphisms) together with the identification of kernel and covariance for relative Cuntz–Pimsner algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Finally we provide a realisation of relative Cuntz–Pimsner algebras as absolute Cuntz–Pimsner algebra by maximal dilation and reveal Katsura’s construction as the canonical dilation given by the maximal covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We further reveal the canonical dilation as a familiar construction from graph algebras and provide an example to illustrate the lack of minimal dilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As an application we provide a systematic approach for an earlier pullback result by Robertson–Szymanski which we extend to the general context in up- coming work with Mariusz Tobolski and Piotr M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Hajac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 1 Correspondences We begin with an introduction to correspondences and their representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In particular, we provide a less formal and more abstract angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This seeks to help to understand their gauge-equivariant representations from an abstract perpsective — and so also to classify the entire lattice of gauge-invariant ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Further, this allows one to better understand dilations and in further work the shift equivalence problem from an abstract angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A Hilbert module is a right module over an operator algebra (seen as coefficient algebra) that comes equipped with a pairing (compatible with the coefficient algebra) which renders the right module complete with respect to the induced norm: ⟨−|−⟩ : X × X → A : ∥x∥2 := ∥ ⟨x|x⟩ ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 2 1 Correspondences Alexander Frei Given a pair of Hilbert modules over a common coefficient algebra one may introduce the notion of adjointable operators as those which admit an adjoint: T : X → Y, T ∗ : Y → X : ⟨T −|−⟩ = ⟨−|T ∗−⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We note that such adjointable operators are automatically continuous which may be seen most easily via the closed graph theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Moreover the class of adjointable operators over pairs of Hilbert modules defines a “consistent system” of Banach spaces with composition and involution L(Y |Z) ◦ L(X|Y ) ⊆ L(X|Z), L(X|Y )∗ = L(Y |X) satisfying the generalized C∗-identity T ∈ L(X|Y ) : ∥T ∥2 = ∥T ∗T ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The vertical separators hereby seek to convey the “Dirac bra–ket notation”, that is we have the following identification which we term Dirac calculus: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The identification (and its conjugate) X = |X⟩ ⊆ L(A|X), X∗ = ⟨X| ⊆ L(X|A) : |x⟩ a := |xa⟩ , ⟨x| y := ⟨x|y⟩ : ⟨x|∗ = |x⟩ define an isometric embedding (and its conjugate), which renders the pairing as ⟨x|y⟩ = ⟨x| ◦ |y⟩ ∈ L(X|A) ◦ L(A|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This further renders the notion of compact operators as K(X, Y ) := span |Y ⟩ ⟨X| ⊆ L(X|Y ) and the notion of adjointable operators as T x = T ◦ |x⟩ , ⟨T x|y⟩ = ⟨x| ◦ T ∗ ◦ |y⟩ = ⟨x|T ∗y⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' All of the above enables one to split expressions as composition of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We first note that the coefficient algebra itself defines a Hilbert module: ⟨−|−⟩ : A × A → A : ⟨x|y⟩ = x∗y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 3 1 Correspondences Alexander Frei One may now verify that the assignments define mutual adjoints ⟨x|∗ = |x⟩ ∈ L(A|X) : � ⟨x|y⟩ ��a � = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' = � y ��xa � = � y �� |x⟩ a � and that the identification defines an isometric embedding X = |X⟩ ⊆ L(A|X) : ∥ |x⟩ ∥2 = ∥ ⟨x| ◦ |x⟩ ∥ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' = ∥ ⟨x|x⟩ ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We leave these as an instructive exercise for the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this identification at hand, a Hilbert module conveniently reads as nothing but a right module together with an abstract pairing given by involution (X ↶ A) XA ⊆ X, (X × X → A) X∗X ⊆ A which we from now on simply indicate as such formal inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We may now turn our attention to the notion of C∗-correspondence: Formally these are given as Hilbert modules together with a representation of the co- efficient algebra as adjointable operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With the viewpoint from above we however obtain the alternative description as a bimodule over the coefficient algebra together with some compatible inner product pairing AX ⊆ X, XA ⊆ X, X∗X ⊆ A (1) satisfying some relations such as (now evident in Dirac formalism) (ax)∗y = x∗(a∗y), x∗(ya) = (x∗y)a, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We meanwhile note that the notion of correspondences has an intrinsic asym- metry by the pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' More precisely, the more traditional notion of Hilbert modules comes equipped with a dual pairing which renders the notion symmet- ric: X∗X ⊆ A, XA ⊆ X, XX∗ ⊆ A, AX ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In fact, this covers the main objective of covariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We return to this aspect in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We continue with a swift introduction to the (internal) tensor product of correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Using our Dirac formalism from above we may now simply introduce those as formal powers such as A(XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Z) ⊆ (AX)Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Z, (XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Z)A ⊆ XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (ZA) 4 1 Correspondences Alexander Frei and where the inner product pairing now simply reads (XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Z)∗(XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Z) = Z∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Y ∗� X∗X ⊆ A � Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Z ⊆ Z∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � Y ∗Y ⊆ B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Z ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ⊆ � Z∗Z ⊆ A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Moreover, this automatically entails the balanced relation as for example � (xa)y − x(ay) �∗� (xa)y − x(ay) � = = y∗(a∗x∗)(xa)y − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + (y∗a∗)(x∗x)(ay) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Let us give an illustrative example for such tensor products: Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2 (Graph correspondences: Tensor powers): Consider a directed graph and regard its graph correspondence X = ℓ2� E = edges � , A = c0 � V = vertices � with action and pairing given by range and source say a b z w : z∗z = a, za = z = bz, z∗z = b, wb = w = bw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand we know by [KPQ12] that every nondegenerate correspon- dence over a “direct sum over a discrete set as vertices” arises as a graph correspondence: A = c0 � vertices = some set � =⇒ X = ℓ2(edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Indeed this follows by some simple counting argument: a, b ∈ vertices : bXa = ℓ2� edges : b ← a � = ℓ2(bEa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' From this argument it furthermore follows that degenerate correspondences arise the same way when allowing for edges with heads pointing into the void such as a b z w and furthermore also any power of a graph correspondence arises as a graph 5 1 Correspondences Alexander Frei correspondence itself and indeed is given by its paths of according length: XX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' X = ℓ2(EE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' E), A = c0(vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In other words, that is by concatenation of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the tensor product may be seen as nothing but a formal power in words and one may equivalently consider also mixed powers with the dual space and its dual pairing (though formally only in the context of operator spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We finish this section with an intrinsic characterization of Hilbert bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we first require the following well-known observation: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Consider a correspondence (in Dirac braket notation) ⟨X|X⟩ ⊆ A, XA ⊆ X, AX ⊆ A and regard the orthogonal complement for the kernel (by left action), ker(A ↷ X) = {aX = 0} ⊆ A : ker(A ↷ X)⊥ = {a ker(A ↷ X) = 0} ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This defines the largest ideal that renders the left action faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In particular one may simultaneously identify the orthogonal complement with its isometric image (by left action) within the space of adjointable operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Before we begin with the actual argument let us make the useful obser- vation that the kernel defines an ideal (closed and two-sided) since A ker(A ↷ X)X = 0, ker(A ↷ X)AX = 0, ker(A ↷ X)X ⊆ ker(A ↷ X)X = 0 and so also a selfadjoint one (which one may also easily observe by hand) � X ��� ker(A ↷ X)∗X � = � ker(A ↷ X)X ���X � = 0 =⇒ ker(A ↷ X)∗X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This kernel determines now precisely those ideals that render the action faitfhul: K ker(A ↷ X) = K ∩ ker(A ↷ X) = 0 ⇐⇒ K ↾ X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such its orthogonal complement defines the largest such ideal: K ker(A ↷ X) = 0 ⇐⇒ K ⊆ ker(A ↷ X)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 6 1 Correspondences Alexander Frei Note that the orthogonal complement (as we have defined from the left only) defines itself an ideal (closed and two-sided) A ker(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' )⊥ ker(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') = 0, ker(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' )⊥A ker(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') = 0, ker(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' )⊥ ker(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') ⊆ ker(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' )⊥ ker(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' )0 and so also a selfadjoint one (which this time is not too obvious).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Building on this, we may now further give an intrinsic characterization of Hilbert bimodules (which we promote as theorem since it’s widely non-standard even until now) and the author would like to thank Adam Skalski and Bar- tosz Kwasniewski for helping with the key detail for this idea: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Consider a correspondence as seen in (1) and regard the orthogonal complement from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3 ker(A ↷ X)⊥ = {a ker(A ↷ X) = 0} ⊆ A and simultaneously identified as in that proposition by their action A ⊇ ker(A ↷ X)⊥ = � ker(A ↷ X)⊥ ↷ X � ⊆ L(X|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Then the correspondence defines a Hilbert bimodule if and only if the compact operators (see Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1) all lie within the orthogonal complement |X⟩⟨X| ⊆ ker(A ↷ X)⊥ ⊆ L(X|X) while the dual pairing is always given by |−⟩⟨−| : X × X → A : |X⟩⟨X| ⊆ ker(A ↷ X)⊥ ⊆ A whence there is no ambiguity left anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the notion of Hilbert bi- modules defines a pure property without any additional structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Before we begin with the proof, we note that the situation above may be easiest pictured in mind (and remembered) by the following illustration: 7 1 Correspondences Alexander Frei A ker(A ↷ X)⊥ L(X|X) |X⟩ ⟨X| As such a correspondence in general may be seen as a Hilbert bimodule with a partial dual pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this picture in mind let us get to the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Instead of Hilbert bimodules, it suffices to consider the case of Hilbert modules (meaning the case of a single action and pairing) say given by ⟨X|X⟩ ⊆ A, XA ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The kernel for the single action (in this case from the right) agrees with the orthgonal complement for the pairing (and so its support ideal): ker(X ↶ A) = ⟨X|X⟩⊥ � = supp(X)⊥� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Indeed using Blanchard factorization [Bla96, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3] this immediately fol- lows from ⟨X|X⟩ a = 0 ⇐⇒ � |X⟩ = |X⟩ ⟨X|X⟩ � a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In turn this observation tells us where to search for the pairing: ⟨X|X⟩ ⊆ ⟨X|X⟩⊥⊥ = ker(X ↶ A)⊥ Replacing the right action from our consideration above by the left action from the proposition we thus just revealed the condition from the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Con- versely, given the condition from the proposition we may simply retrieve the dual pairing using the isometric image of the orthogonal complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Using Dirac calculus from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1 we finally obtain (simply as composition) |x⟩ ◦ � ⟨y| ◦ |z⟩ � = � |x⟩ ◦ ⟨y| � |z⟩ and so the traditional compatibility (between pairings) holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 8 2 Representations Alexander Frei 2 Representations Consider an abstract correspondence as introduced in the first section: X∗X ⊆ A, XA ⊆ X, AX ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This structure is in some sense freely floating, and so we wish to embed this structure as a whole into an ambient operator algebra as illustrated: X A X∗ B X A X∗ A good analogy here is the embedding of Fell bundles into any crossed prod- uct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This is what we understand as a representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' More precisely, that is a representation of both the correspondence and the coefficient algebra into some ambient operator algebra say (X, A) B : τ : X → B, ϕ : A → B so the former being a morphism of vector spaces and the latter a morphism of operator algebras and such that the pair is coherent with the structure in between, which now reads in Dirac formalism (see Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1): ϕ(x∗y) = τ(x)∗τ(y), τ(xa) = τ(x)ϕ(a), τ(ax) = ϕ(a)τ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' It is well-known that the latter follows automatically from the former two: In- deed using the C∗-identity we find (written in Dirac formalism) � ϕ(a)τ(x) − τ(ax) �∗� ϕ(a)τ(x) − τ(ax) � = = ϕ(a)∗τ(x)∗τ(x)ϕ(a) − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + τ(ax)∗τ(ax) = ϕ(x∗a∗ax) − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + ϕ(x∗a∗ax) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Consider now the structure as embedded within the ambient operator algebra X = τ(X) ⊆ B, A = ϕ(A) ⊆ B 9 2 Representations Alexander Frei then we may also view those as subspace and subalgebra which renders the abstract inclusions from the previous section into actual inclusions X∗X ⊆ A ⊆ B, XA ⊆ X ⊆ B, AX ⊆ X ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We note that while these representations are evidently not faitful in general, one may always pass to its quotient correspondence which renders the representation faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This will define the first parameter for the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We now wish to extend a representation to higher and mixed tensor powers: For this we recall the following result by Kajiwara–Pinzari–Watatani: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1 ([KPW98, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Representations canonically extend to the tensor product and compact operators (denoted in Dirac formalism): � τ : X → Y ϕ : A → B � =⇒ � XX → B : XX∗ → B : τ(xy) := τ(x)τ(y) τ(xy∗) := τ(x)τ(y)∗ � The latter further satisfies the relations KL ∈ (XX∗)(XX∗) ⊆ (XX∗) : aKb ∈ A(XX∗)A ⊆ (XX∗) : τ(K)τ(L) = τ(KL) ϕ(a)τ(K)ϕ(b) = τ(aKb) and so defines in particular a morphism of operator algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Furthermore, suppose the morphism is isometric on the coefficient algebra then so also on the correspondence and on compact operators: ∥(A → B)−∥ = ∥−∥ =⇒ ∥(X → B)−∥ = ∥−∥, ∥(A → B)−∥ = ∥−∥ =⇒ ∥(XX∗ → B)−∥ = ∥−∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' By iteration the proposition includes all higher and any other mixed powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We provide the extension to compact operators since it demonstrates the use of Dirac calculus with a neat trick by Kajiwara–Pinzari–Watatani: We need to verify that the formal linear assignment on elementary compact operators remains bounded �����τ �� n xny∗ n ������ ≤ ����� � n xny∗ n �����, ∀xn, yn ∈ X, whence the assignment allows an (a posteriori well-defined) extension to the completion of all compact operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we invoke matrix calculus to 10 2 Representations Alexander Frei reformulate the linear sum as a product of matrices (x)(y∗) := � n xny∗ n = � x1 · · xN � \uf8eb \uf8ec \uf8ec \uf8ed y∗ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' y∗ N \uf8f6 \uf8f7 \uf8f7 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Reformulated, the quite clever trick by Kajiwara Pinzari and Watatani is now to use the C∗-identity (generalized to matrices of adjointable operators): ∥xy∗∥2 = ���(xy∗)(yx∗) = x(y∗y)x ��� = ���x√y∗y ��� = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' = ��� √ x∗x√y∗y ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Note that this automatically invokes matrix inflations since for example x∗x = \uf8eb \uf8ec \uf8ec \uf8ed ⟨x1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ⟨xN| \uf8f6 \uf8f7 \uf8f7 \uf8f8 � |x1⟩ · · |xN⟩ � = \uf8eb \uf8ec \uf8ec \uf8ed ⟨x1|x1⟩ · · ⟨x1|xN⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ⟨xN|x1⟩ · · ⟨xN|xN⟩ \uf8f6 \uf8f7 \uf8f7 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand, this equally applies when invoking the representation and so we obtain the desired bound ∥τ(xy∗)∥ = ���ϕ �√ x∗x√y∗y ���� ≤ ��� √ x∗x√y∗y ��� = ∥xy∗∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In turn this trick also applies for the correspondence and tensor product X → B : ∥τ(x)∥ = ∥ϕ(x∗x)∥ ≤ ∥x∗x∥ = ∥x∥, XX → B : ∥τ(xy)∥ = ∥ϕ(y∗x∗xy)∥ ≤ ∥y∗x∗xy∥ = ∥xy∗∥ as well as on every other type of tensor product such as of operator algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Meanwhile, this moreover infers that isometricity passes from the coefficient algebra to the correspondence (and tensor product) and to compact operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Moreover, the morphism satisfies the relations on compact operators since one easily verifies on elementary compact operators: ϕ(a)τ(xy∗)ϕ(b) = ϕ(a)τ(x)τ(y∗)ϕ(b) = τ(axy∗b), τ(xy∗)τ(zw∗) = τ(x)ϕ(y∗z)τ(w∗) = τ(xy∗zw∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We therefore found the desired relations and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this at hand we may now introduce the notion of covariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We begin for this with the observation that (as in the previous proposition) we could have 11 2 Representations Alexander Frei also defined the morphism (somewhat senseless) A ⊇ ⟨X|X⟩ → B : τ(x∗y) := τ(x)∗τ(y) which agrees with the oringal one on the coefficient algebra (somewhat trivially): ⟨X|X⟩ B A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' τ(−) = ϕ(−) Suppose on the other hand that some elements also act as compact operators: A ∩ |X⟩⟨X| := � a ∈ A ��� a ∈ |X⟩⟨X| � ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Then there is no evidence to believe that the induced morphism from the pre- vious proposition would agree with the morphism for the algebra: A ∩ |X⟩⟨X| B |X⟩⟨X| B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' τ(−) ̸= ϕ(−) Nevertheless the difference of morphisms surprisingly defines a morphism of operator algebras since (using the property from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1) (ϕ − τ)a(ϕ − τ)b = ϕ(a)ϕ(b) − ϕ(a)τ(b) − τ(a)ϕ(b) + τ(a)τ(b) = ϕ(ab) − τ(ab) − τ(ab) + τ(ab) = ϕ(ab) − τ(ab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' and so our representation decomposes into the sum of morphisms � A ∩ |X⟩⟨X| B � = (2) = \uf8eb \uf8ec \uf8ec \uf8ed A ∩ |X⟩⟨X| |X⟩⟨X| B \uf8f6 \uf8f7 \uf8f7 \uf8f8 + \uf8eb \uf8ec \uf8ec \uf8ed A ∩ |X⟩⟨X| B B − A ∩ |X⟩⟨X| |X⟩⟨X| B \uf8f6 \uf8f7 \uf8f7 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we may capture the domain of equality by the covariance ideal: cov � (X, A) → B � := ker � ′′ � ⊴ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (3) 12 3 Kernel and Covariance Alexander Frei We will often drop the dependency on the coefficient algebra for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The covariance and kernel will classify the gauge-equivariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Let us thus take a closer look at kernel morphisms and possible covariances: 3 Kernel and Covariance We begin this section with a characterization of kernel morphisms in the cat- egory of correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this let us introduce the general notion of mor- phisms between correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As for representations, these are given by a pair of morphisms on the correspondence and the coefficient algebra (X, A) (Y, B) : τ : X → Y, ϕ : A → B where the former defines a linear morphism and the latter a morphism of oper- ator algebras and such that the pair is coherent with the structure in between, which conveniently reads in Dirac formalism: ϕ(x∗y) = τ(x)∗τ(y), τ(ax) = ϕ(a)τ(x), τ(xa) = τ(x)ϕ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' While the resulting category fails to be abelian (basically due to the algebraic morphism on the coefficient algebra) it still possesses all kernels and cokernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As the notion of kernels and cokernels is nonstandard however in categories beyond abelian ones, let us give a quick introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Our catgory of corres- pondences and their morphisms has zero morphisms in the sense: (X, A) (Y, B) (Z, C) 0 = (X, A) (Z, C), 0 (X, A) (Y, B) (Z, C) 0 = (X, A) (Z, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 0 A kernel for a morphism is the universal annihilating morphism: (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') (X, A) (Y, B) ker = 0 That is any other annihilating morphism factors uniquely over the kernel: (X′, A′) (X, A) (Y, B) = 0 =⇒ (X′, A′) (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') (X, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ker 13 3 Kernel and Covariance Alexander Frei Dually one may define the cokernel of morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Moreover, we define a short exact sequence denoted by 0 (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') (X, A) (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') 0 whenever each side is the kernel respectively cokernel of the other: (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') (X, A) = ker � (X, A) (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') � (X, A) (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') = coker � (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') (X, A) � We will fill the question marks in the proposition below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' But before we note the following equivalent notions of invariant and hereditary ideals which date all the way back to Pimsner and as explicitely coined by Kajiwara–Pinzari–Watatani (we refrain from the notion of negatively invariant ideals as we will find those from a different perspective later on): Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1 ([Pim97, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='5 following] and [KPW98, section 4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' see further also [FMR03, section 2] and similarly also [Kat07, section 1]): The notion of invariant and hereditary ideals coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' More precisely, one has the characterization for ideals in the coefficient algebra K ⊴ A, XK = {x ∈ X | X∗x ∈ K} = {x ∈ X | x∗x ∈ K} (4) and so it furthermore holds the equivalence X∗KX ⊆ K ⇐⇒ KX ⊆ XK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (5) The former is generally refered to as invariant and the latter as hereditary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Katsura gives a neat proof based on some factorization result by Lance, more precisely [Lan95, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='4], which itself however still requires some rather technical approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Instead we may verify the equivalence with the following fairly elementary observations: Let us first note that both of the right-hand spaces are automatically linear and closed by Cohen–Hewitt: KX = span KX ⊆ X, XK = span XK ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Now we clearly have the forward inclusions since X∗(XK) = (X∗X)K ⊆ AK ⊆ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 14 3 Kernel and Covariance Alexander Frei Conversely, we have using any approximate identity for the ideal: x∗x ∈ K =⇒ (1 − e)x∗x(1 − e) → 0 =⇒ x = lim e (xe) ∈ XK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With the characterization at hand we further obtain the forward direction X∗KX ⊆ K =⇒ KX ⊆ XK : (X∗K∗)(KX) = X∗KX ⊆ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Alternatively, one may verify the forward direction using Blanchard factoriza- tion (whose original proof is very neat and elementary, see [Bla96, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3]): KX = (KX)(KX)∗(KX) = (KX)(X∗KX) ⊆ (KX)K ⊆ XK This however implicitely invokes the yet technical Cohen–Hewitt (see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' If one wishes to refrain from using Cohen–Hewitt altogether, one may also argue in the following elementwise way (formulated in Dirac notation): K |X⟩ ∋ k |x⟩ = |y⟩ ⟨y|y⟩ =⇒ ⟨y|y⟩3 = ⟨x| k∗k |x⟩ ∈ K =⇒ ⟨y|y⟩ = 3� ⟨x| k∗k |x⟩ ∈ K =⇒ k |x⟩ = |y⟩ ⟨y|y⟩ ∈ XK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' All of these variants for the forward direction have their advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For the converse direction we may simply argue KX ⊆ XK =⇒ X∗KX ⊆ X∗XK ⊆ AK ⊆ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So the notion of invariant and hereditary ideals coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Let us give an example to illuminate the notion of hereditary ideals: Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2 (Graph correspondences: hereditary ideals): Consider a graph correspondence as in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2: X = ℓ2� E = edges � , A = c0 � V = vertices � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Then every ideal corresponds to some collection of vertices K = c0 � S = some vertices � ⊴ c0(V ) = A and hereditary ideals become the hereditary collection of vertices c0(S)ℓ2(E) ⊆ ℓ2(E)c0(S) ⇐⇒ SE ⊆ ES 15 3 Kernel and Covariance Alexander Frei which reads written out in words range(some edge) ∈ S =⇒ source(some edge) ∈ S and whence their name: hereditary ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In order to understand kernel morphisms (and so in turn also cokernel mor- phisms) we make the following observation: Suppose a morphism vanishes on the coefficient algebra, then it does so also on the entire correspondence: ( A B ) = 0 =⇒ ( X Y ) = 0 (6) Indeed one easily verifies (in a way using the C∗-identity) τ(x) = 0 =⇒ ϕ(x∗x) = τ(x)∗τ(x) = 0 =⇒ x∗x = 0 =⇒ x = 0 which is equivalently the commutative diagram (see the previous section) ⟨X|X⟩ 0 A 0 ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' τ(−) = ϕ(−) As such one may already expect the kernel of morphisms to involve the kernel on the coefficient algebra in some crucial way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this in mind, we may now give a new intrinsic characterization of kernel and cokernel morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Meanwhile the author would like to note that the idea to consider concrete kernel correspondences (by invariant ideals) dates back to [Pim97] and their quo- tient correspondence to [KPW98] with their representations already appearing in the proof of [KPW98, theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3] and as explicitely in [FMR03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The author identified those as partial results on the intrinsic characterisation of categorical kernel and cokernel morphisms, which further expanded and completed provide the following entire classification of kernel and cokernel morphisms in the category of correspondences, which we promote as theorem due to its usefulness (also in upcoming work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3 (Classification: kernel and cokernel morphisms): The category of correspondences has all kernel and all cokernel morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' More precisely, they are all of the special form 0 (XK, K) (X, A) � X XK , A K � 0 16 3 Kernel and Covariance Alexander Frei precisely for the invariant (and equivalently hereditary) ideals K ⊴ A : X∗KX ⊆ K ( ⇐⇒ KX ⊆ XK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Conversely, given a morphism its kernel is given by (XK, K) (X, A) (Y, B) ker : K = ker(A → B) ⊴ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As a consequence, its cokernel is given as (within the image) (X, A) (Y, B) � Y Y L, B L � coker : L = � n≥0 ⟨Y n| BAB |Y n⟩ ⊴ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Furthermore, the cokernel morphisms are precisely the surjective mor- phisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Before we begin with the proof for the above proposition let us give a quick clarification on the quotient norm (and a rather quite simplified proof thereof): Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='4 (compare with [FMR03, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1] and [Kat07, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='5]): It holds for quotient correspondences: The norm for Hilbert modules agrees with the norm for quotient Banach spaces, ��� x0 XK ��� 2 Hilbert = inf ∥ ⟨x0|x0⟩ + K∥ = inf ∥x0 + XK∥2 = ��� x0 XK ��� 2 Banach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In particular, the quotient correspondence is complete (by Cohen–Hewitt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As compared to both references, we give a simplified proof: On the one hand we have the obvious inclusion ∥x0 + XK∥2 ⊆ ∥ ⟨x0 + XK|x0 + XK⟩ ∥ ⊆ ∥ ⟨x0|x0⟩ + K∥ and as such the bound (when applying infima) ��� x0 XK ��� 2 Hilbert = inf ∥ ⟨x0|x0⟩ + K∥ ≤ inf ∥x0 + XK∥2 = ��� x0 XK ��� 2 Banach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For the converse implication we may invoke the well-known formula for the quotient norm on operator algebras (the simplifying trick) ���� ⟨x0|x0⟩ K ���� = inf ∥ ⟨x0|x0⟩ + K∥ = lim e→1 ∥(1 − e) ⟨x0|x0⟩ (1 − e)∥ where the limit runs over some or any approximate identity for the ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we obtain the converse containment ∥ ⟨x0 − x0e|x0 − x0e⟩ ∥ = ∥x0 − x0e∥2 ∈ ∥x0 + XK∥2 17 3 Kernel and Covariance Alexander Frei and so also the desired converse bound: ��� x0 XK ��� 2 Banach = inf ∥x0 + XK∥2 ≤ lim e→1 ∥(1 − e) ⟨x0|x0⟩ (1 − e)∥ = ��� x0 XK ��� 2 Hilbert So the desired Hilbert module norm and Banach space norm coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Having clarified the completeness for the quotient correspondence, we may now savely get to the desired kernel and cokernel morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof of theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We begin with the statements about kernel morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We first verify that each morphism admits a (categorical) kernel given by (XK, K) = ker( (X, A) (Y, B) ) : K = ker(A → B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Clearly the kernel annihilates the morphism due to our observation (6): � ker(A → B) → A → B � = 0 =⇒ � X ker(A → B) → X → Y � = 0 Meanwhile, let us also note that any such ideal is invariant: ⟨X| ker(A → B) |X⟩ ⊆ ker(A → B) : (A → B) � ⟨X| ker(A → B) |X⟩ � = = (Y ← X) ⟨X| · (A → B) ker(A → B) · (Y ← X) |X⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Conversely, suppose another morphism annihilates from the left (W, D) (X, A) : (W → X → Y ) = 0, (D → A → B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (The annihilation suffices on coefficient algebras due to our observation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') Trivially, we have the desired inclusion at the level of coefficient algebras: (D → A → B) = 0 =⇒ D → ker(A → B) On the other hand, recall that the induced morphism on the support ideal coincides with the morphism at the level of coefficient algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we necessarily also have an inclusion into the kernel, ⟨W|W⟩ ker(A → B) A D ker(A → B) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 18 3 Kernel and Covariance Alexander Frei We however have the equivalance (using the characterization (4)): W → X ker(A → B) ⇐⇒ ⟨W|W⟩ → ker(A → B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we obtain also the desired inclusion at the level of correspondences, D → ker(A → B) =⇒ W → X ker(A → B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So there exists also a unique factorization over the kernel correspondence above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Every morphism thus admits a kernel given by the above kernel morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We next verify that each such morphism is indeed some kernel, namely the kernel from the short exact sequence in the proposition: 0 (XK, K) (X, A) � X XK , A K � 0 : ker � (X, A) � X XK , A K � � = (XK, K) (X, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this let us first verify that the quotient gives a well-defined correspondence � X XK �∗� X XK � ⊆ � A K � , � X XK �� A K � ⊆ � X XK � , � A K �� X XK � ⊆ � X XK � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In other words we need to verify the relations (XK)∗X ⊆ K, X∗(XK) ⊆ K, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' , (K ⊆ A)X ⊆ (XK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The only nontrivial one here (which is not guaranteed automatically) is the last one which precisely calls for hereditary ideals (equivalently invariant ideals) (K ⊆ A)X = KX ⊆ XK ( ⇐⇒ KX ⊆ XK) and so the quotient gives a well-defined correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Let us now get to its kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We already found the unique (categorical) kernel of morphisms above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the desired equality now easily follows from ker � (X, A) � X XK , A K � � = � X ker � A → A K � , ker � A → A K �� = (XK, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So far about kernel morphisms for correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Let us now get to cokernel morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We first verify that each cokernel is also of the special form as above 0 (XK, K) (X, A) � X XK , A K � 0 : coker � (XK, K) (X, A) � = (X, A) � X XK , A K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 19 3 Kernel and Covariance Alexander Frei Clearly the quotient annihilates the kernel (using our obersation (6)): � K → A → A K � = 0 =⇒ (XK → X → X XK ) = 0 Conversely, given any other annihilating morphism say (X, A) (Y, B) (τ,ϕ) : (XK → X → Y ) = 0, (K → A → B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Then both morphisms factor uniquely over the quotient (as linear maps): 0 (XK, K) (X, A) � X XK , A K � 0 0 0 (Y, B) (Y, B) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (τ,ϕ) So it remains to verify that the factorization defines a morphism of corres- pondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' That however basically follows from the original morphism (using congruence classes): ϕ � (x + XK)∗(y + XK) � = ϕ(x∗y) = τ(x)∗τ(y) = τ(x + XK)∗τ(y + XK), τ � (x + XK)(a + K) � = τ(xa) = τ(x)ϕ(a) = τ(x + XK)ϕ(a + K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Recall here that the coherence with the left action follows automatically, =⇒ τ � (a + K)(x + XK) � = ϕ(a + K)τ(x + XK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So we have found the desired cokernel: coker � (XK, K) (X, A) � = � X XK , A K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We now wish to find the cokernel for general morphisms: coker( (X, A) (Y, B) ) = (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') For this we may now make use of the following relation to our advantage: The kernel and cokernel operator satisfy the Galois connection (verbatim from [Lan78, section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1] and confer further [Fre64, chapter 1 and 2]): ker coker ker = ker, ker ker = 0 = coker coker, coker ker coker = coker .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 20 3 Kernel and Covariance Alexander Frei On the other hand, we have already found the form of kernel morphisms: ker � (Y, B) → (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') � = (Y L, L) → (Y, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' as well as the cokernel for kernel morphisms coker( (Y L, L) (Y, B) ) = (Y, B) � Y Y L, B L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We may thus combine these with the Galois connection between the kernel and cokernel operator to find the necessary shape of cokernel morphisms: (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') = coker( (X, A) (Y, B) ) = = coker ker coker( (X, A) (Y, B) ) = � Y Y L, B L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So we are left with finding the invariant ideal which determines the quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Consider for this the factorization over the image (kernel of cokernel) 0 (X, A) (X, A) 0 0 0 (Y L, L) (Y, B) � Y Y L, B L � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the invariant ideal (which determines the kernel) necessarily contains the image of the coefficient algebra (X, A) (Y L, L) =⇒ (A → L → B) and is necessarily also the smallest such ideal (generated by the image) L = BAB + ⟨Y | BAB |Y ⟩ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' = � n≥0 ⟨Y n| BAB |Y n⟩ ⊴ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Indeed any larger invariant ideal defines nothing but a quotient beyond: 0 (Y L, L) (Y, B) � Y Y L, B L � 0 0 (Y L′, L′) (Y, B) � Y Y L′ , B L′ � 0 So we have also found the cokernel for general morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For the final assertion 21 3 Kernel and Covariance Alexander Frei we note that every cokernel is onto as easily seen from their special form � X X XK 0 � and � A A K 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For the converse consider a surjective morphism (X, A) → (Y, B) : � X Y 0 � and � A B 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Recall from above that the kernel and cokernel operator define a Galois connec- tion and so there is no choice for our morphism (to define a cokernel) than to arise as its own coimage (cokernel of kernel) (X, A) → (Y, B) = coker � some morphism � = coker ker coker � some morphism � = coker ker � (X, A) → (Y, B) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Recall for this the factorisation over its coimage (as outlined before): 0 (XK, K) (X, A) � X XK , A K � 0 0 0 (Y, B) (Y, B) 0 ker coker ker So we need to verify that the factorisation defines an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we obtain as our morphism is onto (on the coefficient algebra): 0 K A A/K 0 0 K A B 0 So the factorisation defines an isomorphism on the level of coefficient algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As a consequence, the factorisation is also faithful on the correspondence (recall for instance proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1): ker � A K B � = 0 =⇒ ker � X XK Y � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand, the factorisation is also onto (for the correspondence) im � X XK Y � = im � X X XK Y � = im � X Y � = Y and as such the factorisation defines an isomrphism as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' That is every surjective morphism defines a cokernel morphism as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 22 3 Kernel and Covariance Alexander Frei Let us identify such quotients in the context of directed graphs: Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='5 (Graph correspondences: quotient graphs): Consider a graph correspondence as in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2 X = ℓ2� E = edges � , A = c0 � ver = vertices � and its hereditary ideals given by hereditary collections (as in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2) K = c0 � S ⊆ vertices � ⊴ A : SE ⊆ ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' It is already clear that its quotients themselves arise as a graph, simply since their coefficient algebras define discrete direct sums of vertices as in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2: 0 K = c0(S) A = c0(V ) A K = c0 � T := V \\ S � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' And indeed we may now simply reveal the quotient as 0 XK = ℓ2(ES) X = ℓ2(E) X XK = ℓ2(T E) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Thus its quotients simply arise as the complementary graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We have now found everything about kernel and cokernel morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this in our toolbox, we now proceed to covariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we note that we may render any representation faithful: Indeed we may simply factor any representation over the quotient correspondence 0 (XK, K) (X, A) � X XK , A K � 0 0 0 (B, B) (B, B) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ker The resulting factorization is faithful (isometric) on the coefficient algebra and as such also on the entire correspondence (see proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1): K = ker(A → B) : ker � A K → B � = 0 =⇒ ker � X XK → Y � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So we may always first pass to the (unique) quotient correspondence to ren- der any representation faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This is the first step in the classification of gauge-equivariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The second step is to understand the possi- bly occuring covariances in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This is captured as an important observation 23 3 Kernel and Covariance Alexander Frei by Katsura below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this let us recall Katsura’s ideal max(X, A) = ker(A ↷ X)⊥ ∩ XX∗ ⊴ A (7) which we denoted as maximal ideal for reasons which will become clear in the result below, and we will often drop the dependence on the coefficient algebra for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Note also that we have already encountered this ideal in different context on Hilbert bimodules (see Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this ideal in mind let us get to Katsura’s observation (which we split as a result on faithful representations and further as one on faithful morphisms in general): Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='6 (First part of [Kat04, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3]): Consider an embedding into some ambient operator algebra (X, A) ⊆ B : A ⊆ B ( =⇒ X ⊆ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Then its covariance (3) lies perpendicular to the kernel cov( (X, A) ⊆ B ) ⊥ ker(A ↷ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In particular it holds for general representations (and factored as above) (X, A) � X XK , A K � B : 0 ⊆ cov � � X XK , A K � B � ⊆ max � X XK , A K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So the range of possible covariances is bounded from above by Katsura’s ideal, whence its name and notation as maximal ideal in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We recast the arguments from Katsura in our language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this let us first recall the commutative diagram for the covariance ideal, cov(X → B) ⊆ A B |X⟩⟨X| B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In our case the representation is also faithful (all the horizontal paths): A ⊆ B, X ⊆ B, XX∗ ⊆ B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand, the covariance ideal defines an ideal in the coefficient algebra 24 3 Kernel and Covariance Alexander Frei (as we have observed in the previous section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we also have ker(A ↷ X) cov(X → B) ⊆ cov(X → B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So one may place this expression in the commutative diagram above to obtain ker(A ↷ X) cov(X → B) B ker(A ↷ X)|X⟩⟨X| = 0 B and simply trace back the desired orthogonality (using the top path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The remaining points follow from the discussion preceeding the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Let us reveal the maximal covariance in the case of graph algebras: Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='7 (Graph correspondences: maximal covariance): Consider a graph correspondence as in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2: X = ℓ2� E = edges � , A = c0(vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For these the trivially acting portion (the kernel for the left action) and the compactly acting portion correspond to sources and finite receivers: ker(A ↷ X) = c0 � a : |a edges | = 0 � = c0(sources), A ∩ XX∗ = c0 � a : |a edges | < ∞ � = c0(fin receivers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the orthogonal complement reads together max(X, A) = ker(A ↷ X)⊥ ∩ XX∗ = = c0 � a : 0 < |a edges| < ∞ � = c0(regular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' That is the maximal covariance corresponds to the regular vertices and as such any other covariance ideal corresponds to simply some collection of such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We finish this section with an investigation of covariances for morphisms be- tween correspondences (as opposed to representations into operator algebras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We begin with faithful morphisms between correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As for represent- ations we have (compare Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1): Being faithful passes from the coef- ficient algebra to the correspondence (and to any other power): (X, A) → (Y, B) : A ⊆ B =⇒ X ⊆ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 25 3 Kernel and Covariance Alexander Frei So the faithful morphisms may be seen as nothing but a subcorrespondence: That is those whose inner product already lies in the subalgebra and similar for the action from either side, ⟨Y |Y ⟩ ⊆ B, BY ⊆ Y, Y B ⊆ Y : X ⊆ Y, A ⊆ B : ⟨X|X⟩ ⊆ A, AX ⊆ X, XA ⊆ X (8) while for comparison mixed expressions only satisfy ⟨X|Y ⟩ ⊆ B, BX ⊆ Y, XB ⊆ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Schematically the inclusions for subcorrespondences may look like ⟨X|X⟩ A B AX X = XA Y which compare to mixed expressions as possibly only ⟨X|Y ⟩ ⟨Y |X⟩ A B BX X ̸= XB Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So one may think of a subcorrespondence as some sort of coherent restriction: Think of global actions and their restrictions to possibly partial actions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The action of compact operators on the ambient correspondence further reads (XX∗)Y ⊆ X(Y ∗Y ) ⊆ XB ⊆ Y Altogher, we may thus really think of a subcorrespondence simply as a subspace and subalgebra, which will be quite convenient also further on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In particular one has (for rather trivial reason) (T − T ′)Y = 0 =⇒ (T − T ′)X = 0 and as such also for a ∈ A and k ∈ XX∗, (a − k)Y = 0 =⇒ (a − k)X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (9) 26 3 Kernel and Covariance Alexander Frei This basically trivial implication already verifies Katsura’s second observation, and note how thinking in terms of subspaces and subalgebras paid off: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='8 (Second part of [Kat04, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3]): For a subcorrespondence as in (8) the covariance arises as pullback cov � X ⊆ Y � = � a ∈ A ��� im(a ↷ Y ) ∈ im(XX∗ ↷ Y ) � = (A ↷ Y )−1� im(A ↷ Y ) ∩ im(XX∗ ↷ Y ) � which we usually abbreviate simply as their common intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As a consequence it follows the characterization for trivial covariance cov � (X, A) ⊆ B � = 0 ⇐⇒ im(A → B) ∩ im � XX∗ → B � = 0 which reveals the familiar slogan for the Toeplitz representation: Those whose coefficient algebra has trivial intersection with compact operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This is nothing but the trivial implication (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We continue with the covariance for kernel and cokernel morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We begin for this with the following fairly standard result due to Kajiwara– Pinzari–Watatani which will become particularly useful in later context: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='9 (see [KPW98, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2]): A short exact sequence of correspondences (see theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3) 0 (XK, K) (X, A) � X XK , A K � 0 induces a morphism (of operator algebras) L(X) L � X XK � : T (x + XK) := T x + XK whose kernel admits the equivalent characterization X−1(K) := � X∗T X ⊆ K � = � T X ⊆ XK � = ker � L(X) → L � X XK �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (10) Further the morphism commutes with the left action by the coefficient algebra, 0 K A A AK 0 L(X) L � X XK � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 27 3 Kernel and Covariance Alexander Frei Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In order to verify the induced morphism let us first note the following: Adjointable operators restrict to kernel correspondences: T ∈ L(X) =⇒ T ∈ L(XK) : T (XK) ⊆ (T X)K ⊆ XK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we obtain a commutative diagram and whence the adjointable operator descends also to the quotient, 0 XK X X XK 0 0 XK X X XK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' T The resulting operator is adjointable as one easily verifies ⟨T (x + XK) | y + XK⟩ = ⟨T X | y⟩ = ⟨x | T ∗y⟩ = ⟨x + XK | T ∗(y + XK)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we got the desired morphism (of operator algebras) L(X) L � X XK , A K � : T (x + XK) := T x + XK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We meanwhile note that the morphism is generally not onto!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Moreover, the morphism clearly commutes with the left action by the coefficient algebra as one easily verifies (think in terms of subspaces) (a + K)(x + XK) = ax + XK = a(x + XK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The remaining claim about the kernel immediately follows from (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We continue with the analogous result for compact operators as for example in Fowler–Muhly–Raeburn (for which we provide a slightly simplified proof): Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='10 (see [FMR03, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='6]): A short exact sequence of correspondences (see theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3) 0 (XK, K) (X, A) � X XK , A K � 0 28 3 Kernel and Covariance Alexander Frei induces a short exact sequence at the level of compact operators 0 (XK)(XK)∗ = XKX∗ XX∗ � X XK �� X XK �∗ 0 L(X) L � X XK � which restricts from the morphism of adjointable operators as indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Clearly, the induced morphism on compact operators commutes with the morphism of adjointable operators since (think in terms of subspaces) (x + XK)(y + XK)∗(z + XK) = xy∗z + XK = xy∗(z + XK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Next we already know that the kernel correspondence (as a subcorrespondence) defines an embedding at the level of compact operators: (XK, K) ⊆ (X, A) =⇒ (XK)(XK)∗ ⊆ XX∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For kernel correspondences these now further define an ideal as for example � (XK)(XK)∗ = XKX∗� XX∗ = XK(X∗XX∗) = XKX∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand, the right-hand morphism is also clearly onto since X X XK 0 =⇒ XX∗ � X XK �� X XK �∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Regarding exactness in the middle, we may now invoke the characterization of the kernel from the previous proposition, which reads for compact operators X−1(K) ∩ XX∗ = ker � XX∗ � X XK �� X XK �∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' While the ideal clearly lies inside the kernel (left as an exercise for the reader) the converse inclusion now easily follows from the above description: X−1(K) ∩ XX∗ = XX∗� X−1(K) ∩ XX∗� XX∗ ⊆ X � X∗X−1(K)X � X∗ ⊆ XKX∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So we have found the desired exactness for compact operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We may now easily derive the desired covariance for kernel and cokernel morphisms (with the previous results in mind) 29 3 Kernel and Covariance Alexander Frei Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Kernel and cokernel morphisms have full covariance, 0 (XK, K) (X, A) � X XK , A K � 0 : cov � (XK, K) → (X, A) � = K ∩ XKX∗, cov � (X, A) → � X XK , A K � � = A ∩ XX∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This stands in contrast to the covariance for representations: The covariance for representations is bounded by Katsura’s ideal (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Note that kernel morphisms define a particular type of subcorres- pondences and so we find ourselves in the situation of proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='8: The nontrivial converse of implication (9) holds for kernel correspondences, a ∈ K, k ∈ XKX∗ : (a − k)XKX∗ = 0 =⇒ (a − k)X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Note that we may equivalently verify the implication KX∗(a − k)X∗K = 0 =⇒ X∗(a − k)X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we note the inclusion (due to invariance): X∗(a − k)X ⊆ X∗(K + XKX∗)X = (X∗KX) + (X∗X)K(X∗X) ⊆ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the above implication holds true since for K � X∗(a − k)X ⊆ K � K = 0 =⇒ X∗(a − k)X = 0 so kernel morphisms have full covariance as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Regarding the cokernel morphism, we may combine proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='9 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='10 to obtain the desired full covariance A ∩ XX∗ A K L(X) L � X XK � XX∗ � X XK �� X XK �∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 30 3 Kernel and Covariance Alexander Frei Concluding that kernel and cokernel morphisms have full covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Recall next that we may always render representations faithful by passing to the induced representation on the quotient 0 (XK, K) (X, A) � X XK , A K � 0 0 0 (B, B) (B, B) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ker The previous result allows us now to further clarify the relation between the covariance on the quotient correspondence (the relevant one) and the covariance for the original representation (the author would like to note that he found this special instance as an observation made by Katsura in [Kat07]): Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Consider a cokernel morphism (simply some onto mor- phism as observed in proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3) followed by an arbitrary morphism (X, A) (Y, B) 0 and (Y, B) (Z, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Then the covariance for the composition arises as pullback cov(X → Y → Z) = (A → B)−1 cov(Y → Z) ∩ XX∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In particular the covariance for a representation may be recovered from the in- duced representation on the quotient correspondence, (X, A) � X XK , A K � B : cov � X → B � = � A → A K �−1 cov � X XK → B � ∩ XX∗ which resambles Katsura’s observation [Kat07, lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='10 statement (v)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The result now easily follows from our previous proposition: Indeed as cokernel morphisms have full covariance we obtain a covariance diagram for our cokernel and one for the arbitrary morphism \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed A ∩ XX∗ B ∩ Y Y ∗ XX∗ Y Y ∗ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 and \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed cov(Y → Z) C ∩ ZZ∗ Y Y ∗ ZZ∗ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 31 3 Kernel and Covariance Alexander Frei and so also a commuting diagram for their composition (A → B)−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � ∩ XX∗ cov(Y → Z) C ∩ Y Y ∗ XX∗ Y Y ∗ ZZ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' That is the pullback lies within the covariance (A → B)−1 cov(Y → Z) ∩ XX∗ ⊆ cov(X → Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For the converse inclusion it suffices to establish cov(X → Z) ⊆ (A → B)−1 cov(Y → Z) ⇐⇒ (A → B) cov(X → Z) ⊆ cov(Y → Z) which is to verify the covariance diagram cov(X → Z) (A → B) cov(X → Z) C ∩ ZZ∗ Y Y ∗ ZZ∗ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Indeed this may be easily seen by following the covariance diagram cov(X → Z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' C ∩ ZZ∗ XX∗ Y Y ∗ ZZ∗ and the full covariance for our cokernel morphisms (once more) A ∩ XX∗ B ∩ Y Y ∗ XX∗ Y Y ∗ ZZ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So the covariance for the composition arises from the pullback as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The remaining statement arises now as special instance from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We finish this section with the following negative result about the covari- ance for subcorrespondences: While we have found that kernel and cokernel morphisms have full covariance, this is not the case for subcorrespondences in 32 3 Kernel and Covariance Alexander Frei general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' That is simply the converse of implication (9) fails in general: a ∈ A, k ∈ XX∗ : (a − k)X = 0 ̸=⇒ (a − k)Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we consider the following somewhat minimal example: Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='13 (Subcorrespondence with zero covariance): Consider the direct sum of an operator algebra as both the coefficient algebra and the Hilbert module (which we depict as diagonal operators) Y = � D D � = B =⇒ Y ∗Y ⊆ B, Y B ⊆ Y ✓ but with left action given by the flip automorphism on B = D ⊕ D: � d1 d2 � ↷ � y1 y2 � = �� 1 1 �� d1 d2 �� 1 1 ��� y1 y2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Regard the subcorrespondence given by the subalgebra (noninvariant ideal!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') � X = � D 0 � = A � ⊆ � Y = � D D � = B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Then its left action vanishes identically (and in particular by compact operators): A ↷ X = �� 1 1 �� D 0 �� 1 1 ��� D 0 � = � 0 D �� D 0 � = 0 On the other hand it never vanishes on the ambient correspondence A ↷ Y = �� 1 1 �� D 0 �� 1 1 ��� D D � = � 0 D �� D D � = � 0 D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the covariance diagram is maximally noncommuting: A = A ∩ XX∗ B XX∗ Y Y ∗ ̸= : ker � A ⊆ B ⊆ Y Y ∗ � = 0, ker � A → XX∗ ⊆ Y Y ∗ � = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So one needs to stay cautious about the covariance for subcorrespondences: In worst case one needs to verify a particular covariance by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This finishes our section on kernel and cokernel morphisms on one hand, and the possible covariances on the other hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this at hand we may now proceed to the gauge-equivariant representations and their classification — in other words the classification of relative Cuntz–Pimsner algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 33 4 Relative Cuntz–Pimsner algebras Alexander Frei 4 Relative Cuntz–Pimsner algebras We introduce in this section the relative Cuntz–Pimsner algebras and elaborate why and how these serve to classify the gauge-equivariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We begin with the Toeplitz algebra: That is the universal representation (more precisely the initial representation) as such that any other representation uniquely factors via the universal one, (X, A) (X, A) T (X, A) (B, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Next recall from the previous section that we may render any representation faithful by passing to the quotient correspondence 0 (XK, K) (X, A) � X XK , A K � 0 0 0 (B, B) (B, B) 0 ker and recall from theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3 that any such quotient arises precisely for some invariant ideal (equivalently hereditary ideal) K = ker(A → B) ⊴ A : X∗KX ⊆ K ( ⇐⇒ KX ⊆ XK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand, we found in proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='6 that faithful representations have their covariance bounded from above by the maximal covariance, 0 ⊆ cov � � X XK , A K � B � ⊆ max � X XK , A K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we may aim to classify representations by pairs of some invariant ideal (as possible covariance) and another bounded ideal (as possible covariance): � K ⊴ A : X∗KX ⊆ K ��� I ⊴ A K : I ⊆ max � X XK � � (11) To handle this task, we may consider the class of representations with kernel and covariance at least a given pair of invariant and bounded ideal, (X, A) (B, B) : K ⊆ ker(A → B), I ⊆ cov � X XK → B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Note that any such class contains at least the trivial representations and 34 4 Relative Cuntz–Pimsner algebras Alexander Frei furthermore that representations may well have larger kernel and covariance only (which we cannot exclude at this moment as we will explain below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The universal such representation defines now the relative Cuntz–Pimsner alge- bra for a given kernel–covariance pair as in (11): (X, A) � X XK , A K � � X XK , A K � O(K, I) (B, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' K⊆ker(A→B) I⊆cov � X XK →B � As such we obtain an entire “2-dimensional lattice” of relative Cuntz–Pimsner algebras (as class of universal representations) by first following along the lattice of cokernel morphisms (given by invariant ideals) (X, A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � X X(K∩L), A K∩L � � X XK , A K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � X XL, A L � � X X(K+L), A K+L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (0, 0) 35 4 Relative Cuntz–Pimsner algebras Alexander Frei followed by the lattice of covariance ideals (on the chosen cokernel) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � X XK , A K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' T � X XK , A K � = O(K, 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' O(K, I ∩ J) O(K, I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' O(K, J) O(K, I + J) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' O(K, max) = O � X XK , A K � with Toeplitz algebras and absolute Cuntz–Pimsner algebras as extreme points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We now explain the precise goals for the classification of relative Cuntz– Pimsner algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We will first note that that relative Cuntz–Pimsner al- gebra come equipped with gauge-actions rendering the representation gauge- equivariant (we explain this in more detail in the following section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the possible range of relative Cuntz–Pimsner algebras are the gauge- equivariant representations at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The surprising first goal of their classifica- tion now states that every gauge-equivariant representation indeed arises itself as a relative Cuntz-Pimsner algebra (and in particular every gauge-equivariant representation is itself a universal representation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' More precisely, consider first the induced representation rendering the representation faithful (X, A) � X XK , A K � (B, B) : K = ker(A → B) followed by the covariance for the resulting representation � X XK , A K � (B, B) : I := cov � � X XK , A K � ⊆ B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' To better illuminate the problem let us restrict the representation to its range, that is the operator algebra generated by (the image of) the correspondence, (X, A) � X XK , A K � C∗(X ∪ A) ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 36 4 Relative Cuntz–Pimsner algebras Alexander Frei With this description the first problem states that (X, A) � X XK , A K � O(K, I) = C∗(X ∪ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Or put in other words, the pairs of invariant ideals as kernel and bounded ideals covariance exhaust the gauge-equivariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We will solve this problem via the familiar gauge-invariant uniqueness theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The second goal is to determine that in fact every possible kernel and covari- ance arises itself as an actual kernel and covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' More precisely, consider any pair of invariant ideal as kernel and bounded ideal as covariance: � K ⊴ A : I ⊴ A/K : X∗KX ⊆ K I ⊆ max(X/XK) � Then there simply may be no such representation with precisely the given kernel and covariance ideal (so that not every such pair would arise in nature): (X, A) � X XK , A K � (B, B) : � ker(A → B), cov � X XK → B � � = (K, I) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Or put in other words, two possibly different pairs of kernel and covariance ideal could in principle lead to one and the same relative Cuntz–Pimsner algebra: O(K, I) = O(K′, I′) =⇒ (K, I) = (K′, I′) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We will however find the following relations (both nontrivial!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ): (X, A) � X XK , A K � O(K, I) : � ker � A K → O(K, I) � = 0 ��� cov � X XK → O(K, I) � = I � and as such also the desired kernel–covariance pair � ker(A → O(K, I)) = K ��� cov � X XK → O(K, I) � = I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (12) For this the Fock representation will come into play: Its concrete represent- ation allows us to actually compute its kernel and covariance and so to verify the desired relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such each possible pair of invariant ideal (as kernel) and bounded ideal (as covariance) arises itself as actual kernel and covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Summarizing these goals, the kernel–covariance pairs completely parametrize the entire lattice of gauge-equivariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In other words, the lat- 37 4 Relative Cuntz–Pimsner algebras Alexander Frei tice of relative Cuntz–Pimsner algebras (as schematically given above) classifies the entire lattice of gauge-equivariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this at hand, we then further investigate the connecting morphisms between relative Cuntz–Pimsner algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' More precisely, we will find that our parametrisation given by kernel–covariance pairs defines a lattice isomorphism in the sense that ( K ⊆ L | “I ⊆ J” ) ⇐⇒ O(K, I) ≤ O(L, J) where the latter denotes the factorization (as common notation): (X, A) (X, A) O(K, I) O(L, J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This has been observed already by Katsura in his 2007 article on gauge- invariant ideals using however instead so-called T-pairs (and further O-pairs): We will unravel these as nothing but transformed versions of our kernel– covariance pairs and as such give a natural interpretation for such pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Following we further give precise descriptions for when connecting morphisms exist between cokernel strands (as from where and to where) O(K, I = ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') O(L, J) , O(K, I) O(L, max) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' which generalize results from Katsura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Together we thus obtain a plethora of connecting morphisms between cokernel strands such as � X XK , A K � � X XL, A L � T � X XK , A K � T � X XL, A L � O(K, I) O(L, J) O � X XK , A K � O � X XL, A L � and note that plenty of connecting morphisms also will be missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we will give some further examples from graph algebras to illuminate 38 5 Gauge actions: Fourier spaces Alexander Frei the lack of connecting morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this in mind and without further ado, we now get to the class of gauge-equivariant representations: 5 Gauge actions: Fourier spaces We begin with the following observation to motivate gauge-equivariant representations: For every fixed complex number on the torus we may consider the automorphism which rotates the correspondence z ∈ T : (X, A) (X, A) : z ↷ x = zx, z ↷ a = a which together define a circle action T ↷ (X, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Clearly these do not affect the kernel of representations as for (X, A) B : ker( A A B 1 ) = ker( A B ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Similarly they do not affect the covariance as they act trivially on compacts, zz∗ = 1 : XX∗ XX∗ : x(zz∗)y∗ = xy∗ : A ∩ XX∗ A ∩ XX∗ ⊆ A B XX∗ XX∗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' = =?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such every relative Cuntz–Pimsner algebra (as universal representation for some relations) admits a unique gauge action which renders its representation gauge-equivariant (think in terms of generators and relations): ∃T ↷ O(K, J) : � T ↷ (X, A) � � T ↷ O(K, I)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such at best we may hope to classify the relative Cuntz–Pimsner algebras amongst those representations which come along with a gauge-action rendering the representation equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' More precisely, that is written out (ϕ, τ) : � T ↷ (X, A) � � T ↷ B � : z ↷ τ(x) = τ(z ↷ x) = τ(zx), z ↷ ϕ(a) = ϕ(z ↷ a) = ϕ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 39 5 Gauge actions: Fourier spaces Alexander Frei Equivalently these are the conventional gauge-equivariant representations of op- erator algebras (from any of the preceeding relative Cuntz–Pimsner algebras) π : � T ↷ O(K, I) � � T ↷ B � : z ↷ π(−) = π(z ↷ −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Consider now the relative Cuntz–Pimsner algebras (as universal representations) which allow factorizations for our gauge-equivariant representation: (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) O(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) B ⇐⇒ � K ⊆ ker(A → B) ��� I ⊆ cov � X XK → B � � As such — if the gauge-equivariant representation has a chance of being a uni- versal representation for some kernel–covariance pair — then certainly the best chance is given by the kernel–covariance pair for the representation itself: (K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) = � ker(A → B) ��� cov � X XK → B � � : (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) O(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) C∗(X ∪ A) ⊆ B ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This is what the gauge-equivariant uniqueness theorem will establish in the next section and so also the first goal in the classification of relative Cuntz–Pimsner algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So let us prepare ourselves a bit more for this by taking a closer look at gauge-equivariant representations and their Fourier spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' It is well known that every operator algebra equipped with a circle action (such as our gauge-equivariant representations) comes along with Fourier spaces B(n ∈ Z) = � b ��� � b(z) := z ↷ b � = znb � ⊆ B and in particular its fixed point algebra B(n = 0) = � b ��� � b(z) := z ↷ b � ≡ b � ⊆ B which define a Fell bundle over the integers B(m)B(n) ⊆ B(m + n),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' B(n)∗ = B(−n) together with a conditional expectation onto its fixed point algebra E : B B(0) : E(b) = � T � b(z) = z ↷ b � dz 40 5 Gauge actions: Fourier spaces Alexander Frei and more generally with projections onto any of its Fourier spaces En : B B(n) : En(b) = � T z−nb(z) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Recall that conditional expectations given by averaging are automatically faith- ful since the state space separates the positive elements (and since averaging runs as Bochner integral): b ≥ 0 =⇒ � b(z) = z ↷ b � ≥ 0 =⇒ SB � b(z) = z ↷ b � ≥ 0 : E(b ≥ 0) = 0 =⇒ SB(E(b)) = SB �� b(z) dz � = � SB(b(z)) dz = 0 =⇒ SB � b(z) = z ↷ b � ≡ 0 =⇒ SB � b = 1 ↷ b � = 0 =⇒ b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We meanwhile note that the Fourier spaces densely span the operator algebra by [Exe94, proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='5] (which seems a rather intuitive yet nontrivial relation) B = � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + B(−1) + B(0) + B(1) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � = � n B(n) and we encourage the reader to have a look into the beautiful proof by Exel: It does not require any Cesaro approximations and instead only invokes the elementary isomorphism (as the basic version of Coburn) C∗(Z) = C∗(u∗u = 1 = uu∗) = C( σu | u∗u = 1 = uu∗) = C(T) =⇒ C∗(Z) = � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + Cu∗ + Ce + Cu + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � = C(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A short digression: One may meanwhile wonder how it may be possible that any element may be approximated by Fourier sums while the series of its Fourier coefficients only converges in Cesaro mean (and generally diverges in norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The answer to this question becomes evident when replacing for instance the torus with an open disk and the trigonometric with polynomial sums {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' , 1/z, 1, z, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='} ⊆ C(T) ⇝ {1, z, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='} ⊆ C(D) It is well-known here that any continuous function may be approximated in norm by polymoials (by Stone–Weierstrass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand, the convergent power series correspond to the class of Taylor analytic functions f(z) = � n anzn ⇐⇒ f ∈ C∞(D) 41 5 Gauge actions: Fourier spaces Alexander Frei The difference lies in the fact that a polynomial approximation of continuous functions generally changes each of the coefficient in the sequence — also the previously already set coefficients!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such one may think of the convergent Fourier series as a generalization of analytic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We now further restrict our attention on the range of the representation (also since the ambient rest does not reflect the correspondence): More precisely, that is the operator algebra generated by (the image of) the correspondence, (X, A) C∗(X ∪ A) ⊆ B ⇝ B = C∗(X ∪ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Recall that a correspondence (and so its image under the representation) satisfies its defining inclusions as explained in more detail in section 1: X∗X ⊆ A, XA ⊆ X, AX ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the the range (as generated operator algebra) admits “a fine structure” C∗(X ∪ A) = span � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + (X∗ + XX∗X∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' )+ +(A + XX∗ + XXX∗X∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') + (X + XXX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � which we organised in groups according to the Fourier spaces they belong to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we also obtain “a fine structure” for the fixed point algebra B(n = 0) = E � B = C∗(X ∪ A) � = span � A + XX∗ + XXX∗X∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � which allows us to deduce the gauge-invariant uniqueness theorem by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Note further that for any correspondence (including possibly degenerate ones!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') X∗X ⊆ A, XA ⊆ X, AX ⊆ X =⇒ XA = X, XXA = XX, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the Fourier spaces further satisfy B(n ≥ 1) = En � B = C∗(X ∪ A) � = span � Xn + XnXX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � = Xn� A + XX∗ + XXX∗X∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � = XnB(0) where we left the linear spans and closures implicit for more pointy statements, and by involution also for negative Fourier spaces and as such altogether B(n ≥ 1) = XnB(0), B(n ≤ −1) = B(0)X−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 42 5 Gauge actions: Fourier spaces Alexander Frei As a consequence, the Fell bundle (restricted on the range) further satisfies B(n ≥ 1) = B(1)n : B(n) = XnB(0) ⊆ B(1)nB(0) ⊆ B(1)n ⊆ B(n) and similarly for its negative Fourier spaces, and as such defines altogether a semi-saturated Fell bundle (see for instance [Exe94, proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='8]): B(m ≥ 0)B(n ≥ 0) = B(m + n), B(m ≤ 0)B(n ≤ 0) = B(m + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' These relations are quite useful for bipartite inflations as introduced in [MPT08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With the knowledge about Fourier spaces at hand, we may now get back to the classification problem from above: (K, I) = � ker(A → B) ��� cov � X XK → B � � : (X, A) O(K, I) C∗(X ∪ A) ⊆ B ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' At first we may pass to the quotient correspondence (X, A) � X XK , A K � O(K, I) B to obtain an honest embedding since K = ker(A → B) =⇒ A/K ⊆ B =⇒ X/XK ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we may assume that our correspondence lies faithfully in the ambient operator algebra (upon replacing the original with the quotient correspondence) � X XK , A K � ⇝ (X, A) =⇒ O(K, I) ⇝ O(0, I) =⇒ X ⊆ B, A ⊆ B (13) and at the same time restrict our attention to its range: C∗(X ∪ A) ⊆ B ⇝ C∗(X ∪ A) = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We may thus think of both ambient algebras as faithful completions for the correspondence — but under a priori possibly different norm topologies: O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) = C∗(X ∪ A) X ∪ A C∗(X ∪ A) = B As another valuable perspective, one may also think of the above prob- lem as gauge-equivariant envelopes (comparable to the maximal and minimal 43 5 Gauge actions: Fourier spaces Alexander Frei C∗-envelope of non-selfadjoint operator algebras): T (X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' : (X, A) ⊆ (T ↷ Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The author would like to thank Elias Katsoulis for bringing closer this idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Next as our representation is gauge-equivariant it restricts in particular to fixed point algebras and further commutes with conditional expectations: O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) B O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I)(0) B(0) O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) B O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I)(0) B(0) As such it suffices to verify the above equality on fixed point algebras: π : O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I)(0) B(0) =⇒ π : O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) B (14) Indeed this follows as standard argument from the faithful conditional expecta- tion (sufficiently for the relative Cuntz–Pimsner algebra): π(a) = 0 =⇒ π(a∗a) = 0 =⇒ π(E(a∗a)) = E(π(a∗a)) = 0 =⇒ E(a∗a) = 0 =⇒ a∗a = 0 =⇒ a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Another interesting less commonly known argument due to Exel basically exploits that continuous functions, whose Fourier coefficients all vanish, already vanish identically themselves (see [Exe94, proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='9]): En(a)En(a)∗ ∈ O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) � n − n = 0 � : π(a) = 0 =⇒ En(πa)En(πa)∗ = π � En(a)En(a)∗� ≡ 0 =⇒ En(a)En(a)∗ ≡ 0 =⇒ En(a) ≡ 0 =⇒ a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For the fixed point algebra we however found the “fine structure” O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I)(n = 0) = span � A + XX∗ + XXX∗X∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � so the fixed point algebras arises as an increasing union (inductive limit) O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I)(n = 0) = closure �� N span � A + XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XNX−N�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 44 6 Uniqueness theorem Alexander Frei As such we may verify the faithfulness by induction along n ∈ N: O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) ⊇ span(A + XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XnX−n) B ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (15) The point here is not that each summand embeds separately (which is trivial) A B, XX∗ B, XXX∗X∗ B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' but instead that the summands will have plenty of correlation among each other (namely precisely the amount of covariance) which causes lots of sums to collapse within the representation such as the covariance itself, cov(X → B) = ker \uf8ee \uf8ef\uf8ef\uf8f0 \uf8eb \uf8ec \uf8ec \uf8ed cov(X → B) B B \uf8f6 \uf8f7 \uf8f7 \uf8f8 − \uf8eb \uf8ec \uf8ec \uf8ed cov(X → B) XX∗ B \uf8f6 \uf8f7 \uf8f7 \uf8f8 \uf8f9 \uf8fa\uf8fa\uf8fb =⇒ � (a) + (k = −a) ∈ A + XX∗ ���� a ∈ cov(X → B) � ⊆ ker � A + XX∗ B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In fact, proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='8 tells us that the covariance precisely captures this kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As an interesting question to the encouraged reader: can you figure out why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Before we proceed, we would like to note that the idea for the above in- duction and its proof are due to Evgenios Kakariadis as in [Kak16] and note that this simplifies the original proof by Katsura substantially!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We will not go further into how these approaches compare (since this would not benefit our current work) but we would like to note that Katsuras approach analysing cores may be quite valuable after all in the more general context of product systems and so we refer the curious reader to [Kat04, section 5] for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this we may now proceed to the gauge-invariant uniqueness theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 6 Uniqueness theorem We may now state and proof our version of the gauge-invariant uniqueness theorem for arbitrary gauge-equivariant representations which arise as relative Cuntz–Pimsner algebra along kernel–covariance pairs: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1 (Gauge-invariant uniqueness theorem: The general version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Consider a gauge-equivariant representation (as in the previous section) � T ↷ (X, A) � � T ↷ B � 45 6 Uniqueness theorem Alexander Frei and choose the kernel–covariance pair for the representation (K, I) = � ker(A → B) ��� cov � X XK → B � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Then the quotient from the relative Cuntz–Pimsner algebra is faithful: (X, A) O(K, I) C∗(X ∪ A) ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Thus each gauge-equivariant defines a relative Cuntz–Pimsner algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand, each relative Cuntz–Pimsner algebra defines itself a gauge- equivariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the gauge-equivariant representations agree with relative Cuntz–Pimsner algebras (the range of kernel–covariance pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We have basically already proven the theorem: Indeed we have al- ready reduced the general version to the case of faithful representations (see the previous section) and the faithful case is a classical result due to Kat- sura from [Kat04] resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' the simplified version due to Kakariadis in [Kak16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For convenience of the reader we review the simplified version: We begin for this with a couple of observations and useful relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' At first consider the increasing subalgebras from the induction problem (15) (which exhaust the fixed point algebra): span(A + XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XnX−n) ⊆ O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) In order to simplify their induction we need to address a technical detail: Note that while the sum of ideals is already closed, this fails in general for the sum of subalgebras, A ⊆ B, A′ ⊆ B : (A + A′) ⊆ (A + A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' However the sum of an ideal and a subalgebra is closed nevertheless: A ⊆ B, J ⊴ B : (A + J) = (A + J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Quick proof for an algebra and ideal (as for pairs of ideals): Consider the short exact sequences of possibly incomplete algebras, 0 J (A + J) (A + J)/J 0 0 J (A + J) (A + J)/J 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' π 46 6 Uniqueness theorem Alexander Frei Note that the morphism between quotients defines an embedding since J ∩ (A + J) = J =⇒ (A + J)/J ⊆ (A + J)/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Since the range of ∗-homomorphisms is always closed we obtain also π(A + J) = π(A) = π(A) = π(A + J) = π � A + J � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the quotients agree and so (by basic homological algebra) (A + J)/J = (A + J)/J =⇒ (A + J) = (A + J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (16) So the sum of an algebra and ideal defines a closed subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this at hand we find in our case (using the obvious inclusion) AXX∗ ⊆ XX∗ =⇒ span(A + XX∗) = A + span(XX∗) and similarly on larger sums span � A + XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XnX−n� = A + span � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XnX−n� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This finishes our discussion on the technical detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this the induction problem asks for the kernel ker � O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) ⊇ A + span � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XnX−n� B � and so for the coefficient algebra that intersects compact operators A ∩ � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XnX−n� ⊆ B where we drop from now on the closed linear span for more pointy statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We would like to better understand this intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we first discuss the following result, which gives an interesting algebraic description for non- commutative Cartan subalgebras from [Exe11] as the nondegenerate ones: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A subalgebra contains some approximate identity for the ambient algebra if and only if the subalgebra is nondegenerate: A ⊆ B : AB = B ⇐⇒ (1 − e)B → 0 for A ∋ e → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand, the nondegeneracy holds equivalently if the subalgebra reaches 47 6 Uniqueness theorem Alexander Frei the entire ambient algebra hereditarily (with implicit Cohen–Hewitt) AB = B ⇐⇒ her(A) = ABA = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Further, the subalgebra remains nondegenerate on intermediate subalgebras: A ⊆ B0 ⊆ B : AB = B =⇒ AB0 = B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The analogous statements also hold when replacing the subalgebra and the am- bient algebra by any pair of operator algebras (after suitable modifications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The forward direction is obvious since for any approximate unit A ∋ e → 1 : (1 − e)B = (1 − e)AB → 0 where we implicitly use Cohen–Hewitt as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The converse is also obvious as evidentily (1 − e)B → 0 =⇒ B ⊆ AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Regarding the hereditary subalgebra we have: AB = B ( =⇒ B = B∗ = (AB)∗ = B∗A∗ = BA) =⇒ B = (AB) = A(BA) = ABA =⇒ B = ABA ⊆ AB Regarding intermediate algebras we have: AB = B (1 − e)B → 0 for A ∋ e → 1 AB0 = B0 (1 − e)B0 → 0 for A ∋ e → 1 For arbitrary pairs of operator algebras (instead of a subalgebra): One needs to replace the equality by an inclusion and an additional closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With the nondegeneracy in mind we may now unravel the intersection above: For this we first note that the compact operators are nondegenerate in the sum of higher order compact operators, � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XnX−n� = XX∗� XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XnX−n� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Indeed we may simply pull out a rabbit from each summand (using Blanchard): XX∗ = (XX∗)XX∗, XXX∗X∗ = (XX∗)XXX∗X∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 48 6 Uniqueness theorem Alexander Frei As a consequence we obtain the inclusion (similarly as in lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2): A ∩ � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XnX−n� ⊆ � A ∩ (XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XnX−n) � XX∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The latter however lies inside the algebra of compact operators since � A ∩ (XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XnX−n) � XX∗ ⊆ AXX∗ ⊆ XX∗ and as such inside the covariance for the representation (see proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As the covariance cannot decrease we obtain the combined relation: A ∩ � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XnX−n� = cov(X → B) = cov � X → O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In particular the kernel arises as sum of compact operators: ker � A + � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XXnX−nX∗� B � ⊆ XX∗ + � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XXnX−nX∗� ⊆ O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So far our introductory observations and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With these in mind we may now get to the proof of the gauge-equivariant uniqueness theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof of theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1 (Gauge-invariant uniqueness theorem): We have already reduced the general version to the case of faithful representations in (13): � X XK , A K � ⇝ (X, A) =⇒ O(K, I) ⇝ O(0, I) =⇒ X ⊆ B, A ⊆ B Furthermore the problem reduces to fixed point algebras as in (14) O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I)(0) B(0) =⇒ O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) B which may be solved as an induction along its fine structure as in (15): O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) ⊇ A + � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XnX−n� B ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this at hand we may now begin the proof by induction from [Kak16]: The base case simply states the inclusion that we are already well aware of, O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) = C∗(X ∪ A) A C∗(X ∪ A) = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Before we begin with the induction combine the observations from above Con- 49 6 Uniqueness theorem Alexander Frei sider now the induction step (while assuming the induction hypothesis): ker � A + � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + Xn+1X−n−1� B � = 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We may reduce this problem to the induction hypothesis via compression X∗ ker � A + � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + Xn+1X−n−1� B � X ⊆ ker � X∗X + X∗� XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + Xn+1X−n−1� X B � ⊆ ker � A + � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XnX−n� B � = 0 from which we infer that the kernel vanishes for the compression XX∗ ker � A + � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + X(XnX−n)X∗� B � XX∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We however found that the kernel lies inside the sum of compact operators: ker � A + � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XXnX−nX∗� B � ⊆ XX∗ + � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XXnX−nX∗� ⊆ O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' At the same time, the compact operators define a nondegenerate subalgebra within higher order compact operators and as such have trivial annihilator: � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XXnX−nX∗� = XX∗� XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XXnX−nX∗� =⇒ (XX∗)⊥ ∩ � XX∗ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' + XXnX−nX∗� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So the kernel above necessarily also vanishes without compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Concluding the current and previous section, we have achieved the first half in the classification of relative Cuntz–Pimsner algebras: that is we have found that every gauge-equivariant representation arises itself as relative Cuntz– Pimsner algebra for some kernel–covariance pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In short, that is the kernel– covariance pairs exhaust the gauge-equivariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We may now proceed with the second half on the classification of rela- tive Cuntz–Pimsner algebras: The kernel–covariance pairs classify the rela- tive Cuntz–Pimsner algebra and equivalently every possible kernel–covariance pair arises itself in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we need to construct sufficiently many representations (as well as any nontrivial one whatsoever).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This is where the 50 7 Fock representation Alexander Frei Fock representation and the quotients thereof will come into play: 7 Fock representation We begin with the following problem: Note that up until now we have not come across any representation whatsoever (put aside sufficiently many) except the trivial representation B = 0 : X 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A 0 and as such it could be in principle that the relative Cuntz–Pimsner algebras (for some correspondences) all coincide as the only trivial representation (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) T (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' = O(K, I) = O(K′, I′) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' = 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the question arises whether a correspondence admits always a nontrivial representation (and further also sufficiently many faithful ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we consider a failing attempt which leads us in turn to the Fock representation: As a first guess we may consider the linking algebra associated to our correspondence (seen as Hilbert module only) B = � A X �� A∗ X∗ � = � A X∗ X XX∗ � ⊆ L �� A X �� together with the representation given by the canonical embedding � A 0 � ⊆ � A X∗ X XX∗ � = B, � 0 X � ⊆ � A X∗ X XX∗ � = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Now while this representation respects the Hilbert module structure (basically by construction) it does not respect the left module structure: � x∗ 0 �� 0 y � = � x∗y 0 � , � 0 x �� a 0 � = � 0 xa � , � a 0 �� 0 x � = � 0 0 � ̸= � 0 ax � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The solution is to simply extend these down the diagonal to infinity which brings us straight to the Fock space representation: More precisely, the Fock space is 51 7 Fock representation Alexander Frei the infinite direct sum of increasing tensor powers (as in section 1) F(X) := A ⊕ X ⊕ XX ⊕ XXX ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed A X XX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 and the representation as diagonal action and as right shift respectively: A → L(FX) : a � A ⊕ X ⊕ XX ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � := aA ⊕ aX ⊕ aXX ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' X → L(FX) : x � A ⊕ X ⊕ XX ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � := 0 ⊕ xA ⊕ xX ⊕ xXX ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We visualize them in matrix notation (using the formal left and right shift): X⊗R = X \uf8eb \uf8ed 0 1 0 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8, A⊗1 = A \uf8eb \uf8ed 1 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8, X∗⊗L = X∗ \uf8eb \uf8ed 0 1 0 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Further the Fock representation comes with the (inner) circle action T ↷ L(FX) : (z ↷ −) := \uf8eb \uf8ed 1 z z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8 − \uf8eb \uf8ed 1 z z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8 ∗ which renders the representation gauge-equivariant: \uf8eb \uf8ed 1 z z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8A \uf8eb \uf8ed 1 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8 \uf8eb \uf8ed 1 z z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8 ∗ = 1A \uf8eb \uf8ed 1 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8, \uf8eb \uf8ed 1 z z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8X \uf8eb \uf8ed 0 1 0 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8 \uf8eb \uf8ed 1 z z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8 ∗ = zX \uf8eb \uf8ed 0 1 0 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Furthermore the representation of compact operators reads XX∗ �→ X \uf8eb \uf8ed 0 1 0 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8 \uf8eb \uf8ed 0 1 0 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8X∗ = XX∗ \uf8eb \uf8ed 0 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' \uf8f6 \uf8f8 With these preliminary computations ready we may now reveal the Fock representation as the universal representation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' the Toeplitz algebra): Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1 (Fock representation = Toeplitz algebra): 52 7 Fock representation Alexander Frei The Fock representation has trivial kernel and covariance ker � A L(FX) � = 0, cov � X L(FX) � = 0 and as such defines the universal representation (by theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Clearly the representation is faithful (and so has trivial kernel) as � a a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' �� A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � = � aA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � = � 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � =⇒ a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As a consequence, we may further compute the covariance as the intersection within the representation (as the special instance from proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='8) cov � X L(FX) � = A � 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � ∩ XX∗ � 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So the covariance is trivial as well and the proposition follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Note that more importantly we established the existence of any non- trivial representation whatsoever (equivalently the universal representation is different from the trivial representation) and further also the existence of faithful representations (equivalently the universal representation is faithful).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Put in other words, we may now distinguish Toeplitz algebras along the lattice of quotient correspondences (the first dimension of kernel–covariance pairs): T � X XK � = O(K, 0) = O(K′, 0) = T � X XK′ � =⇒ K = K′ Indeed we have even found the stronger kernel relation from (12): ker � A T (X) � = 0 =⇒ ker � A A K T � X XK � � = K We are still left with the question (which we get to next) O(K, I) = O(K′, I′) =⇒ ( K = K′ | I = I′ ) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we may now consider the Fock representation as a concrete realization for the Toeplitz algebra and as such further construct every relative Cuntz– Pimsner algebra (along kernel–covariance pairs) as a concrete quotient thereof: For example given first a quotient correspondence (for some invariant ideal) K ⊴ A : X∗KX ⊆ K : (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) � X XK ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A K � we may construct the corresponding Toeplitz algebra as quotient by the kernel 53 7 Fock representation Alexander Frei (more precisely its ideal generated within the original Toeplitz representation) 0 (XK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' K) (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) � X XK ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A K � 0 0 T X(K ⊆ A)T X T X T � X XK � 0 where we omit as usual the closed linear span for more pointy statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Before continuing we replace for convenience the original correspondence by the quotient correspondence, � X XK , A K � ⇝ (X, A) =⇒ T � X XK � ⇝ T (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Consider next an ideal (as possible covariance) bounded from above by the maximal covariance (as in proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='6), I ⊴ A : I ⊆ max(X, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The bound from above is necessary as larger covariances force an additional kernel and as such would factor over some further quotient correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Similarly as for relative Toeplitz algebras, the relative Cuntz–Pimsner algebra arises now as coequalizer for the chosen covariance O(K, I) = coeq \uf8eb \uf8ec \uf8ec \uf8ed I ⊆ A ∩ XX∗ T X XX∗ T X \uf8f6 \uf8f7 \uf8f7 \uf8f8 and as such also as quotient by their difference as in (2): 0 T X � (ϕ − τ)I � T X T X O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) 0 : (ϕ − τ) = \uf8eb \uf8ec \uf8ed A ∩ XX∗ T X T X \uf8f6 \uf8f7 \uf8f8 − \uf8eb \uf8ec \uf8ed A ∩ XX∗ XX∗ T X \uf8f6 \uf8f7 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We now invoke the Fock representation as Toeplitz algebra: Recall that we have already found here the concrete embedding for the coefficient algebra as well as the concrete embedding of compact operators (see above) T X ⊆ L �� A X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' �� : A �→ A � 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � , XX∗ �→ XX∗ � 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � 54 7 Fock representation Alexander Frei and as such their difference reads A ∩ XX∗ �→ A ∩ XX∗ � 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We therefore found the relative Cuntz–Pimsner algebra as quotient 0 T X � I 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � T X T X O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (17) This was originally established by Muhly and Solel in [MS98, theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We now wish to verify that the induced quotient representation remains faithful: That is we note that the quotient could in principle introduce new kernel, I ⊆ A ∩ XX∗ : ker � A T X O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) � = 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Indeed we note that the quotient does introduce new kernel as soon as the covariance exceeds the maximal covariance (as we have noted also above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we have to make sure this does not happen as long as the covariance lies below the maximal covariance from proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we note that the trivial kernel above may be equivalently verified now as the trivial intersection I ⊆ max(X, A) : A � 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � ∩ T X � I 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � T X = 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (18) On the other hand, we wish to also verify the covariance relation from (12): cov � A T X O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) � = I ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The problem here is that the covariance could in principle increase as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Indeed the construction (and even our very definition) of relative Cuntz–Pimsner algebras guarantees just a least covariance for the provided covariance ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This becomes more evident as follows: Consider for this the difference morphism which factors by the universal property via the Toeplitz algebra: A ∩ XX∗ A ∩ XX∗ 0 T X � I 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � T X T X O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ϕ − τ 55 7 Fock representation Alexander Frei As such the covariance may be read off from the common intersection as � A ∩ XX∗ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � ∩ T X � I 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � T X = � I 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (19) Note this meanwhile also highlights how the covariance could increase: how?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Let us begin to verify that the representation remains faithful along the quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we begin with the following well-known relation (which goes all the way back to an observation by Joachim Cuntz made in [Cun77]): Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The ideal generated by the covariance as in (17) coincides with the ideal of compact operators of the form T X � I 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � T X = � A X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � I( A X∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ) = K �� A X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � I � with implicit closed linear spans for more pointy statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Sketch of proof: The result follows most easily using the formal right and left shift operators (as further above) from which the covariance ideal reads � I 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � = I �� 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � − � 0 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' �� = I ⊗ (1 − RL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Recall however that these satisfy the well-known relation LR = 1 =⇒ L(1 − RL) = 0 = (1 − RL)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we obtain as the only contributions for the ideal T X � I 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � T X = � mn � X � 0 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ��m� I 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' �� X∗ � 0 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ��n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand they generate the system of matrix units such as � 0 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' �m� 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' �� 0 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' �n = � 0 0 1 0 0 � = � 0 1 0 � ( 0 1 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we obtain for the above ideal T X � I 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � T X = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' = � mn � 0 Xn 0 � I( 0 X−n 0 ) = F(X)F(X)∗ which is the desired relation for the covariance ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In order to handle the kernel for the induced representation on the quotient 56 7 Fock representation Alexander Frei we need to take a closer look into the ideal of compact operators from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2: We begin for this with the well-known approximation by “finite rank matrices”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' More precisely, one has for compact operators on Fock space and S ⊆ N: � 0 K[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='] 0 � ∋ � 0 k(S) 0 � � k00 k01 k10 k11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � ∈ K �� A X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' S→N Indeed one easily verifies this (using Dirac calculus from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1): � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 0 1 �� A X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � (A∗ X∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') 0 � A X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � (A∗ X∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 0 1 � In particular we obtain for the diagonal compact operators as in 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2: K �� A X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � I � ∩ � L(A) L(X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � = � I XIX∗ → 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Meanwhile the author would like to take a moment to thank his previous tutor Dominic Enders for highlighting this perspective during personal discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The idea is now to use the previous relation in contrast to the following observation by Katsura which we reformulate in our language: Consider for this the diagonal operators on Fock space � L(A) L(X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � ⊆ L � FX = � A X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' �� and the representation between such diagonal operators: \uf8eb \uf8ed 0 L(Xn) 0 0 \uf8f6 \uf8f8 \uf8eb \uf8ed 0 0 L(Xn) ⊗ 1 0 \uf8f6 \uf8f8 ⊆ \uf8eb \uf8ed 0 0 L(Xn ⊗ X) 0 \uf8f6 \uf8f8 This generally fails to define a faithful representation: one may for instance consider the graph correspondence for any finite acyclic graph such as X = ℓ2� E = • � =⇒ XX = ℓ2� EE = ∅ � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Katsura’s crucial obervation tells us now that this becomes faithful when re- stricted to the subspace of compact operators by our covariance ideal: Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3 ([Kat04, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The representation above defines an 57 7 Fock representation Alexander Frei embedding when restricted to the maximal covariance as in proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2, \uf8eb \uf8ed 0 |Xn⟩ max(X, A) ⟨Xn| 0 0 \uf8f6 \uf8f8 ⊆ \uf8eb \uf8ed 0 0 |Xn⟩ max(X, A) ⟨Xn| ⊗ 1 0 \uf8f6 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the representation defines also an embedding for any covariance ideal below the maximal covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof from [Kat04]: We note for the kernel (in Dirac braket notation) ⟨Xn| ker � |Xn⟩ max(X, A) ⟨Xn| ↷ |Xn⟩ ⊗ |X⟩ � |Xn⟩ ⊆ ker � ⟨Xn|Xn⟩ max(X, A) ⟨Xn|Xn⟩ ↷ |X⟩ � ⊆ ker(A ↷ X) ∩ ker(A ↷ X)⊥ = 0 where we have used the obvious inclusion ⟨Xn|Xn⟩ max(X, A) ⟨Xn|Xn⟩ ⊆ max(X, A) ⊆ ker(A ↷ X)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (Note the intersection reflects also the first level as in proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') For an ideal such as the kernel above it holds however ⟨Xn| � ker[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='] = ker[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ]∗ ker[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='] � |Xn⟩ = 0 =⇒ ker[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='] |Xn⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we found the inclusion ker � |Xn⟩ max(X, A) ⟨Xn| ↷ |Xn⟩ ⊗ |X⟩ � ⊆ ker � |Xn⟩⟨Xn| ↷ |Xn⟩ � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In particular, the same holds true for any covariance below the maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With Katsura’s observation at hand we may now verify the desired kernel and covariance relation as in (12) which will resolve the second half of our classification of relative Cuntz–Pimsner algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We note here that the kernel relation is already due to Katsura as established in [Kat04]: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='4 (Relative Cuntz–Pimsner algebras: Kernel and Covariance): For the relative Cuntz–Pimsner algebra as above it holds ker � A → T → O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) � = 0 cov � X → T → O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) � = I As a consequence, the kernel–covariance pairs are also classifying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 58 7 Fock representation Alexander Frei Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We begin with the kernel relation due to Katsura from [Kat04]: As in the beginning discussion we need to verify the relation (18): I ⊆ max(X, A) : A � 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � ∩ T X � I 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � T X = 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we first revealed in proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2 that the ideal generated by our covariance (within the Toeplitz algebra) agrees with compact operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the intersection with the coefficient algebra reads A � 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � ∩ T X � I 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � T X = A � 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � ∩ � I |X⟩ I ⟨X| → 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In contrast, Katsura’s observation from proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3 tells us that these embed along the diagonal and as such the norm remains also constant throughout, � a a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � ∈ � I |X⟩ I ⟨X| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � : ∥a∥ = ∥a ↷ X∥ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' = ∥a ↷ Xn∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Both the vanishing of compact operators along the diagonal and the constant norm are only possible for the trivial intersection and as such the trivial kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We continue with the covariance relation from (12): For this we may now simply verify the common intersection as in (19) also using proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2: � A ∩ XX∗ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � ∩ T X � I 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � T X = = � A ∩ XX∗ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � ∩ � I IX∗ XI XIX∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � = � I 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the covariance does not increase and the theorem is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We have thus established also the second half in our classification: More precisely, we have first found that the class of relative Cuntz–Pimsner algebras exhausts the gauge-equivariant representations (which was the content of the gauge-invariant uniqueness theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand we now found that the parametrisation via kernel–covariance pairs is also classifying: O(K, I) = O(K′, I′) =⇒ (K, I) = (K′, I′) ✓ Altogether we have thus found: the lattice of kernel–covariance pairs parametrises the entire lattice of gauge-equivariant representations (as points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 59 8 Lattice structure Alexander Frei 8 Lattice structure While our discussion (so far) captured the lattice of gauge-equivariant represent- ations as individual points along the lattice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' this still leaves open how the lattice structure of kernel–covariance pairs reflects the lattice structure of gauge-equivariant representations (among each other) to which we now get: Recall for this that our kernel–covariance pairs encode the covariance for the quotient correspondence (which rendered the representation faithful) (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) B : K = ker � A → B � =⇒ I = cov � X XK → B � and its intrinsic characterisation on the quotient (as bounded ideal) � K ⊴ A ��� X∗KX ⊆ K � =⇒ � I ⊴ A/K ��� I ⊆ max � X XK � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We therefore begin with a translation of our kernel–covariance pairs which lives on the original correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This allows us to give an intrinsic or- der on kernel–covariance pairs reflecting the lattice structure of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Along this translation we further reveal Katsura’ mysterious T-pairs as nothing but our original kernel–covariance pairs (with maximal covariance in disguise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For our translation we first recall that the covariance for an embedding (faithful representation) may be simply read off as common intersection within the ambient algebra (as in proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='8) (X, A) ⊆ B : cov(X → B) = im(A → B) ∩ im(XX∗ → B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' More precisely, one may take the portion of the coefficient algebra cov(X → B) = � a ∈ A ��� a ∈ im(XX∗ → B) � = A ∩ im(XX∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' That is however the same amount of information as the actual intersection, as long as one keeps track of the embedding for the coefficient algebra: 60 8 Lattice structure Alexander Frei B XX∗ A im(XX∗) im(XX∗) im(A) With this picture in mind, we continue on some general representation (X, A) � X XK , A K � B : K = ker(A → B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We first note that for a quotient (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' surjective mapping) there is absolutely no loss of generality when pulling back any ideal along the quotient since for I ⊴ A/K ⇝ � A → A K �−1I ⊴ A : I = � A → A K �� A → A K �−1I and so we may use the equivalent intrinsic definition of kernel–covariance pairs as those with covariance below the maximal covariance within the quotient: (I + K) ⊴ A : I/K ⊆ max � X XK � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (20) We wrote the covariance ideal (here and below) as sum with the kernel ideal simply to guarantee that the ideal arises indeed as pullback from the quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand, note that the amount of covariance (as described above) does not change either in the sense of how much common intersection the coef- ficient algebra has with compact operators since (see also proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='10): im � A A K B � = im � A K B � , im � XX∗ � X XK �� X XK �∗ B � = im � � X XK �� X XK �∗ B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So there is really no loss of generality from this perspective either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we obtain another equivalent extrinsic definition of kernel–covariance pairs as those with covariance ideal describing the amount of common intersec- tion between the coefficient algebra and compact operators: (X, A) B : (I + K) = im(A → B) ∩ im(XX∗ → B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 61 8 Lattice structure Alexander Frei For comparison between kernel–covariance pairs we however take from now on the portion within the coefficient algebra as above, that is (I + K) = � a ∈ A ��� a ∈ im(XX∗ → B) � = A ∩ im(XX∗ → B) and we note this agrees with our intrinsic definition (somewhat obvious now).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Meanwhile we also keep in mind the viewpoint on the covariance as amount of common intersection as it provides an interesting perspective on representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With both these definitions at hand (the intrinsic and the extrinsic) we may now get to the lattice of gauge-equivariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Given a pair of representations we define the usual order of representations as � (X, A) → B � ≤ � (X, A) → B′ � : (X, A) B B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' More precisely, that is the representation factors over the other and note that the sole existence of such a factorisation entails a unique such as the representations are all generated as an operator algebra by (the image of) the correspondence: B = C∗(A ∪ X) =⇒ B B′ uniquely ✓ Given a factorisation we now easily infer for their kernel and covariance ker(A → B) ⊆ ker(A → B → B′), A ∩ im(XX∗ → B) ⊆ A ∩ im(XX∗ → B → B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Indeed the latter may be easily seen as (somewhat trivially) im(a → B) ∈ im(XX∗ → B) =⇒ im(a → B → B′) ∈ im(XX∗ → B → B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Schematically the amount of intersection could look something like: B B′ So we have found the following converse direction (using theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='4): � K ⊆ L ��� I + K ⊆ J + L � ⇐= O(K, I) ≤ O(L, J) ✓ What about the forward direction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' That is assume we have an inclusion of 62 8 Lattice structure Alexander Frei kernel–covariance pairs as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As we have an inclusion of kernel ideals we obtain in particular for their quotient correspondence (X, A) � X XK , A K � � X XL, A L � O(K, I) O(L, J) and so we may replace our correspondence as usual by the quotient � X XK , A K � ⇝ (X, A) =⇒ O(K, I) ⇝ O(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Recall that the relative Cuntz–Pimsner algebra satisfies (by definition) (X, A) O(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) B ⇐⇒ I ⊆ cov( X → B ) and as such we need to verify the least amount of covariance I ⊆ cov � (X, A) � X XL, A L � O(L, J) � ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This basically follows now from our study of kernel morphisms and covariance ideals from section 3, which we recall now for more clarity in our context: At first we found that kernel and cokernel morphisms have full covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' That is in our context the fully commutative diagram for the quotient morphism and on the other hand the covariance diagram for the quotient representation: A ∩ XX∗ A/L XX∗ � X XL �� X XL �∗ and J/L O(L, J) � X XL �� X XL �∗ O(L, J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Combing these with our assumption on covariance ideals we obtain (I + 0) J/L O(L, J) XX∗ � X XL �� X XL �∗ O(L, J) 63 8 Lattice structure Alexander Frei and as such the desired amount of covariance, (I + 0) ⊆ (J + L) =⇒ I ⊆ cov � (X, A) → O(L, J) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we also found the forward direction and so the order isomorphism, which is the main conclusion of this article: Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1 (Kernel–covariance pairs: Order isomorphism): The kernel–covariance pairs as in (20) define the order isomorphism � K ⊆ L ��� I + K ⊆ J + L � ⇐⇒ O(K, I) ≤ O(L, J) (21) and as such the lattice of kernel–covariance pairs with its natural order by inclusion describes the entire lattice of gauge-equivariant representations, equivalently the entire lattice of gauge-invariant ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Let us give an example of our result for some graph algebra, also to illustrate the ease of working with such kernel–covariance pairs: Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2 (Graph correspondences: gauge-invariant ideals): Consider a graph correspondence as in example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2, X = ℓ2� E = edges � , A = c0(vertices) and recall its quotient graphs as in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='5 (given by hereditary ideals as in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2) as well as their covariance ideals given by their sets of regular vertices as in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As an example consider the following graph and its quotient graphs /2 K = 0 : \uf8eb \uf8ec \uf8ec \uf8ed a b \uf8f6 \uf8f7 \uf8f7 \uf8f8 K = (a) : \uf8eb \uf8ec \uf8ec \uf8ed b \uf8f6 \uf8f7 \uf8f7 \uf8f8 K = (a ∪ b) : � ∅ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 64 8 Lattice structure Alexander Frei Then its lattice of gauge-equivariant representations reads (as Hasse diagram) K = 0 : K = (a) : K = (a ∪ b) : I = 0 I = (a) I = (a) I = (b) I = (a ∪ b) I = (a ∪ b) I = (a ∪ b) and so equivalently also the lattice of gauge-invariant ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Note how we now easily read off the order via kernel–covariance pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Before we continue let us make a few remarks on the order isomorphism: In particular the following discussion on connecting morphisms will basically cover the notion of suprema and infima as addressed in the following remark: Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3 (Lattice isomorphism: suprema and infima): Note that as both lattices are order isomorphic they will be also lattice isomor- phic as unions and intersections (finite or arbitrary) as well as top and bottom elements are determined as suprema and infima respectively order iso � sup s as � = sup s � order isoas � which aside their existence depend only on the given order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Put in other words the notion of a lattice is really just a pure property and defines no additional structure so there is really no difference between the lattice of gauge-equivariant representations and the lattice of kernel–covariance pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We will however later discover that arbitrary suprema and infima of kernel–covariance pairs do not necessarily always arise as intersections and sums of their kernel and covariance ideals, that is we only have inf s (Ks|Is) ≤ � � s Ks ���� � s Is � and sup s (Ks|Is) ≥ � � s Ks ���� � s Is � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Indeed while the intersection und sum of invariant ideals remain invariant ⟨X| �� Ks � |X⟩ ⊆ � Ks and ⟨X| �� Ks � |X⟩ ⊆ � Ks 65 8 Lattice structure Alexander Frei the intersection and sum of covariance ideals may not always end up below the maximal covariance, � Is ⊆ max � X X(� Ks) � and � Is ⊆ max � X X(� Ks) � ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Schematically that is the next possible kernel–covariance pair may lie just further beyond (as seen within the lattice of any pairs of ideals) such as � K : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ��� I ⊆ max( X XK ) ✓ � � (K1 + K2) : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ��� (I1 + I2) ⊈ max � X X(K1+K2) � � � Ks : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ��� Is ⊆ max � X XKs � ✓ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For example the requirement to have at least as much covariance as all the given covariance ideals can force larger kernel than just the sum of given kernel ideals, or put more drastically there may exist no connecting morphism from each relative Cuntz–Pimsner algebra to the cokernel strand over the sum of kernel ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We will see such an example in 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='5 below involving already even just a pair of kernel–covariance pairs: (X, A) � X XK1 , A K1 � � X XK2 , A K2 � � X XK , A K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' O(K1, I1) O(K2, I2) O(K, I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' × × Furthermore we note that from either description the order defines a partial order (as opposed to just a preorder) as easily seen from � (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) C∗(X ∪ A) = B � and � (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) C∗(X ∪ A) = B′ � : \uf8eb \uf8ec \uf8ec \uf8ed (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) B B′ B \uf8f6 \uf8f7 \uf8f7 \uf8f8 = \uf8eb \uf8ec \uf8ec \uf8ed (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) B B \uf8f6 \uf8f7 \uf8f7 \uf8f8 so one is a retract of the other and similarly the other way around,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' or one may 66 8 Lattice structure Alexander Frei equivalently also argue using their kernel from the Toeplitz algebra � T X B B′ � =⇒ ker � T X → B � ⊆ ker � T X → B′� from which they had been already the same quotient ker � T X → B � = ker � T X → B′� =⇒ � T X B = B′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Or one may now also argue using kernel–covariance pairs, � K ⊆ L ⊆ K ��� I ⊆ J ⊆ I � =⇒ � K = L ��� I = J � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Altogether we remark that the lattice of gauge-equivariant representations co- incides entirely with the lattice of kernel–covariance pairs, while suprema and infima of kernel–covariance pairs may lay only beyond of just the intersection and sum of their kernel and covariance ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this in mind we continue on the remaining questions from section 4: More precisely, we wish to find suitable characterisations when morphisms exists between different quotient strands (based on kernel–covariance pairs) � X XK , A K � O(K, 0) O(K, I) O(K, max) � X XL, A L � O(L, 0) O(L, J) O(L, max) and we warn ahead that these won’t always exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Now at first we note that clearly the existence infers the inclusion of kernel ideals and we may for simplicity replace the original correspondence with the quotient (X, A) := � X XK , A K � O(K = 0, 0) O(K = 0, I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In particular there always exist connecting morphism from at least the Toeplitz algebra and so also some further relative Cuntz-Pimsner algebras T � X XK � = O(K, 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' O(K, I =?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' T � X XL � = O(L, 0) O(L, J) O(L, max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ✓ ✓ ✓ × As such the first question is to find the smallest relative Cuntz–Pimsner algebra 67 8 Lattice structure Alexander Frei from which connecting morphisms exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This may be now easily solved as O(K, I) O(L, J) ⇐⇒ (I + K) ⊆ (J + L) and as such the largest covariance ideal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' within the maximal covariance) simply arises as intersection with the given covariance from the quotient, I = max � X XK � ∩ � A/K → A/L �−1J = max � X XK � ∩ (J + K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Schematically the intersection can look something like this: A/K J ∩ max max � X XK � A/L J max � X XL � The reader may easily find some examples with (using graph algebras as above): max � X XK � ̸= 0 : (J ∩max) = 0 / 0 ̸= (J ∩max) ̸= max / (J ∩max) = max ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Moreover one may now easily guess the meet of kernel–covariance pairs: � s ( Ks | Is ) = � K = � Ks ���� I = � Is ∩ max � X XK � � So far about connecting morphisms from preceeding cokernel strands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The other direction however is more interesting: That is a relative Cuntz– Pimsner does not necessarily connect to every following quotient correspondence and in there not even beginning at every relative Cuntz-Pimsner algebra either, O(K, 0) O(K, I) O(K, max) O(L =?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', 0) O(L =?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', J =?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=') O(L =?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Note this also describes the lattice of gauge-invariant ideals for the given relative Cuntz–Pimsner algebra (simply as each such defines a quotient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We note however that as soon as it connects to another relative Cuntz–Pimsner algebra (in some following quotient correspondence) then it certainly also does so to the absolute Cuntz–Pimsner algebra for that quotient correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 68 8 Lattice structure Alexander Frei As such this introduces an obstruction which may be now handle using (21): O(K, I) ≤ O(L, max) ⇐⇒ Jmin := � A/K → A/L � I ⊆ max � X XL � This condition fails from time to time (we provide a simple example below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In case this condition is met we obtain as smallest solution O(K, 0) O(K, I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' O(L, 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' O(L, Jmin) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' O(L, max) × ✓ ✓ ✓ while in case the condition fails then there simply is no connecting morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we also obtain the lattice for any relative Cuntz-Pimsner algebra which is really just our main result restated (while this also generalizes O-pairs): Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='4 (Relative Cuntz-Pimsner algebra: gauge-invariant ideals): Consider a relative Cuntz-Pimsner algebra (as described in section 4) (X, A) � X XK , A K � O(K, 0) O(K, I) for some kernel–covariance pair as in (20) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Then its lattice of gauge-invariant ideals simply runs over pairs as in (21) � K ⊆ L ��� I + K ⊆ J + L � ⇐⇒ � K ⊆ L ��� Jmin ⊆ J ⊆ max � X XL � � or in words simply over all larger kernel–covariance pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Let us give an example for when there is no connecting morphism: Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='5 (Graph correspondences: no connecting morphism): Consider as an example the following graph and its quotient graphs /2 K = 0 : � a → b � K = (a) : � b � K = (a ∪ b) : � ∅ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Then there is no connecting morphism for the absolute Cuntz-Pimsner algebras 69 8 Lattice structure Alexander Frei between the first and second quotient (by simply reading off covariance ideals): T � X = ℓ2(a → b) � T � X = ℓ2(b) � T (X = 0) O � X = ℓ2(a → b) � O � X = ℓ2(b) � O(X = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' × Indeed the obstruction fails for the covariance ideal: I1 = (b) = reg � a → b � and K2 = (a) ⊆ her(a → b) : (A → A/K2)I1 = (b) ⊈ reg � quotient graph = b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The issue here is that the hereditary ideal is simply not saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Alternatively one may note that the first defines the simple algebra of 2×2 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In particular, we obtain for the join of kernel–covariance pairs � K1 = 0 ��� I1 = (b) � ∨ � K2 = (a) ��� I2 = (a) � = � K = (a ∪ b) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In other words, the join as next possible kernel–covariance pair lies only beyond of just the sum of kernel and covariance ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So we found an example for the issue (mentioned further above) that suprema and infima will be generally beyond just intersections and sums of ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We finish this section with a widely missed identification of Katsura’s work: That is we clarify how Katsura’s T-pairs (and O-pairs) are nothing but the pullback version of our kernel–covariance pairs from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' More precisely, we elaborate Katsura’s cryptic requirement J(K) := � a ∈ A ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' and aX−1(K) ⊆ K � : K ⊆ I ⊆ J(K) and how this defines a translation of the constraint from proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='6: That is any covariance for an embedding into an operator algebra is necessarily orthogonal to the kernel (for its left action) which read in our case cov � X XK → B � ⊥ ker � A K ↷ X XK � and as such these cannot exceed the maximal covariance (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Katsura’s ideal) cov � X XK → B � ⊆ �� X XK �� X XK �∗ ∩ ker � A K ↷ X XK �⊥� = max � X XK � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='Lattice structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='Alexander Frei ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='As such we found instead our kernel–covariance pairs as given by invariant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='ideals as kernel (which defines some sort of discrete range for kernel ideals) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='K ⊴ A : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='X∗KX ⊆ K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='together with ideals bounded from above as covariance (which defines an ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='upper bound on the range of covariance ideals) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='I ⊴ A/K : ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='0 ⊆ I ⊆ max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='XK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='while we found in the second ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='half ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='our ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='classification that each ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='such kernel–covariance pair indeed arises itself (more precisely theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In order to establish Katsura’s requirement we first note the obvious I = � A → A/K �−1 J ⇐⇒ K ⊆ I and one the other hand the inclusion (for quotient maps) J ⊆ max � X XK � ⇐⇒ � A → A/K �−1 J ⊆ � A → A/K �−1 max � X XK � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such Katsura’s condition simply states (see [Kat07, lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2]) J(K) = (A → A/K)−1 max � X XK � for which it further suffices to verify (see also [Kat07, lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2]) � aX−1(K) ⊆ K � = � A → A/K �−1 ker � A K ↷ X XK �⊥ since the dotted condition represents nothing but compactly acting coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This is however now easily verified: Consider for this the pullback (which con- tains the same information) ker � A ↷ X XK � = � A → A/K �−1 ker � A K ↷ X XK � : � A → A/K �� A → A/K �−1 ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ) = ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ) and which further reads (as in proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='9) ker � A ↷ X XK � = � a X XK = 0 � = {aX ⊆ XK} = {X∗aX ⊆ K} = X−1(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Put together we obtain the desired relation for the orthogonal complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We note that the relation has been worked out by Katsura in [Kat07, lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2] 71 9 Pimsner dilations Alexander Frei which however has been not continued further on: Katsura chose to work with the cryptic requirement instead of pursuing their kernel–covariance counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Possibly because they got only partially recognized as covariance ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Finally the author notes that the results here arose from a more detailed study of [Kat07] which builds on [FMR03] and further [KPW98] and [Pim97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' More precisely, the author realized the relations drawn in [Kat07, lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='10] (which extend [FMR03, lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='9]) as a partial result on categorical kernel and cokernel morphisms which led to their intrinsic characterization (in theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3) and so also on the range of possible kernel ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand the author realized the first observation made in [Kat04, proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3] as an intrinsic characterisation on the range of possi- ble covariance ideals for the induced representation on the quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' These let the author to systematically employ such kernel–covariance pairs, which allowed on one hand to handle the general version of the gauge- invariant uniqueness-theorem by reduction to the faithful case which follows from the sleek and simplifying proof by Evgenios Kakariadis in [Kak16] (which draws from the second observation made in [Kat04, proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3]) and on the other hand the critical observation made by Takeshi Katsura in [Kat04, lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='7] in his seminal paper from 2004, which led the author to retrieve kernel–covariance pairs from their relative Cuntz-Pimsner algebra (in theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The main difference however is that we didn’t need to build any ad-hoc semi-kind-of categorical pushout for correspondences as was handled in [Kat07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Instead it is all based on the simple idea of reduction to faithful representations using kernel and cokernel morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 9 Pimsner dilations This final section introduces the notion of dilations and verifies the existence of the maximal dilation as Hilbert bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We further reveal Katsura’s construction as a particular nonmaximal dilation and illustrate the lack of minimal dilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Meanwhile, the author would like to take this opportunity to thank Ralf Meyer for sharing his enlightening perspective on the Pimsner dilation as maximal dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We begin with the concept of dilations: More precisely that is any gauge- equivariant factorisation over some intermediate correspondence such as (Y, B) (X, A) O(K, I) 72 9 Pimsner dilations Alexander Frei where the gauge-equivariance boils down to simply Y O(K, I)(1) and B O(K, I)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As the original correspondence generates the relative Cuntz–Pimsner algebra, so does also the intermediate one C∗(X ∪ A) = O(K, I) =⇒ C∗(Y ∪ B) = O(K, I) whence the factorisation defines a relative Cuntz–Pimsner algebra itself: (Y, B) O(K, I) = O � Y, B ��� L =?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' J =?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � As such the task is now to find dilations which generate the relative Cuntz– Pimsner algebra as an absolute Cuntz–Pimsner algebra: (X, A) � Y =?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=', B =?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' � O(K, I) = O � Y, B ��� L = 0, J = max � ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As the kernel should be trivial we have no choice than to look within the rel- ative Cuntz–Pimsner algebra itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Also we may as usual assume that our original correspondence embeds itself (simply by replacing our original corres- pondence by its quotient correspondence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such our intermediate correspon- dence necessarily arises as an intermediate subspace X ⊆ Y ⊆ O(K = 0, I)(1) and A ⊆ B ⊆ O(K = 0, I)(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand we found (as a well-known description) that the absolute Cuntz–Pimsner algebra arises as the smallest gauge-equivariant quotient for which the coefficient algebra faithfully embeds into (and so also the correspon- dence) or in other words the coefficient algebra detects the gauge-invariant ideals within the absolute Cuntz–Pimsner algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' So we aim to find an intermediate subspace which detects the remaining gauge-invariant ideals J ⊴ O(K = 0, I) : B ∩ J = 0 =⇒ J = 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' By our main result (theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='4) we found a parametrisation for the entire 73 9 Pimsner dilations Alexander Frei lattice of gauge-equivariant representations O(K = 0, I = 0) O(K ̸= 0, I = 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' O(K = 0, I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' O(K = 0, max) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' given by kernel–covariance pairs � K ⊴ A : X∗KX ⊆ K ��� I ⊴ A K : I ⊆ max � X XK � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' and so also of gauge-invariant ideals (within the Toeplitz algebra) whence also for the relative Cuntz-Pimsner algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Our original coefficient algebra however already detects the kernel component: A ∩ T (X, A)(K ⊆ A)T (X, A) = 0 =⇒ K = 0 ✓ As such the only gauge-invariant ideals which our original coefficient algebra cannot detect are precisely the covariance ideals (with trivial kernel component) I ⊆ max(X, A) : A ∩ T (X, A) � I 0 � T (X, A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Furthermore we have already taken a quotient for some covariance (which brought us to our relative Cuntz–Pimsner algebra) and so the remaining gauge- invariant ideals arise as remaining quotients O(K = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I = 0) O(K = 0, I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' O(K = 0, max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we need to find an intermediate coefficient algebra which detects all of the remaining covariance ideals beyond the given one I ⊆ J ⊆ max(X, A): A ⊆ B ⊆ O(K, I)(0) : B ∩ O(K, I) � J 0 � O(K, I) ̸= 0 (22) with an intermediate subspace as correspondence (as described in section 1) X ⊆ Y ⊆ O(K, I)(1) : Y ∗Y ⊆ B, BY ⊆ Y, Y B ⊆ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such one may first choose an intermediate subalgebra which detects the re- maining covariance ideals (and if desired also a chosen subspace) and from there 74 9 Pimsner dilations Alexander Frei simply enlarge the chosen subalgebra to form a correspondence, for instance as the smallest correspondence above: �� � Y0 ⊆ Y ��B0 ⊆ B � ��� Y ∗Y ⊆ B, BY ⊆ Y, Y B ⊆ Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Indeed the intersection of any class of correspondences forms a correspondence: � Y = � Yn ��� B = � Bn � : Y ∗ n Yn ⊆ Bn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' =⇒ Y ∗Y ⊆ B, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We meanwhile need to verify that there always exists one above: For this we simply consider the maximal dilation (also known as Pimsner dilation) � Y = O(K, I)(1) ��� B = O(K, I)(0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Indeed the maximal dilation defines even a Hilbert bimodule and so also a correspondence simply as Fourier spaces define Fell bundles (confer section 5): O(K, I)(−1)O(K, I)(1) ⊆ O(K, I)(−1 + 1 = 0), O(K, I)(0)O(K, I)(1) ⊆ O(K, I)(0 + 1 = 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand the maximal dilation is also easily seen to detect all of the remaining covariance simply as each is generated from the fixed point algebra: � J 0 � ⊆ A � 1 1 � + XX∗� 0 1 � ⊆ O(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I)(0) =⇒ � J 0 � ⊆ O(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I)(0) ∩ � O(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) � J 0 � O(K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' I) � (23) In fact one may argue more generally: Given any operator algebra with a given circle action and consider its fixed point algebra (as in section 5) T ↷ B : B(n = 0) = � b ��� � b(z) := z ↷ b � ≡ b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Then its fixed point algebra detects every gauge-invariant subalgebra: � A = T ↷ A � ⊆ B : A ∩ B(0) = 0 =⇒ A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Indeed this follows using the conditional expectation (confer section 5) T ↷ B : E(b) = � T � b(z) = z ↷ b � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As the subalgebra is gauge-invariant the conditional expectation does not leave 75 9 Pimsner dilations Alexander Frei the subalgebra (using its construction as Bochner integral) � A = T ↷ A � =⇒ E(A) = � T � z ↷ A ⊆ A � dz ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand, every operator algebra is spanned by its positive portion, A = pos(A) − pos(A) + i pos(A) − i pos(A), pos(A) := {0 ≤ a ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The conditional expectation (given by averaging) is however faithful on the positive portion and as such we have have found the detection E(pos A) ⊆ A ∩ B(0) = 0 =⇒ pos(A) = 0 =⇒ A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This well-known technique is quite worthwhile in other context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we have found the following familiar result with ease (note there was basically nothing left to prove anymore): Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1 (Maximal dilation: absolute Cuntz–Pimsner algebra): The maximal dilation realises relative Cuntz–Pimsner algebras as absolute one O(K, I) = O � Y = O(K, I)(1) ��� B = O(K, I)(0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' and further defines the maximal Hilbert bimodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This is now an immediate consequence from (23) satisfying (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This dilation however is rather large in the sense that there is not much control over its behavior (besides its universal description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For instance one may think of the maximal dilation similar to maximal Furstenberg boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Instead we therefore seek for some dilation small enough to be tractable com- binatorially while large enough to detect covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As explained above, one may for this simply begin with a small subalgebra which detects covariance and simply enlarge the chosen subalgebra to form a correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In practice one may for instance attempt to run the algorithm Y = BX + X + XB + BXB =⇒ B′ = B + Y ∗Y =⇒ Y ′ = B′Y + Y + Y B′ =⇒ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' (24) with implicit closed linear span as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' While there always exist a smallest correspondence above (as we found above) this process may never halt and whence leave us clueless about its combina- torial behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In good cases however, the algorithm halts and thus allows 76 9 Pimsner dilations Alexander Frei for its combinatorial description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This happens in particular for the canonical subalgebra given by the maximal covariance itself: A ⊆ � B = A + � max(X, A) 0 � � ⊆ O(K, I)(0) (25) whose sum defines a subalgebra since � max(X, A) 0 � A � 1 1 � ⊆ � max(X, A)A 0 � ⊆ � max(X, A) 0 � which is nothin but the relation (from proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1) (ϕ − τ) max(X, A) · ϕ(A) = (ϕ − τ) � max(X, A)A � ⊆ (ϕ − τ) max(X, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand its left action keeps the space invariant � max(X, A) 0 � X ⊆ (A + XX∗)X ⊆ X (26) and as such the algorithm halts right after the first round, Y = X + X � max(X, A) 0 � = XB : Y ∗Y = BX∗XB ⊆ B and as such we got the canonical dilation as a combinatorial object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Using the left and right shift (as in section 7) we further even note � max(X, A) 0 � ↷ X = max(X, A)(1 − RL) · XR = 0 (27) which is nothing but the obvious relation (see also proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1) τ(XX∗)τ(X) = τ(XX∗X) =⇒ (ϕ − τ) � A ∩ XX∗� τ(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This resambles Katsura’s construction from [Kat04] and so we refer to the canon- ical dilation given by the maximal covariance as Katsura dilation (and note also here that there was basically nothing left to prove anymore): Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2 (Katsura dilation: absolute Cuntz–Pimsner algebra): The canonical dilation given by the maximal covariance realises a relative Cuntz– Pimsner algebra as an absolute Cuntz–Pimsner algebra O(K = 0, I) = O � Y = X + X � max(X, A) 0 � ��� B = A + � max(X, A) 0 � � and the analogous dilation for kernel–covariance pairs with kernel ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This dilation may well fail to define a minimal dilation (detecting covariance) 77 9 Pimsner dilations Alexander Frei and even if minimal, it generally fails to be the only minimal dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This is now an immediate consequence from (25) satisfying (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We further provide examples for the failure of minimality in example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3 (Katsura dilation: intrinsic description): The canonical dilation given by the maximal covariance allows the intrinsic description as the operator algebra freely generated by their abstract copies � A = A � 1 1 � � ∪ � max(X, A)/I = � max(X, A) 0 � � with multiplication given by A � 1 1 � � max(X, A) 0 � ⊆ � A max(X, A) 0 � ⊆ � max(X, A) 0 � and similarly for the correspondence itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The analogous expression holds for kernel–covariance pairs with kernel ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This further recovers the particular description from [Kat07, definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' By part of our main result (the nontrivial part of theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='4) we found that the relative Cuntz–Pimsner algebra does not introduce additional kernel which is the faithful copy of the coefficient algebra (as a familiar result): (X, A) O(K = 0, I) : A = A � 1 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand the relative Cuntz–Pimsner algebra also does not introduce additional covariance (asides the already given covariance) which reads � A ∩ XX∗ 0 � ∩ T (X, A) � I 0 � T (X, A) = � I 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' and as such the added maximal covariance defines a faithful copy up to max(X, A)/I = � max(X, A) 0 � ⊆ O(K = 0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Note that the maximal covariance absorbs the coefficient algebra and as such their sum already defines a closed and thus complete operator algebra (see the quick proof (16) on the sum of an algebra and ideal from section 6): A � 1 1 � + � max(X, A) 0 � = A � 1 1 � + � max(X, A) 0 � ⊆ O(K = 0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 78 9 Pimsner dilations Alexander Frei In fact this holds in any representation as also their universal: C∗� A ∪ max(X, A)/I ��� A max(X, A)/I ⊆ max(X, A)/I � = A + max(X, A)/I = A + max(X, A)/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such any concrete representation which provides faithful disjoint copies for the coefficient algebra and the maximal covariance (mod covariance ideal) de- fines a faithful representation for their freely generated operator algebra: A ∩ max(X, A)/I = 0 ⊆ B =⇒ C∗(A ∪ max(X, A)/I) ⊆ B Indeed this simply follows by some basic linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' This holds in particular for their copy in the relative Cuntz–Pimsner algebra A � 1 1 � ∩ � max(X, A) 0 � = � ker(A ↷ X) 0 � ∩ � max(X, A) 0 � = 0 where we have used their trivial intersection ker(A ↷ X) ∩ max(X, A) ⊆ ker(A ↷ X) ∩ ker(A ↷ X)⊥ = 0 and as such the dilation arises as universal representation C∗� A ∪ max(X, A)/I ��� A max(X, A)/I ⊆ max(X, A)/I � = A � 1 1 � + � max(X, A) 0 � ⊆ O(K = 0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Note this defines a quite general argument which applies also in other context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Finally Katsura’s description is nothing but the isomorphism C∗� A ∪ M ��� AM ⊆ M � = � a ⊕ (im a + m) ∈ A ⊕ (im A + M ⊆ B) � which simply enforces faithful disjoint copies for any A → B: A = A(1 ⊕ 1), A(1 ⊕ 1) ∩ (0 ⊕ M) = 0, M = 0 ⊕ (M ⊆ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' There is however nothing special about this choice of formal description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Instead it is the universal description as freely generated copies with one ab- 79 9 Pimsner dilations Alexander Frei sorbing which captures its properties: C∗� A ∪ max(X, A)/I ��� A max(X, A)/I ⊆ max(X, A)/I � = A � 1 1 � + � max(X, A) 0 � ⊆ O(K = 0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The reader may now similarly argue for the correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We may now easily recover the classical result that any gauge-equivariant quotient for some (possibly relative) graph algebra remains a graph algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we first recall that any quotient correspondence (as kernel component) arises as a quotient graph (confer example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='5) A/K = c0(V )/c0 � H = hereditary � = c0 � W = V \\ H � X/XK = ℓ2(E)/ℓ2(EH) = ℓ2� F := WE � and as such we may replace the original graph by the quotient graph X = ℓ2(E := F), A = c0(V := W) =⇒ O(K = 0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand recall that any covariance ideal for a graph (in our case the quotient graph) arises simply as a regular set of vertices (confer example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='7): max(X, A) = c0 � regular � =⇒ I = c0 � R ⊆ regular � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With this notation in mind we may now find the canonical dilation as a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We note for this that as it was given by the algorithm above we may re- cover the canonical dilation as a combinatorial object from the original data, which boils down in our case to the canonical dilation arising as a graph: Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='4 (Katsura dilation: graph correspondences): The canonical dilation given by the maximal covariance as in (25) � XB = X + X � max(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) 0 � ��� B = A + � max(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' A) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='arises as the following canonical graph (with notation from above): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='Its coefficient algebra arises as the orthogonal sum of vertices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='W = singular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='∪ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='regular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='∪ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� regular \\ R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='Pimsner dilations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='Alexander Frei ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='together with the correspondence given by the graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='EW = E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='singular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='∪ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='regular ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='∪ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� regular \\ R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='and its left action given by (whence defining the range of edges) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='EW = (aE)W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='EW = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='EW = (bE)W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='while trivially acting for the left over last summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the graph reads in more classical terms W = � all vertices = singular ∪ regular � ∐ � regular \\ R � EW = E � all vertices � ∐ E � regular \\ R � with range and source map given by s(− ∐ ∅) = s(−) ∐ ∅ and s(∅ ∐ −) = ∅ ∐ s(−), r(− ∐ ∅) = r(−) ∐ ∅ = r(∅ ∐ −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Note that the combined summands basically recover the original graph whereas the last provides an additional copy to make up for the maximal covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand the edges all point into the original copy of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such this recovers the familiar construction for graph algebras: Any gauge- equivariant quotient arises as a graph algebra itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In order to find the canonical dilation as a graph it suffices to recover its coefficient algebra as an orthogonal sum of vertices (see example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In our case the coefficient algebra already reads as a sum of vertices A � 1 1 � + � max(X, A) 0 � = c0 � vertices �� 1 1 � + � c0 � regular � 0 � which we may decompose now further into an orthogonal sum: First the singular vertices (that is the nonregular ones) are trivially disjoint from the regular ones and as such define an orthogonal summand, vertices = singular ∪ regular : singular � 1 1 � ⊥ � regular regular � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand the sum on regular vertices may be also taken as regular � 1 1 � + � regular 0 � = � 0 regular � + � regular 0 � 81 9 Pimsner dilations Alexander Frei whose summands belong to the relative Cuntz–Pimsner algebra since τ(XX∗) = � 0 XX∗ � and (ϕ − τ) � A ∩ XX∗� = � A ∩ XX∗ 0 � which is available only for the compactly acting portion!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The latter summand vanishes precisely for the given covariance (see theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='4 or corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='3) � regular \\ R 0 � ̸= 0 and � R 0 � = 0 and as such the sum reduces to the non-zero vertices � 0 regular � + � regular 0 � = � 0 regular � + � regular \\ R 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the coefficient algebra for our dilation decomposes into vertices � 1 1 � + � regular 0 � = singular � 1 1 � + � 0 regular � + � regular \\ R 0 � and so we have found the vertices for our graph correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We may now simply read off the edges from our correspondence as X + X � max(X, A) 0 � = ℓ2(E) � singular � 1 1 � + � 0 regular � + � regular \\ R 0 � � with the source of edges as evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand its left action reads (using the induced morphism on compact operators from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='1) � a a � EW = ϕ(a)τ(E)W = τ(aE)W = (aE)W � 0 b � EW = τ(b)τ(E)W = τ(bE)W = (bE)W � c 0 � EW = (ϕ − τ)cτ(E)W = 0 and so we have found the desired graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We illustrate the canonical dilation for the following prominent graph: Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='5 (Katsura dilation: Toeplitz graph): Consider the correspondence given by the single loop � X = ℓ2 � a x � = Cx ����� A = c0 � vertices � = Ca � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 82 9 Pimsner dilations Alexander Frei Its Toeplitz algebra recovers the traditional Toeplitz algebra since T (X, A) = C∗� x ∪ a ��� x∗x = a, ax = x � = C∗� x∗x = 1 � = T and as such its suggestive name as Toeplitz graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' On the other hand its absolute Cuntz–Pimsner algebra recovers the traditional circle algebra O(X, A) = C∗� x ∪ a ��� x∗x = a = xx∗ � = C∗� x∗x = 1 = xx∗ � = C(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' These already define all its gauge-equivariant representations: Indeed there is only a single covariance ideal given by the single vertex max(X, A) = c0 � regular = a = vertices � = A and no further quotient graph (except the trivial one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we found the entire lattice of gauge-equivariant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Further we may now compute the graph for the canonical dilation: For this we may now simply read off the graph as (confer corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='4) � W = (a) ∐ (a \\ ∅) = a ∐ a ��� EW = (xa) ∐ (xa) = x ∐ x � : \uf8eb \uf8ed s(x ∐ ∅) = a ∐ ∅, s(∅ ∐ x) = ∅ ∐ a r(x ∐ ∅) = a ∐ ∅ = s(∅ ∐ x) \uf8f6 \uf8f8 =⇒ \uf8eb \uf8ec \uf8ec \uf8ed a ∐ ∅ x ∐ ∅ ∅ ∐ a ∅ ∐ x \uf8f6 \uf8f7 \uf8f7 \uf8f8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Put together we found the realisation for the Toeplitz algebra: T \uf8eb \uf8ed \uf8f6 \uf8f8 = O \uf8eb \uf8ed \uf8f6 \uf8f8 The latter appears sometimes as well under the name Toeplitz graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We now continue with the issue about the existence of minimal dilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we begin with the following positive result for relative graph algebras: Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='6 (Katsura dilation: minimal dilation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For graph correspondences the canonical dilation is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' That is roughly speaking, there is no smaller graph which realises the relative graph algebra as an absolute one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Consider a subalgebra detecting each covariance B ⊆ c0(regular): B ∩ c0 � S ⊆ regular � = 0 =⇒ c0 � S ⊆ regular � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' 83 9 Pimsner dilations Alexander Frei Then the subalgebra necessarily contains each summand and so also c0 � S = {r} � = C(r) ⊆ B =⇒ B = � rC(r) = c0 � regular � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the maximal covariance defines a minimal dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' The previous observation suggests now the following result for correspon- dences over spaces (as coefficient algebra) and in particular for integer actions: Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='7 (Katsura dilation: correspondences over spaces): It holds for correspondences with abelian maximal covariance (and so also for correspondences over spaces): The canonical dilation given by the maximal co- variance defines a minimal dilation if and only if the maximal covariance has discrete spectrum (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' the maximal covariance defines a discrete subspace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Maximal covariances with totally disconnected spectra are not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Note that the entire problem deals with an abelian operator algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' By Stone–Weierstrass it thus suffices to focus on any ideal over some point in the spectrum and any subalgebra over some pair of points of the form ker � ω : B → C � = � b(ω) = 0 � and eq � ωi : B ⇒ C � = � b(ω1) = b(ω2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For the ideal we first observe for a pair of closed subspaces E, F ⊆ X = ΓB: ker(E) ∩ ker(F) = ker(E ∪ F) = 0 ⇐⇒ E ∪ F = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the only ideals some non-isolated point cannot detect are ker(ω) ∩ ker(F) = 0 : ker � F = X − ω = X � = 0 whence the non-isolated point would already detect every ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Suppose on the other hand that the spectrum defines a discrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Then every subalgebra as above fails detect even two ideals: eq � ω1, ω2 � ∩ ker(X − ωi) = � b(X − ωi) = 0 = b(ωi) � = 0 The corollary now follows from combining these two observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' We leave the final statement about totally disconnected spaces to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' With the previous corollary in mind, we may now illustrate some negative examples on the existence of minimal dilations altogether: 84 9 Pimsner dilations Alexander Frei Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='8 (Minimal dilations: failures of existence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' For this we may now simply consider any integer action as our correspondence, Z ↷ C0(space) =⇒ X = C0(space) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Note that any integer action defines a regular correspondence and as such the coefficient algebra defines the maximal covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Consider now some contin- uous space such as for instance the real line for which we thus obtain: a) The canonical dilation fails to be minimal as for instance the following ideals already detect each covariance C0(R − r) = ker(r) ⊆ C0(R) : C0(R − r) ∩ I = 0 =⇒ I = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' b) On the other hand none of those is minimal either since furthermore detection of covariance already happens for those with discrete complement such as for C0 � R − {r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' , rn} � = ker � r1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ∪ rn � ⊆ C0(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Consider now some enumeration of rational numbers {q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='} = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such we obtain some decreasing sequence of ideals with detection: ker(q1) ⊇ ker(q1 ∪ q2) ⊇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ⊇ ker(q1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' ∪ qn) ⊇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Their intersection however fails to detect any covariance since ker � Q = {q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='} � = ker � {q1, q2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content='} = R � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' As such the given sequence of ideals admits no minimal dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' In fact we have even found that the axiom of choice fails to apply!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Acknowledgements The author would like to thank his supervisor Søren Eilers for his kind support and encouragements, as well as Evgenios Kakariadis for pointing out to [Kat07] as a valuable follow-up article by Takeshi Katsura, which inspired the author on finding the missing components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE1T4oBgHgl3EQfTwPq/content/2301.03083v1.pdf'} +page_content=' Moreover, the author acknowledges the support under the 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b/ANFQT4oBgHgl3EQfMDYd/content/tmp_files/2301.13266v1.pdf.txt @@ -0,0 +1,2001 @@ +arXiv:2301.13266v1 [cs.DC] 30 Jan 2023 +Stream-based Decentralized Runtime Verification +Ritam Ganguly +Department of Computer Science and Engineering +Michigan State University +East Lansing, USA +gangulyr@msu.edu +Borzoo Bonakdarpour +Department of Computer Science and Engineering +Michigan State University +East Lansing, USA +borzoo@msu.edu +Abstract—Industrial Control Systems (ICS) are often built +from geographically distributed components and often use pro- +grammable logic controllers for localized processes. Since ver- +ification of such systems is challenging because of both time +sensitivity of the system specifications and the inherent asyn- +chrony in distributed components, developing runtime assurance +that verifies not just the correctness of different components, +but also generates aggregated statistics of the systems is of +interest. In this paper, we first present a general technique for +runtime monitoring of distributed applications whose behavior +can be modeled as input/output streams with an internal compu- +tation module in the partially synchronous semantics, where an +imperfect clock synchronization algorithm is assumed. Second, +we propose a generalized stream-based decentralized runtime +verification technique. We also rigorously evaluate our algorithm +on extensive synthetic experiments and several ICS and aircraft +SBS message datasets. +I. INTRODUCTION +Industrial Control Systems (ICS) are information systems +to control industrial processes such as manufacturing, product +handling, distribution, etc. It includes supervisory control +and data acquisition systems used to control geographically +dispersed assets and distributed control systems using a pro- +grammable logic controller for each of the localized processes. +A typical programmable logic controller (PLC) receives data +produced by a large number of sensors, fitted across the +system. The data produced by these components are often the +target of cyber and ransom-ware attack putting the security +of the system in jeopardy. Since these systems are linked +to essential services, any attack on these facilities put the +users life on the front line. The integrity of the data produced +from these distributed components is very important as the +PLC’s behavior is dictated by it. Recent attacks have shown +that an attack on a company’s ICS costs the company around +$5 million and 50 days of system down time. Additionally, +according to a recent report [1], it takes the effected company +around 191 days to fully recover and around 54% of all +organization are vulnerable to such attacks. +In this paper, we advocate for a runtime verification (RV) +approach, to monitor the behavior of a distributed system with +respect to a formal specification. Applying RV to multiple +components of an ICS can be viewed as the general problem of +distributed RV, where a centralized or decentralized monitor(s) +observe the behavior of a distributed system in which the +processes do not share a global clock. Although RV deals with +finite executions, the lack of a common global clock prohibits +it from having a total ordering of events in a distributed setting. +. +x +y +x + y +3 +5 +6 +9 +1 +3 +5 +7 +2(ǫ − 1) +2(ǫ − 1) +2(ǫ − 1) +1 +3 +4 +6 +9 +11 +{4} +{4, 6, 8} +{8} +{8, 10, 9, 11} +{11} +{11, 13, 14, 16} +{16} +Fig. 1: Partially Synchronous LOLA +In other words, the monitor can only form a partial ordering +of events which may yield different evaluations. Enumerating +all possible interleavings of the system at runtime incurs in +an exponential blowup, making the approach not scalable. To +add to this already complex task, a PLC often requires time +sensitive aggregation of data from multiple sources. +We propose an effective, sound and complete solution +to distributed RV for the popular stream-based specification +language LOLA [2]. Compared to other temporal logic, LOLA +can describe both correctness/failure assertions along with +statistical measures that can be used for system profiling and +coverage analysis. To present a high level of LOLA example, +consider two input streams x and y and a output stream, sum +as shown in Fig. 1. Stream x has the value 3 until time instance +2 when it changes to 5 and so on. +input x:int +input y:int +output sum := x+y +We consider a fault proof decentralized set of monitors +where each monitor only has a partial view of the system and +has no access to a global clock. In order to limit the blow- +up of states posed by the absence of the global clock, we +make a practical assumption about the presence of a bounded +clock skew ǫ between all the local clocks, guaranteed by a +clock synchronization algorithm (like NTP [3]). This setting is +known to be partially synchronous. As can be seen in Fig. 1, +any two events less than ǫ = 2 time apart is considered to +be concurrent and thus the non-determinism of the time of +occurrence of each event is restricted to ǫ − 1 on either side. +1 + +When attempting to evaluate the output stream sum, we need +to take into consideration all the possible time of occurrence +of the values. For example, when evaluating the value of sum +at time 1, we need to consider the value of x (resp. y) as 3 +and 5 (resp. 1 and 3) which evaluates to 4, 6 and 8. The same +can be observed for evaluations across all time instances. +Our first contribution in this paper is introducing a partially +synchronous semantics for LOLA. In other words, we define +LOLA which takes into consideration a clock-skew of ǫ when +evaluating a stream expression. Second, we introduce an SMT- +based associated equation rewriting technique over a partially +observable distributed system, which takes into consideration +the values observed by the monitor and rewrites the associated +equation. The monitors are able to communicate within them- +selves and are able to resolve the partially evaluated equations +into completely evaluated ones. +We have proved the correctness of our approach and the +upper and lower bound of the message complexity. Addi- +tionally, we have completely implemented our technique and +report the results of rigorous synthetic experiments, as well +as monitoring correctness and aggregated results of several +ICS. As identified in [4], most attacks on ICS components +try to alter the value reported to the PLC in-order to make +the PLC behave erroneously. Through our approach, we were +able to detect these attacks in-spite of the clock asynchrony +among the different components with deterministic guarantee. +We also argue that our approach was able to evaluate system +behavior aggregates that makes studying these system easier +by the human operator. Unlike machine learning approaches +(e.g., [5]–[7]), our approach will never raise false negatives. +We put our monitoring technique to test, studying the effects +of different parameters on the runtime and size of the message +sent from one monitor to other and report on each of them. +Organization: Section II presents the background con- +cepts. Partially synchronous LOLA and the formal problem +statement are introduced in Section III. Our RV technique is +collectively presented in Sections Section IV – VII followed +by the experimental results in Section VIII. Related work is +discussed in Section IX before we make concluding remarks +in Section X. Details of syntax of LOLA, proofs of correctness +and more details about the ICS case studies can be found in +the Appendix XI. +II. PRELIMINARIES – STREAM-BASED SPECIFICATION +LANGUAGE (LOLA) [2] +A LOLA [2] specification describes the computation of +output streams given a set of input streams. A stream α of +type T is a finite sequence of values, t ∈ T. Let α(i), where +i ≥ 0, denote the value of the stream at time stamp i. We +denote a stream of finite length (resp. infinite length) by T∗ +(resp. Tω). +Definition 1: A LOLA specification is a set of equations +over typed stream variables of the form: +s1 = e1(t1, · · · , tm, s1, · · · , sn) +... +... +sn = en(t1, · · · , tm, s1, · · · , sn) +where s1, s2, · · · , sn are called the dependent variables, +t1, t2, · · · , tm +are +called +the +independent +variables, +and +e1, e2, · · · , en +are +the +stream +expressions +over +s1, · · · , sn, t1, · · · , tm. ■ +Typically, Input streams are referred to as independent +variables, whereas output streams are referred as dependent +variable. For example, consider the following LOLA specifi- +cation, where t1 and t2 are independent stream variables of +type boolean and t3 is an independent stream variable of type +integer. +s1 = true +s2 = t1 ∨ (t3 ≤ 1) +s3 = ite(s3, s4, s4 + 1) +s4 = s9[−1, 0] + (t3 +mod 2) +where, ite is the abbreviated form of if-then-else and stream +expressions s7 and s8 refers to the stream t1 with an offset of ++1 and −1, respectively. Due to space constrains we present +the full syntax of LOLA in Appendix XI-A. +The semantics of LOLA specifications is defined in terms +of the evaluation model, which describes the relation between +input and output streams. +Definition 2: Given a LOLA specification ϕ over indepen- +dent variables, t1, · · · , tm, of type, T1, · · · , Tm, and depen- +dent variables, s1, · · · , sn with type, Tm+1, · · · , Tm+n, let +τ1, · · · , τm be the streams of length N + 1, with τi of type +Ti. The tuple ⟨α1, · · · , αn⟩ of streams of length N + 1 is +called the evaluation model, if for every equation in ϕ +si = ei(t1, · · · , tm, s1, · · · , sn) +⟨α1, · · · , αn⟩ satisfies the following associated equations: +αi(j) = val(ei)(j) +for (1 ≤ i ≤ n) ∧ (0 ≤ j ≤ N) +where val(ei)(j) is defined as follows. For the base cases: +val(c)(j) = c +val(ti)(j) = τi(j) +val(si)(j) = αi(j) +For the inductive cases, where f is a function (e.g., arithmetic): +val +� +f(e1, · · · , ek) +� +(j) = f +� +val(e1)(j), · · · , val(ek)(j) +� +val +� +ite(b, e1, e2) +� +(j) = if val(b)(j) then val(e1)(j) +else val(e2)(j) +val(e[k, c])(j) = +� +val(e)(j + k) +if 0 ≤ j + k ≤ N +c +otherwise +■ +The set of all equations associated with ϕ is noted by ϕα. +2 + +Definition 3: A dependency graph for a LOLA specification, +ϕ is a weighted and directed graph G = ⟨V, E⟩, with vertex +set V = {s1, · · · , sn, t1, · · · , tm}. An edge e : ⟨si, sk, w⟩ +(resp. e : ⟨si, tk, w⟩) labeled with a weight w is in E iff the +equation for αi(j) in ϕα contains αk(j +w) (resp. τk(j +w)) +as a subexpression. Intuitively, an edge records that si at a +particular position depends on the value of sk (resp. tk), offset +by w positions. +Given a set of synchronous input streams {α1, α2, · · · , αm} +of respective type T = {T1, T2, · · · , Tm} and a LOLA +specification, ϕ, we evaluate the LOLA specification, given +by: +(α1, α2, · · · , αm) |=S ϕ +given the above semantics, where |=S denotes the synchronous +evaluation. +III. PARTIALLY SYNCHRONOUS LOLA +In this section, we extend the semantics of LOLA to one +that can accommodate reasoning about distributed systems. +A. Distributed Streams +Here, we refer to a global clock which will act as the “real” +timekeeper. It is to be noted that the presence of this global +clock is just for theoretical reasons and it is not available to +any of the individual streams. +We assume a partially synchronous system of n streams, +denoted by A = {α1, α2, · · · , αn}. For each stream αi, +where i ∈ [1, |A|], the local clock can be represented as a +monotonically increasing function ci : Z≥0 → Z≥0, where +ci(G) is the value of the local clock at global time G. Since +we are dealing with discrete-time systems, for simplicity and +without loss of generality, we represent time with non-negative +integers Z≥0. For any two streams αi and αj, where i ̸= j, +we assume: +∀G ∈ Z≥0. | ci(G) − cj(G) |< ǫ, +where ǫ > 0 is the maximum clock skew. The value of ǫ is +constant and is known (e.g., to a monitor). This assumption is +met by the presence of an off-the-shelf clock synchronization +algorithm, like NTP [3], to ensure bounded clock skew among +all streams. The local state of stream αi at time σ is given by +αi(σ), where σ = ci(G), that is the local time of occurrence +of the event at some global time G. +Definition 4: A distributed stream consisting of A = +{α1, α2, . . . , αn} streams of length N + 1 is represented by +the pair (E, ⇝), where E is a set of all local states (i.e., +E = ∪i∈[1,n],j∈[0,N]αi(j)) partially ordered by Lamport’s +happened-before (⇝) relation [8], subject to the partial syn- +chrony assumption: +• For every stream αi, 1 ≤ i ≤ |A|, all the events +happening on it are totally ordered, that is, +∀i, j, k ∈ Z≥0 : (j < k) → (αi(j) ⇝ αi(k)) +• For any two streams αi and αj and two corresponding +events αi(k), αj(l) ∈ E, if k+ǫ < l then, αi(k) ⇝ αj(l), +where ǫ is the maximum clock skew. +• For events, e, f, and g, if e ⇝ f and f ⇝ g, then +e ⇝ g. ■ +Definition 5: Given a distributed stream (E, ⇝), a subset of +events C ⊆ E is said to form a consistent cut if and only if +when C contains an event e, then it should also contain all +such events that happened before e. Formally, +∀e, f ∈ E.(e ∈ C) ∧ (f ⇝ e) → f ∈ C. ■ +The frontier of a consistent cut C, denoted by front(C) is the +set of all events that happened last in each stream in the cut. +That is, front(C) is a set of αi(last) for each i ∈ [1, |A|] and +αi(last) ∈ C. We denote αi(last) as the last event in αi such +that ∀αi(σ) ∈ C.(αi(σ) ̸= αi(last)) → (αi(σ) ⇝ αi(last)). +B. Partially Synchronous LOLA +We define the semantics of LOLA specifications for partially +synchronous distributed streams in terms of the evaluation +model. The absence of a common global clock among the +stream variables and the presence of the clock synchronization +makes way for the output stream having multiple values at any +given time instance. Thus, we update the evaluation model, so +that αi(j) and val(ti)(j) are now defined by sets rather than +just a single value. This is due to nondeterminism caused by +partial synchrony, i.e., the bounded clock skew ǫ. +Definition 6: Given a LOLA [2] specification ϕ over in- +dependent variables, t1, · · · , tm of type T1, · · · , Tm and de- +pendent variables, s1, · · · , sn of type Tm+1, · · · , Tm+n and +τ1, · · · , τm be the streams of length N + 1, with τi of type +Ti. The tuple of streams ⟨α1, · · · , αn⟩ of length N + 1 with +corresponding types is called the evaluation model in the +partially synchronous setting, if for every equation in ϕ: +si = ei(t1, · · · , tm, s1, · · · , sn), +⟨α1, · · · , αn⟩ satisfies the following associated equations: +αi(j) = +� +val(ei)(k) | max{0, j−ǫ+1} ≤ k ≤ min{N, j+ǫ−1} +� +where val(ei)(j) is defined as follows. For the base cases: +val(c)(j) = {c} +val(ti)(j) = +� +τi(k) | max{0, j − ǫ + 1} ≤ k ≤ min{N, j + ǫ − 1} +� +val(si)(j) = αi(j) +For the inductive cases: +val +� +f(e1, · · · , ep) +� +(j) = +� +f(e′ +1, · · · , e′ +p) | e′ +1 ∈ val(e1)(j), · · · , +e′ +p ∈ val(ep)(j) +� +val(ite(b, e1, e2))(j) = +� +val(e1)(j) +true ∈ val(b)(j) +val(e2)(j) +false ∈ val(b)(j) +val(e[k, c])(j) = +� +val(e)(j + k) +if 0 ≤ j + k ≤ N +c +otherwise +■ +Example 1: Consider the LOLA specification, ϕ, over the +independent boolean variables read and write: +3 + +read +write +count(read) +count(write) +check +1 +2 +3 +4 +5 +6 +7 +{0} +{0} +{0} +{0, 1} +{0, 1} +{1, 2} +{1, 2, 3} +{2, 3} +{0} +{0, 1} +{0, 1, 2} +{1, 2} +{2, 3} +{2, 3, 4} +{3, 4} +{4} +{true} +{true} +{true} +{true} +{true, false} +{true, false} +{true, false} +{true} +Fig. 2: Partially Synchronous LOLA Example +input read:bool +input write:bool +output countRead := ite(read, countRead[-1,0] ++ 1, countRead[-1,0]) +output countWrite := ite(write, +countWrite[-1,0] + 1, countWrite[-1,0]) +output check := (countWrite - countRead) <= 2 +In Fig. 2, we have two input stream read and write which +denotes the time instances where the corresponding events +take place. It can be imagined that read and write are streams +of type boolean with true values at time instances 4, 6, 7 +and 2, 3, 5, 6 and false values at all other time instances +respectively. We evaluate the above mentioned LOLA speci- +fication considering a time synchronization constant, ǫ = 2. +The corresponding associated equations, ϕα, are: +countRead(j) = + + + + + + + +ite(read, 1, 0) +j = 0 +ite +� +read, countRead(j− +1) + 1, countRead(j) +� +j ∈ [1, N) +countWrite(j) = + + + + + + + +ite(write, 1, 0) +j = 0 +ite +� +write, countWrite(j− +1) + 1, countWrite(j) +� +j ∈ [1, N) +check(j) = +� +countWrite(j) − countRead(j) +� +≤ 2 +Similar to the synchronous case, evaluation of the partially +synchronous LOLA specification involves creating the depen- +dency graph. +Definition 7: A dependency graph for a LOLA specification, +ϕ is a weighted directed multi-graph G = ⟨V, E⟩, with vertex +set V = {s1, · · · , sn, t1, · · · , tm}. An edge e : ⟨si, sk, w⟩ +(resp. e : ⟨si, tk, w⟩) labeled with a weight w = {ω | p − ǫ < +ω < p+ǫ} is in E iff the equation for αi(j) contains αk(j+p) +(resp. τk(j + p)) as a sub-expression, for some j and offset p. +■ +a +b1 +b2 +{0, 1, 2} +{0, 1, 2}, {−2, −1, 0}, {−3, −2, −1} +{−2, −1, 0} +Fig. 3: Dependency Graph Example +Intuitively, the dependency graph records that evaluation of +a si at a particular position depends on the value of sk (resp. +tk), with an offset in w. It is to be noted that there can be more +than one edge between a pair of vertex (si, sk) (resp. (si, tk)). +Vertices labeled by ti do not have any outgoing edges. +Example 2: Consider the LOLA specification over the inde- +pendent integer variable a: +input a : uint +output b1 := b2[1, 0] + ite(b2[-1,7] <= a[1, +0], b2[-2,0], 6) +output b2 := b1[-1,8] +Its dependency graph, shown in Fig. 3 for ǫ = 2, has 1 edge +from b1 to a with a weight {0, 1, 2}. Similarly, there are 3 +edges from b1 to b2 with weights {0, 1, 2}, {−2, −1, 0} and +{−3, −2, −1} and 1 edge from b2 to b1 with a weight of +{−2, −1, 0} +Given +a +set +of +partially +synchronous +input +streams +{α1, α2, · · · , α|A|} +of +respective +type +T += +{T1, T2, · · · , T|A|} and a LOLA specification, ϕ, +the evaluation of ϕ is given by +(α1, α2, · · · , α|A|) |=P S ϕ +where, |=P S denotes the partially synchronous evaluation. +IV. DECENTRALIZED MONITORING ARCHITECTURE +A. Overall Picture +We consider a decentralized online monitoring system com- +prising of a fixed number of |M| reliable monitor processes +M = {M1, M2, · · · , M|M|} that can communicate with each +other by sending and receiving messages through a complete +point-to-point bidirectional communication links. Each com- +munication link is also assumed to be reliable, i.e., there is +no loss or alteration of messages. Similar to the distributed +system under observation, we assume the clock on the indi- +vidual monitors are asynchronous, with clock synchronization +constant = ǫM. +Throughout this section we assume that the global dis- +tributed stream consisting of complete observations of |A| +streams is only partially visible to each monitor. Each monitor +process locally executes an identical sequential algorithm +which consists of the following steps (we will generalize +this approach in Section VII). In other words, an evaluation +iteration of each monitor consists of the following steps: +1) Reads the a subset of E events (visible to Mi) along +with the corresponding time and valuation of the events, +which results in the construction of a partial distributed +stream; +4 + +Algorithm 1 Behavior of a Monitor Mi, for i ∈ [1, |M|] +1: for j = 0 to N do +2: +Let (Ei, ⇝i)j be the partial distributed stream view of Mi +3: +LS j ← �(E, ⇝) |=P S ϕα +� +4: +Send: broadcasts symbolic view LSj +5: +Receive: Πj ← {LS k +j | 1 ≤ k ≤ M} +6: +Compute: LSj+1 ← LC (Πj) +7: end for +2) Each monitor evaluates the LOLA specification ϕ given +the partial distributed stream; +3) Every monitor, broadcasts a message containing rewrit- +ten associated equations of ϕ, denoted LS, and +4) Based on the message received containing associated +equations, each monitor amalgamates the observations of +all the monitors to compose a set of associated equations. +After a evaluation iteration, each monitor will have the +same set of associated equations to be evaluated on the +upcoming distributed stream. +The message sent from monitor Mi at time π to another +monitor Mj, for all i, j ∈ [1, |M|], during a evaluation +iteration of the monitor is assumed to reach latest by time +π + ǫM. Thus, the length of an evaluation iteration k can be +adjusted to make sure the message from all other monitors +reach before the start of the next evaluation iteration. +B. Detailed Description +We now explain in detail the computation model (see Algo- +rithm 1). Each monitor process Mi ∈ M, where i ∈ [1, |M|], +attempts to read e ∈ E, given the distributed stream, (E, ⇝). +An event can either be observable, or not observable. Due +to distribution, this results in obtaining a partial distributed +stream (Ei, ⇝) defined below. +Definition 8: Let (E, ⇝) be a distributed stream. We say that +(E′, ⇝) is a partial distributed stream for (E, ⇝) and denote it +by (E′, ⇝) ⊑ (E, ⇝) iff E′ ⊆ E (the happened before relation +is obviously preserved). ■ +We now tie partial distributed streams to a set of decentral- +ized monitors and the fact that decentralized monitors can only +partially observe a distributed stream. First, all un-observed +events is replaced by ♮, i.e., for all αi(σ) ∈ E if αi(σ) ̸∈ Ei +then Ei = Ei ∪ {αi(σ) = ♮}. +Definition 9: Let (E, ⇝) be a distributed stream and +M = {M1, M2, · · · , M|M|} be a set of monitors, where +each monitor Mi, for i ∈ [1, |M|] is associated with a +partial distributed stream (Ei, ⇝) ⊑ (E, ⇝). We say that these +monitor observations are consistent if +• ∀e ∈ E.∃i ∈ [1, |M|].e ∈ Ei, and +• ∀e ∈ Ei.∀e′ ∈ Ej.(e = e′ ∧ e ̸= ♮) ⊕ +� +(e = ♮ ∨ e′ = ♮) +� +, +where ⊕ denoted the exclusive-or operator. +In a partially synchronous system, there are different order- +ing of events and each unique ordering of events might eval- +uate to different values. Given a distributed stream, (E, ⇝), +a sequence of consistent cuts is of the form C0C1C2 · · · CN, +where for all i ≥ 0: (1) Ci ⊆ E, and (2) Ci ⊆ Ci+1. +Given the semantics of partially-synchronous LOLA, evalu- +ation of output stream variable si at time instance j requires +events αi(k), where i ∈ [1, |A|] and k ∈ +� +π | max{0, j − +ǫ + 1} ≤ π ≤ {N, j + ǫ − 1} +� +. To translate monitoring of a +distributed stream to a synchronous stream, we make sure that +the events in the frontier of a consistent cut, Cj are αi(k). +Let C denote the set of all valid sequences of consistent +cuts. We define the set of all synchronous streams of (E, ⇝) +as follows: +Sr(E, ⇝) = +� +front(C0)front(C1) · · · | C0C1 · · · ∈ C +� +Intuitively, Sr(E, ⇝) can be interpreted as the set of all possi- +ble “interleavings”. The evaluation of the LOLA specification, +ϕ, with respect to (E, ⇝) is the following : +� +(E, ⇝) |=P S ϕ +� += +� +(α1, · · · , αn) |=S ϕ | (α1, · · · , αn) ∈ +Sr(E, ⇝) +� +This means that evaluating a partially synchronous distributed +stream with respect to a LOLA specification results in a +set of evaluated results, as the computation may involve +several streams. This also enables reducing the problem from +evaluation of a partially synchronous distributed system to the +evaluation of multiple synchronous streams, each evaluating to +unique values for the output stream, with message complexity +O +� +ǫ|A|N|M|2� +Ω(N|M|2) +C. Problem Statement +The overall problem statement requires that upon the termi- +nation of the Algorithm 1, the verdict of all the monitors in +the decentralized monitoring architecture is the same as that of +a centralized monitor which has the global view of the system +∀i ∈ [1, m] : Resulti = +� +(E, ⇝) |=P S ϕ +� +where (E, ⇝) is the global distributed stream and ϕ is the +LOLA specification with Resulti as the evaluated result by +monitor Mi. +V. CALCULATING LS +In this section, we introduce the rules of rewriting LOLA as- +sociated equations given the evaluated results and observations +of the system. In our distributed setting, evaluation of a LOLA +specification involves generating a set of synchronous streams +and evaluating the given LOLA specification on it (explained +in Section VI). Here, we make use of the evaluation of LOLA +specification into forming our local observation to be shared +with other monitors in the system. +Given the set of synchronous streams, (α1, α2, · · · , α|A|), +the symbolic locally computed result LS (see Algorithm 1) +consists of associated LOLA equations, which either needs +more information (data was unobserved) from other monitors +to evaluate or the concerned monitor needs to wait (positive +offset). In either case, the associated LOLA specification is +shared with all other monitors in the system as the missing data +5 + +can be observed by either monitors. We divide the rewriting +rules into three cases, depending upon the observability of +the value of the independent variables required for evaluating +the expression ei for all i ∈ [1, n]. Each stream expression +is categorized into three cases (1) completely unobserved, (2) +completely observed or (3) partially observed. This can be +done easily by going over the dependency graph and checking +with the partial distributed stream read by the corresponding +monitor. +Case 1 (Completely Observed). +Formally, a completely +observed stream expression si can be identified from the +dependency graph, G = ⟨V, E⟩, as for all sk (resp. tk) +⟨si, sk, w⟩ ∈ E (resp. ⟨si, tk, w⟩ ∈ E), sk(j + w) ̸= ♮ (resp. +tk(j + w) ̸= ♮) are observed for time instance j. If yes, this +signifies, that all independent and dependent variables required +to evaluate si(j), is observed by the monitor M, there by +evaluating: si(j) = ei(s1, · · · , sn, t1, · · · , tm) and rewriting +si(j) to LS. +Case 2 (Completely Unobserved). +Formally, we present +a completely unobserved stream expression, si from the +dependency graph, G = ⟨V, E⟩, as for all sk (resp. tk), +⟨si, sk, w⟩ ∈ E (resp. ⟨si, tk, w⟩ ∈ E), sk(j + w) = ♮ (resp. +tk(j + w) = ♮) are unobserved, for time instance j . This +signifies that the valuation of neither variables are known to the +monitor M. Thus, we rewrite the following stream expressions +s′ +k(j) = +� +sk(j + w) +0 ≤ j + w ≤ N +default +otherwise +t′ +k(j) = +� +tk(j + w) +0 ≤ j + w ≤ N +default +otherwise +for all ⟨si, sk, w⟩ ∈ E and ⟨si, tk, w⟩ ∈ E, and include the +rewritten associated equation for evaluating si(j) as +si(j) = ei(s′ +1, · · · , s′ +n, t′ +1, · · · , t′ +m) +It is to be noted that the default value of a stream variable, +sk (resp. tk), depends on the corresponding type Tk (resp. +Tm+k) of the stream. +Case 3 (Partially Observed). Formally, we present a partially +observed stream expression, si from the dependency graph, +G = ⟨V, E⟩, as for all sk (resp. tk), they are either observed +or unobserved, for time instance j. In other words, we can +represent a set Vo = {sk | ∃sk(j + w) ̸= ♮} of all observed +stream variable and a set Vu = {sk | sk(j + w) = ♮} of all +unobserved dependent stream variable for all ⟨si, sk, w⟩ ∈ E. +The set can be expanded to include independent variables as +well. For all sk ∈ Vu (resp. tk ∈ Vu) that are unobserved, are +replaced by: +su +k(j) = +� +sk(j + w) +0 ≤ j + w ≤ N +default +otherwise +tu +k(j) = +� +tk(j + w) +0 ≤ j + w ≤ N +default +otherwise +a +b +a +b +1 +2 +3 +4 +5 +6 +1 +7 +5 +4 +4 +7 +3 +5 +9 +3 +5 +1 +Fig. 4: Example of generating LS +and for all sk ∈ Vo (resp. tk ∈ Vo) that are observed, are +replaced by: +so +k(j + w) = value +to +k(j + w) = value +and there by partially evaluating si(j) as +si(j) = ei(so +1, · · · , so +n, to +1, · · · , to +m, su +1, · · · , su +n, tu +1, · · · , tu +m) +followed by adding the partially evaluated associated equation +for si(j) to LS. It is to be noted, that a consistent partial +distributed stream makes sure that for all sk (resp. tk), can +only be either observed or unobserved and not both or neither. +Example 3: Consider the LOLA specification mentioned +below and the stream input of length N = 6 divided into +two evaluation rounds and ǫ = 2 as shown in Fig. 4 with the +monitors M1 and M2. +input a : uint +input b : uint +output c := ite(a[-1,0] <= b[1, 0], a[1,0], +b[-1, 0]) +The associated equation for the output stream is: +c = + + + + + + + + + +ite(0 ≤ b(i + 1), a(i + 1), 0) +i = 1 +ite(a(i − 1) ≤ b(i + 1), a(i + 1), +b(i − 1)) +2 ≤ i ≤ N − 1 +ite(a(i − 1) ≤ 0, 0, b(i − 1)) +i = N +Let +the +partial +distributed +stream +read +by +monitor +M1 +include +{a, (1, 1), (3, 5)}, {b, (2, 5), (3, 9)} +and +the +partial distributed stream read by monitor M2 +include +{a, (1, 1), (2, 7)}, {b, (1, 3), (3, 9). +Monitor +M1 +evaluates +c(2) = 5 and partially evaluates c(1) and c(3). Thus LS 1 +1 = +{c(1) = a(2), c(2) = 5, c(3) = ite(a(2) ≤ b(4), a(4), 5)}. +Monitor M2 partially evaluates all c(1), c(2) and c(3) +and thus LS 2 +1 = {c(1) = ite(0 ≤ b(2), a(2), 0), c(2) = +a(3), c(3) = ite(7 ≤ b(4), a(4), b(2))}. +Let +the +partial +distributed +stream +read +by +monitor +M1 include {a, (4, 4), (5, 4)}, {b, (4, 3), (6, 1)} and the par- +tial +distributed +stream +read +by +monitor +M2 +include +{a, (5, 4), (6, 7)}, {b, (4, 3), (5, 5)}. Monitor +M1 +evaluates +c(4) = 9 and c(5) = 3 and partially evaluates c(6). Thus +LS 1 +2 = {c(4) = 9, c(5) = 3, c(6) = b(5)}. Monitor M2 +evaluates c(6) = 5 and partially evalues c(4) and c(5) and +thus LS 2 +2 = {c(4) = ite(a(3) ≤ 5, 4, 9), c(5) = ite(a(4) ≤ +b(6), 7, 3), c(6) = 5}. +It is to be noted, the after the first round of evaluation, the +corresponding local states, LS 1 +1 and LS 2 +1 will be shared which +6 + +will enable evaluating the output stream for few of the partially +evaluated output stream (will be discussed in Section VII-A). +These will be included in the local state of the following +evaluation round. +Note that generating LS takes into consideration an ordered +stream. One where the time of occurrence of events and values +are comparable. It can be imagined that generating the same +for the distributed system involves generating it for all possible +ordering of events. This will be discussed in details in the +following sections.s. +VI. SMT-BASED SOLUTION +A. SMT Entities +SMT entities represent (1) LOLA equations, and (2) vari- +ables used to represent the distributed stream. Once we have +generated a sequence of consistent cuts, we use the laws +discussed in Section V, to construct the set of all locally +computer or partially computed LOLA equations. +Distributed Stream. In our SMT encoding, the set of events, +E, is represented by a bit vector, where each bit corresponds +to an individual event in the distributed stream, (E, ⇝). The +length of the stream under observation is k, which makes +|E| = k × |A| and the length of the entire stream is N. +We conduct a pre-processing of the distributed stream where +we create a E × E matrix, hbSet to incorporate the happen- +before relations. We populate hbSet as hbSet[e][f] = 1 iff +e ⇝ f, else hbSet[e][f] = 0. In order to map each event to +its respective stream, we introduce a function, µ : E → A. +We introduce a valuation function, val : E → T (whatever +the type is in the LOLA specification), in order to represent +the values of the individual events. Due to the partially +synchronous assumption of the system, the possible time of +occurrence of an event is defined by a function δ : E → Z≥0, +where ∀α(σ) ∈ E.∃σ′ ∈ [max{0, σ − ǫ + 1}, min{σ + ǫ − +1}, N].δ +� +α(σ) +� += σ′. We update the δ function when referring +to events on output streams by updating the time synchroniza- +tion constant to ǫM. This accounts for the clock skew between +two monitors. Finally, we introduce an uninterpreted function +ρ : Z≥0 → 2E that identifies a sequence of consistent cuts for +computing all possible evaluations of the LOLA specification, +while satisfying a number of given constrains explained in +Section VI-B. +B. SMT Constrains +Once we have defined the necessary SMT entities, we move +onto the SMT constraints. We first define the SMT constraints +for generating a sequence of consistent cuts, followed by the +ones for evaluating the given LOLA equations ϕα. +Constrains for consistent cuts over ρ: In order to make +sure that the uninterpreted function ρ identifies a sequence +of consistent cuts, we enforce certain constraints. The first +constraint enforces that each element in the range of ρ is in +fact a consistent cut: +∀i ∈ [0, k].∀e, e′ ∈ E. +� +(e ⇝ e′) ∧ (e′ ∈ ρ(i)) +� +→ (e ∈ ρ(i)) +Next, we enforce that each successive consistent cut consists +of all events included in the previous consistent cut: +∀i ∈ [0, k − 1].ρ(i) ⊆ ρ(i + 1) +Next, we make sure that the front of each consistent cut +constitutes of events with possible time of occurrence in +accordance with the semantics of partially-synchronous LOLA: +∀i ∈ [0, k].∀e ∈ front(ρ(i)).δ(e) = i +Finally, we make sure that every consistent cut consists of +events from all streams: +∀i ∈ [0, k].∀α ∈ A.∃e ∈ front(ρ(i)).µ(e) = α +Constrains for LOLA specification: These constraints will +evaluate the LOLA specifications and will make sure that ρ +will not only represent a valid sequence of consistent cuts but +also make sure that the sequence of consistent cuts evaluate +the LOLA equations, given the stream expressions. As is +evident that a distributed system can often evaluate to multiple +values at each instance of time. Thus, we would need to +check for both satisfaction and violation for logical expressions +and evaluate all possible values for arithmetic expressions. +Note that monitoring all LOLA specification can be reduce +to evaluating expressions that are either logical or arithmetic. +Below, we mention the SMT constraint for evaluating different +LOLA equations at time instance j: +ti[p, c] = +� +val(e) +0 ≤ j + p ≤ N +c +otherwise +� +∃e ∈ front(ρ(j + p)).(µ(e) = αi) +� +si(j) = true front(ρ(j)) |= ϕα +(Logical expression, satisfaction) +si(j) = ei(∀e ∈ front(ρ(j)).val(e)) +(Arithmetic expression, evaluation) +The previously evaluated result is included in the SMT in- +stance as a entity and a additional constrain is added that only +evaluates to unique value, in order to generate all possible +evaluations. The SMT instance returns a satisfiable result iff +there exists at-least one unique evaluation of the equation. This +is repeated multiple times until we are unable to generate a +sequence of consistent cut, given the constraints, i.e., generate +unique values. It is to be noted that stream expression of the +form ite(si, sk, sj) can be reduced to a set of expressions +where we first evaluate si as a logical expression followed by +evaluating sj and sk accordingly. +VII. RUNTIME VERIFICATION OF LOLA SPECIFICATIONS +Now that both the rules of generating rewritten LOLA +equations (Section V) and the working of the SMT encoding +(Section VI) have been discussed, we can finally bring them to- +gether in order to solve the problem introduced in Section IV. +7 + +Algorithm 2 Computation on Monitor Mi +1: LS i +1[0] = ∅ +2: for r = 1 to ⌈N/k⌉ do +3: +(Ei, ⇝i)r ← r-th Consistent partial distributed stream +4: +j = 0 +5: +do +6: +j = j + 1 +7: +(α1, α2, · · · , α|A|) ∈ Sr(Ei, ⇝i) +8: +LSi +r[j] ← LSi +r[j − 1] ∪ +� +(α1, α2, · · · , α|A|) |=S ϕα +� +9: +while (LS i +r[j] ̸= LS i +r[j − 1]) +10: +Send: broadcasts symbolic view LSi +r[j] +11: +Receive: Πi +r ← {LS k +r | 1 ≤ k ≤ M} +12: +Compute: LSi +r+1[0] ← LC(Πi +r) +⊲ Section VII-A +13: end for +14: Resulti ← � +r∈[1,⌈N/k⌉+1] LSi +r[0] +A. Computing LC +Given a set of local states computed from the SMT encod- +ing, each monitor process receives a set of rewritten LOLA +associated equations, denoted by LS i +j, where i ∈ [1, |M|] for +j-th computation round. Our idea to compute LC from these +sets is to simply take a prioritized union of all the associated +equations. +LC (Πi +j) = +� +i∈[1,|M|] +LS i +j +The intuition behind the priority is that an evaluated LOLA +equation will take precedence over a partially evaluat- +ed/unevaluated LOLA equation, and two partially-evaluated +LOLA equation will be combined to form a evaluated or +partially evaluated LOLA equation. For example, taking the lo- +cally computed LS 1 +1 and LS 2 +1 from Example 3, LC (LS 1 +1, LS 2 +1) +is computed to be {c(1) = a(2), c(2) = 5, c(3) = ite(7 ≤ +b(4), a(4), 5)} at Monitor M1 and {c(1) += +7, c(2) += +5, c(3) = ite(7 ≤ b(4), a(4), 5)} at Monitor M2. Subse- +quently, LC (LS 1 +2, LS 2 +2) is computed to be {c(4) = 9, c(5) = +3, c(6) = 5} at Monitor M1 and {c(4) = 9, c(5) = 3, c(6) = +5} at Monitor M2. +B. Bringing it all Together +As stated in Section IV-A, the monitors are decentralized +and online. Since, setting up of a SMT instance is costly (as +seen in our evaluated results in Section VIII), we often find it +more efficient to evaluate the LOLA specification after every k +time instance. This reduces the number of computation rounds +to ⌈N/k⌉ as well as the number of messages being transmitted +over the network as well with an increase to the size of the +messages. We update Algorithm 1 to reflect our solution more +closely to Algorithm 2. +Each evaluation round starts by reading the r-th partial +distributed system which consists of events occurring between +the time max{0, (r − 1) × ⌈N/k⌉} and min{N, r × ⌈N/k⌉} +(line 3). We assume that the partial distributed system is +consistent in accordance with the assumption that each event +has been read by atleast one monitor. To account for any +concurrency among the events in (r − 1)-th computation +round with that in the r-th computation round, we expand +the length by ǫ time, there-by making the length of the r-th +computation round, max{0, (r − 1) × ⌈N/k⌉ − ǫ + 1} and +min{N, r × ⌈N/k⌉}. +Next, we reduce the evaluation of the distributed stream +problem into an SMT problem (line 7). We represent the +distributed system using SMT entities and then by the help +of SMT constraints, and we evaluate the LOLA specification +on the generated sequence of consistent cuts. Each sequence of +consistent cut presents a unique ordering of the events which +evaluates to a unique value for the stream expression (line 8). +This is repeated until we no longer can generate a sequence of +consistent cut that evaluates ϕα to unique values (line 9). Both +the evaluated as well as partially evaluated results are included +in LS as associated LOLA equations. This is followed by the +communication phase where each monitor shares its locally +computed LS i +r, for all i ∈ [1, |M|] and r evaluation round +(line 10-11). +Once, the local states of all the monitors are received, we +take a prioritized union of all the associated equation and +include them into LS i +r+1 set of associated equations (line +12). Following this, the computation shifts to next computation +round and the above mentioned steps repeat again. Once we +reach the end of the computation, all the evaluated values are +contained in Resulti +Lemma 1: Let A = {S1, S2, · · · , Sn} be a distributed sys- +tem and ϕ be an LOLA specification. Algorithm 1 terminates +when monitoring a terminating distributed system. +Theorem 1: Algorithm 2 solves the problem stated in +Section IV. +Theorem 2: Let ϕ be a LOLA specification and (E, ⇝) be +a distributed stream consisting of |A| streams. The message +complexity of Algorithm 2 with |M| monitors is +O +� +ǫ|A|N|M|2� +Ω(N|M|2) +VIII. CASE STUDY AND EVALUATION +In this section, we analyze our SMT-based decentralized +monitoring solution. We note that we are not concerned about +data collections, data transfer, etc, as given a distributed +setting, the runtime of the actual SMT encoding will be the +most dominating aspect of the monitoring process. We evaluate +our proposed solution using traces collected from synthetic ex- +periments (Section VIII-A) and case studies involving several +industrial control systems and RACE dataset (Section VIII-B). +The implementation of our approach can be found on Google +Drive(https://tinyurl.com/2p6ddjnr). +A. Synthetic Experiments +1) Setup: Each experiment consists of two stages: (1) +generation of the distributed stream and (2) verification. For +data generation, we develop a synthetic program that randomly +generates a distributed stream (i.e., the state of the local +computation for a set of streams). We assume that streams +are of the type Float, Integer or Boolean. For the +streams of the type Float and Integer, the initial value is +a random value s[0] and we generate the subsequent values +by s[i-1] + N(0, 2), for all i ≥ 1. We also make sure +that the value of a stream is always non-negative. On the other +8 + +hand, for streams of the type Boolean, we start with either +true or false and then for the subsequent values, we stay +at the same value or alter using a Bernoulli distribution of +B(0.8), where a true signifies the same value and a false +denotes a change in value. +For the monitor, we study the approach using Bernoulli +distribution B(0.2), B(0.5) and B(0.8) as the read distribution +of the events. A higher readability offers each event to be +read by higher number of monitors. We also make sure that +each event is read by at least one monitor in accordance with +the proposed approach. To test the approach with respect to +different types of stream expression, we use the following +arithmetic and logical expressions. +input a1 : uint +input a2 : uint +output arithExp := a1 + a2 +output logicExp := (a1 > 2) && (a2 < 8) +2) Result - Analysis: We study different parameters and +analyze how it effects the runtime and the message size in our +approach. All experiments were conducted on a 2017 Mac- +Book Pro with 3.5GHz Dual-Core Intel core i7 processor and +16GB, 2133 MHz LPDDR3 RAM. Unless specified otherwise +all experiments consider number of streams, |A| = 3, time +synchronization constant, ǫM = ǫ = 3s, number of monitors +same as the number of streams, computation length, N = 100, +with k = 3 with a read distribution B(0.8). +Time Synchronization Constant. Increasing the value of +the time synchronization constant ǫ, increases the possible +number of concurrent events that needs to be considered. This +increases the complexity of evaluating the LOLA specification +and there-by increasing the runtime of the algorithm. In addi- +tion to this, higher number of ǫ corresponds to higher number +of possible streams that needs to be considered. We observe +that the runtime increases exponentially with increasing the +value of ǫ in Fig. 5a, as expected. An interesting observation +is that with increasing the value of k, the runtime increases at +a higher rate until it reaches the threshold where k = ǫ. This +is due to the fact, that the number of streams to be considered +increases exponentially but ultimately gets bounded by the +number of events present in the computation. +Increasing the value of the time synchronization constant +is also directly proportional to the number of evaluated re- +sults at each instance of time. This is because, each stream +corresponds to a unique value being evaluated until it gets +bounded by the total number of possible evaluations, as can +be seen in Fig. 6a. However, comparing Figs. 5a and 6a, we +see that the runtime increases at a faster rate to the size of the +message. This owes to the fact that initially a SMT instance +evaluates unique values at all instance of time. However, as +we start reaching all possible evaluations for certain instance +of time, only a fraction of the total time instance evaluates +to unique values. This is the reason behind the size of the +message reaching its threshold faster than the runtime of the +monitor. +Type of Stream Expression. Stream expressions can be +divided into two major types, one consisting of arithmetic op- +erations and the other involving logical operations. Arithmetic +operations can evaluate to values in the order of O(|A|.ǫ), +where as logical operations can only evaluate to either true +or false. When the monitors have high readability of the +distributed stream, it is mostly the case, that the monitor was +able to evaluate the stream expression. Thus, we observe in +Fig. 5c that the runtime grows exponentially for evaluating +arithmetic expressions but is linear for logical expressions. +However, with low readability of the computation, irrespective +of the type of expression, both takes exponential time since +neither can completely evaluate the stream expression. So, +each monitor has to generate all possible streams. +Similarly, for high readability and logical expressions, the +message size is constant given the monitor was was able to +evaluate the stream expression. However with low readability, +message size for evaluating logical expressions matches with +that of its arithmetic counterpart. This can be seen in Fig. 6c +and is due to the fact, that with low readability, complete +evaluation of the expression is not possible at a monitor and +thus needs to send the rewritten expression with the values +observed to the other monitors where it will be evaluated. +Number of Streams. As the number of streams increases, +the number of events increase linearly and thereby making +exponential increase in the number of possible synchronous +streams (due to interleavings). This can be seen in Fig. 5b, +where the runtime increases exponentially with increase in +the number of streams in the distributed stream. Similarly, +in Fig. 6b, increase in the number of streams linearly effects +the number of unique values that the LOLA expression can +evaluate to and there-by increasing the size of the message. +B. Case Studies: Decentralized ICS and Flight Control RV +We put our runtime verification approach to the test with +respect to several industrial control system datasets that in- +cludes data generated by a (1) Secure Water Treatment plant +(SWaT) [9], comprising of six processes, corresponding to +different physical and control components; (2) a Power Dis- +tribution system [10] that includes readings from four phaser +measurement unit (PMU) that measures the electric waves on +an electric grid, and (3) a Gas Distribution system [11] that +includes messages to and from the PLC. In these ICS, we +monitor for correctness of system properties. Additionally we +monitor for mutual separation between all pairs of aircraft +in RACE [12] dataset, that consists of SBS messages from +aircrafts. For more details about each of the systems along +with the LOLA specifications refer to the Appendix XI-C. +For our setting we assume, each component has its own +asynchronous local clock, with varying time synchronization +constant. Next we discuss the results of verifying different ICS +with respect to LOLA specifications. +Result Analysis: We employed same number of monitors +as the number of components for each of the ICS case-studies +and divided the entire airspace into 9 different ones with one +monitor responsible for each. We observe that our approach +9 + +1 +2 +3 +4 +5 +1 +5 +10 +50 +100 +500 +1,000 +Time Synchronization Constant (sec.) ε +Runtime (sec.) +k = 5 +k = 4 +k = 3 +k = 2 +k = 1 +(a) Epsilon +2 +3 +4 +5 +7 +10 +1 +5 +10 +50 +100 +500 +1,000 +500 +10,000 +50,000 +Number of Streams |A| +Runtime (sec.) +k = 5 +k = 4 +k = 3 +k = 2 +k = 1 +(b) Number of Streams +2 +3 +4 +5 +7 +10 +1 +5 +10 +50 +100 +500 +1,000 +Number of Streams |A| +Runtime (sec.) +arithExp, B(0.8) +logicExp, B(0.8) +arithExp, B(0.5) +logicExp, B(0.5) +arithExp, B(0.2) +logicExp, B(0.2) +(c) Different LOLA Specification +Fig. 5: Impact of different parameters on runtime for synthetic data. +1 +2 +3 +4 +5 +5 +10 +50 +100 +Time Synchronization Constant (sec.) ε +Size of Messages (bytes) +k = 5 +k = 4 +k = 3 +k = 2 +k = 1 +(a) Epsilon +2 +3 +4 +5 +7 +10 +5 +10 +50 +100 +500 +1,000 +Number of Streams |A| +Size of Messages (bytes) +k = 5 +k = 4 +k = 3 +k = 2 +k = 1 +(b) Number of Streams +2 +3 +4 +5 +7 +10 +10 +50 +100 +500 +1,000 +Number of Streams |A| +Size of Messages (bytes) +arithExp, B(0.8) +logicExp, B(0.8) +arithExp, B(0.5) +logicExp, B(0.5) +arithExp, B(0.2) +logicExp, B(0.2) +(c) Different LOLA Specification +Fig. 6: Impact of different parameters on message size for synthetic data. +0.1 +0.5 +1 +2 +3 +100 +100.7 +101 +101.3 +101.6 +Time-Synchronization constant ǫ +Average % of False-Positives +SWaT +Power Distribution +Gas Distribution +RACE +Fig. 7: False-Positives for ICS Case-Studies +does not report satisfaction of system property when there +has been an attack on the system in reality (false-negative). +However, due to the assumption of partial-synchrony among +the components, our approach may report false positives, i.e., +it reports a violation of the system property even when there +was no attack on the system. As can be seen in Fig. 7, with +decreasing time synchronization constant, the number of false- +positives reduce as well. This is due to the fact that with +decreasing ǫ, less events are considered to be concurrent by +the monitors. This makes the partial-ordering of events as +observed by the monitor closer to the actual-ordering of events +taking place in the system. +We get significantly better result for aircraft monitoring +with fewer false-positives compared to the other dataset. This +can be attributed towards Air Traffic Controllers maintaining +greater separation between two aircrafts than the minimum +that is recommended. As part of our monitoring of other ICS, +we would like to report that our monitoring approach could +successfully detect several attacks which includes underflow +and overflow of tank and sudden change in quality of water in +SWaT, differentiate between manual tripping of the breaker +from the breaker being tripped due to a short-circuit in +Power Distribution and Single-point data injection in Gas +distribution. +IX. RELATED WORK +Online predicate detection for both centralized and de- +centralized monitoring setting have been extensively studies +in [13], [14]. Extensions to more expressive temporal operators +are introduced in [15], [16]. Monitoring approaches introduced +in [13], [15], [16] considers a fully asynchronous distributed +system. An SMT-based predicate detection solution has been +introduced in [17]. Runtime Verification for synchronous dis- +tributed system has been studied in [18]–[20]. The assumption +of a common global clock shared among all the components +act as a major shortcoming of this approach. Finally, fault- +tolerant monitoring, where monitors can crash, has been inves- +tigated in [21] for asynchronous and in [22] for synchronized +distributed processes. +10 + +Runtime Verification of stream-based specification was in- +troduced in [2], [23], where the occurrence of the events +was assumed to be synchronous. To extend the stream-based +runtime verification to more complex systems, one where the +occurrence of events is asynchronous, a real-time based logic +was introduced in [24]–[26]. However, these methods fall short +to verify large geographically separated distributed system, due +to their assumption regarding the presence of a shared global +clock. On the contrary, we assume the presence of a clock +synchronization algorithm which limits the maximum clock +skew among components to a constant. This is a realistic +assumption since different components of a large industrial +system have their own clock and it is certain to have a skew +between them. A similar SMT-based solution was studied +for LTL and MTL specifications in [27], [28] respectively, +which we extend to include a more expressive stream-based +specification. +X. CONCLUSION +In this paper, we studied distributed runtime verifica- +tion w.r.t. to the popular stream-based specification language +LOLA. We propose a online decentralized monitoring approach +where each monitor takes a set of associated LOLA specifica- +tion and a partial distributed stream as input. By assuming +partial synchrony among all streams and by reducing the +verification problem into an SMT problem, we were able to +reduce the complexity of our approach where it is no longer +dependent on the time synchronization constant. 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Available: +https://doi.org/10.1007/978-3-030-32079-9 16 +11 + +[26] M. Leucker, C. S´anchez, T. Scheffel, M. Schmitz, and A. Schramm, +“Runtime verification of real-time event streams under non-synchronized +arrival,” Software Quality Journal, vol. 28, no. 2, pp. 745–787, 2020. +[27] R. +Ganguly, +A. +Momtaz, +and +B. +Bonakdarpour, +“Distributed +Runtime Verification Under Partial Synchrony,” in 24th International +Conference +on +Principles +of +Distributed +Systems +(OPODIS +2020), +vol. +184, +2021, +pp. +20:1–20:17. +[Online]. +Available: +https://drops.dagstuhl.de/opus/volltexte/2021/13505 +[28] R. Ganguly, +Y. Xue, A. Jonckheere, +P. Ljung, B. Schornstein, +B. Bonakdarpour, and M. Herlihy, “Distributed runtime verification +of +metric +temporal +properties +for +cross-chain +protocols,” +in +2022 +IEEE +42nd +International +Conference +on +Distributed +Computing +Systems +(ICDCS). +Los +Alamitos, +CA, +USA: +IEEE +Computer +Society, +jul +2022, +pp. +23–33. +[Online]. +Available: +https://doi.ieeecomputersociety.org/10.1109/ICDCS54860.2022.00012 +XI. APPENDIX +A. LOLA Syntax +A stream expression is constructed as follows: +• If c is a constant of type T, then c is an atomic stream +expression of type T +• If s is a stream variable of type T, then s is an atomic +stream expression of type T. +• If f : T1 × T2 × · · · Tk → T is a k-ary operator and +for 1 ≤ i ≤ k, ei is an expression of type Ti, then +f(e1, e2, · · · , ek) is a stream expression of type T +• If b is a stream expression of type boolean and e1, e2 +are stream expressions of type T, then ite(b, e1, e2) is a +stream expression of type T, where ite is the abbreviated +form of if-then-else. +• If e is a stream expression of type T, c is a constant of +type T and i is an integer, then e[i, c] is a stream expres- +sion of type T. e[i, c] refers to the value of the expression +e offset by i positions from the current position. In case +the offset takes it beyond the end or before the beginning +of the stream, then the default value is c. +Furthermore, LOLA can be used to compute incremental +statistics, where a given a stream, α, a function, fα(v, u), +computes a measure, where u represents the measure thus +far and v, the current value. Given a sequence of values, +v1, v2, · · · , vn, with a default value d, the measure over the +data is given as +u = fα(vn, fα(vn−1, · · · , fα(v1, d))) +Example of such functions include count, fcount(v, u) = u+1, +sum, fsum(v, u) = u + v, max, fmax(v, u) = max{v, u}, +among others. Aggregate functions like average, can be de- +fined using two incremental functions, count and sum. +B. Proofs +Lemma 2: Let A = {S1, S2, · · · , Sn} be a distributed sys- +tem and ϕ be an LOLA specification. Algorithm 1 terminates +when monitoring a terminating distributed system. +Proof 1: First, we note that our algorithm is designed for +terminating system, also, note that a terminating program only +produces a finite distributed computation. In order to prove the +lemma, let us assume that the system send out a stop signal to +all monitor processes when it terminates. When such a signal +is received by a monitor, it starts evaluating the output stream +expression using the terminal associated equations. This might +arise to two cases. One where all the values required for +the evaluation has been observed or one where the values +required for the evaluation has not been observed. Although +the termination of the monitor process for the first case is +trivial, the termination of the monitor process for the second +case is dependent upon replacing such unobserved stream +value by the default value of the stream expression. Thus, +terminating the monitor process eventually. +Theorem 3: Algorithm 2 solves the problem stated in +Section IV. +Proof 2: We prove the soundness and correctness of Al- +gorithm 2, by dividing it into three steps. In the first step +12 + +we prove that given a LOLA specification, ϕ, the values +of the output stream when computed over the distributed +computation, (E, ⇝), of length N is the same as when +the distributed computation is divided into +N +k computation +rounds of length k each. Second, we prove that for all time +instances the stream equation is eventually evaluated after the +communication round. Finally we prove the set of all evaluated +result is consistent over all monitors in the system. +Step 1: +From our approach, we see that the value of a +output stream variable, is evaluated on the events present in +the consistent cut with time j. Therefore, we can reduce the +proof to: +Sr(E, ⇝) = Sr(E1.E2 · · · E N +k , ⇝) +• (⇒) Let Ck be a consistent cut such that Ck is in Sr(E, ⇝) +, but not in Sr(E1.E2 · · · E N +k , ⇝), for some k ∈ [0, |E|]. +This implies that the frontier of Ck, front(Ck) ̸⊆ E1 and +front(Ck) ̸⊆ E2 and · · · and front(Ck) ̸⊆ E N +k . However, +this is not possible, as according to the computation round +construction in Section VII-B, there must be a Ei, where +1 ≤ i ≤ N +k such that front(Ck) ⊆ Ei. Therefore, such +Ck cannot exist, and (α1, α2, · · · , αn) ∈ Sr(E, ⇝) =⇒ +(α1, α2, · · · , αn) ∈ Sr(E1.E2 · · · E N +k , ⇝). +• (⇐) Let Ck be a consistent cut such that Ck is in +Sr(E1.E2 · · · E N +k , ⇝) but not in Sr(E, ⇝) for some k ∈ +[0, |E|]. This implies, front(Ck) ⊆ Ei and front(Ck) ̸⊆ E +for some i ∈ [1, N +k ]. However, this is not possible due to +the fact that ∀i ∈ [1, N +k ].Ei ⊂ E. There, such Ck cannot +exist, and (α1, α2, · · · , αn) ∈ Sr(E1.E2 · · · E N +k , ⇝) =⇒ +(α1, α2, · · · , αn) ∈ Sr(E, ⇝). +Therefore, Sr(E, ⇝) = Sr(E1.E2 · · · E N +k , ⇝). +Step 2: +Given a output stream expression si and the +dependency graph G = ⟨V, E⟩, for each ⟨si, sk, w⟩ ∈ E, +evaluating the value at time instance j ∈ [1, N], αk(j+w) ̸= ♮ +or αk(j + w) = ♮ or αk(w + j) not observed. +• If αk(j +w) ̸= ♮, then we evaluate the stream expression +• If αk(j + w) = ♮, there exists at-least one other monitor +where αk(j + w) ̸= ♮. Thereby evaluating the stream +expression, followed by sharing the the evaluated result +with all other monitors +• If αk(w+j) not observed, then at some future evaluation +round and at some monitor αk(j + w) ̸= ♮ and there-by +evaluating the stream expression si +Similarly, it can be proved for ⟨si, tk, w⟩ ∈ E. +Step 3: Each monitor in our approach is fault-proof with +communication taking place between all pairs of monitors. +We also assume, all messages are eventually received by the +monitors. This guarantees all observations are either directly +or indirectly read by each monitor. +Together with Step 1 and 2, soundness and correctness of +Algorithm 1 is proved. +Theorem 4: Let ϕ be a LOLA specification and (E, ⇝) be +a distributed stream consisting of |A| streams. The message +complexity of Algorithm 2 with |M| monitors is +O +� +ǫ|A|N|M|2� +Ω(N|M|2) +Proof 3: +We analyze the complexity of each part of Algorithm 2. +The algorithm has a nested loop. The outer loop iterates for +⌈N/k⌉ times, that is O(N). The inner loop is dependent on +the number of unique evaluations of the stream expression. +• Upper-bound +Due +to +our +assumption +of +partial- +synchrony, each event’s time of occurrence can be off +by ǫ. This makes the maximum number of unique eval- +uations in the order of O(ǫ|A|). +• Lower-bound The minimum number of unique evalua- +tions is in the order of Ω(1). +In the communication phase, each monitor sends |M| +messages to all other monitors and receives |M| messages +from all other monitors. That is |M|2. Hence the message +complexity is +O +� +ǫ|A|N|M|2� +Ω(N|M|2) +As a side note, we would like to mention that in case of high +readability of the monitors and evaluation of logical expres- +sion, the complexity is closer to the lower-bound, whereas with +low readability and arithmetic expressions, the complexity is +closer to the upper bound. +C. Industrial Control Systems +a) SWaT Dataset: Secure Water Treatment (SWaT) [9] +utilizes a fully operational scaled down water treatment plant +with a small footprint, producing 5 gallons/minute of doubly +filtered water. It comprises of six main processes correspond- +ing to the physical and control components of the water +treatment facility. It starts from process P1 where it takes raw +water and stores it in a tank. It is then passed through the +pre-treatment process, P2, where the quality of the water is +assessed and maintained through chemical dosing. The water +then reaches P3 where undesirable materials are removed +using fine filtration membranes. Any remaining chlorine is +destroyed in the dechlorination process in P4 and the water is +then pumped into the Reverse Osmosis system (P5) to reduce +inorganic impurities. Finally in P6, water from the RO system +is stored ready for distribution. +The dataset classifies different attack on the system into +four types, based on the point and stage of the attack: Single +Stage-Single Point, Single Stage-Multi Point, Multi Stage- +Single Point and Multi Stage-Multi Point. We for the scope +of this paper are the most interested in the attacks either +covering multiple stages or multiple points. Few of the LOLA +specifications used are listed below. +input FIT-101 : uint +input MV-101 : bool +input LIT-101 : uint +input P-101 : bool +input FIT-201 : uint +output inflowCorr := ite(MV-101 == true, +FIT-101 > 0, FIT-101 == 0) +output outflowCorr := ite(P-101 == true, +FIT-201 > 0, FIT-201 == 0) +output tankCorr := ite(MV-101 == true || +P-101 == true, LIT-101 = LIT-101[-1, 0] + +FIT-101[-1, 0] - FIT-201[-1, 0]) +13 + +where FIT-101 is the flow meter, measuring inflow into +raw water tank, MV-101 is a motorized valve that controls +water flow to the raw water tank, LIT-101 is the level +transmitter of the raw water tank, P-101 is a pump that pumps +water from raw water tank to the second stage and FIT-201 +is the flow transmitter for the control dosing pumps. The +above LOLA specification checks the correctness of the inflow +meter and valve pair (resp. outflow meter and pump pair) in +inflowCorr (resp. outflowCorr) output expressions. On +the other hand, tankCorr checks if the water level in the +tank adds up to the in-flow and out-flow meters. +input AIT-201 : uint +input AIT-202 : uint +input AIT-203 : uint +output numObv := numObv[-1, 0] + 1 +output NaClAvg := (NaClAvg[-1, 0] * +numObv[-1, 0] + AIT-201) / numObv +output HClAvg := (HClAvg[-1, 0] * numObv[-1, +0] + AIT-202) / numObv +output NaOClAvg := (NaOClAvg[-1, 0] * +numObv[-1, 0] + AIT-202) / numObv +where AIT-201, AIT-202 and AIT-203 represents the +NaCl, HCl and NaOCl levels in water respectively and +NaClAvg, HClAvg and NaOClAvg keeps a track of the +average levels of the corresponding chemicals in the water, +where as numObv keeps a track of the total number of +observations read by the monitor. +b) Power System Attack Dataset: Power System Attack +Dataset [10] consists of three datasets developed by Missis- +sippi State University and Oak Ridge National Laboratory. +It consists of readings from four phaser measurement unit +(PMU) or synchrophasor that measures the electric waves on +an electric grid. Each PMU measures 29 features consisting of +voltage phase angle, voltage phase magnitude, current phase +angle, current phase magnitude for Phase A-C, Pos., Neg. and +Zero. It also measures the frequency for relays, the frequency +delta for relay, status flag for relays, etc. Apart from these 116 +PMU measurements, the dataset also consists of 12 control +panel logs, snort alerts and relay logs of the 4 PMU. +The dataset classifies into either natural event/no event or +an attack event. Few of the LOLA specifications used are listed +below. The first attempts to detect a single-line-to-ground +(1LG) fault. +input R1-I : float +input R2-I : float +input R1-Relay : bool +input R2-Relay : bool +output R1-I-low := R1-I < 200 +output R1-I-high := R1-I > 1000 +output R2-I-low := R2-I < 200 +output R2-I-high := R2-I > 1000 +output 1LG := R1-I-high && R2-I-high && +R1-Relay[+2, false] && R2-Relay[+2, +false] && R1-I-low[+4, false] && +R2-I-low[+4, false] +where R1-I and R2-I represents the current measured at +the R1 and R2 PMU respectively. Additionally, R1-Relay +and R2-Relay keeps a track of the state of the corresponding +relay. As a part of the 1LG attack detection, we first categorize +the current measured as either low or high depending upon the +amount of the current measured. We categorize an attack as +1LG if both R1 and R2 detects high current flowing followed +by the relay tripping followed by low current. +input R1-PA1-I : float +input R1-PA2-I : float +input R1-PA3-I : float +output phaseBal := (R1-PA1-I - R1-PA2-I) <= +10 && (R1-PA2-I - R1-PA3-I) <= 10 && +(R1-PA3-I - R1-PA1-I) <= 10 +where R1-PA1-I, R1-PA2-I and R1-PA3-I are the +amount of current measured by R1 PMU at Phase A, B and +C respectively. The monitor helps us to check if the load on +three phases are equally balanced. +c) Gas Distribution System: Gas Distributed System [11] +is a collection of labeled Remote Terminal Unit (RTU) teleme- +try streams from a Gas pipeline system in Mississippi State +University’s Critical Infrastructure Protection Center with col- +laboration from Oak Ridge National Laboratory. The telemetry +streams includes messages to and from the Programmable +Logic Controller (PLC) under normal operations and attacks +involving command injection and data injection attack. The +feature set includes the pipeline pressure, setpoint value, +command data from the PLC, response to the PLC and the +state of the solenoid, pump and the Remote Terminal Unit +(RTU) auto-control. +One of the most common data injection attack is Fast +Change. Here the reported pipeline pressure value is succes- +sively varied to create a lack of confidence in the correct op- +eration of the system. The corresponding LOLA specification +monitoring against such attack is mentioned below: +input PipePress : float +input response : bool +output fastChange := ite(response, +mod(PipePress - PipePress[-1, 1000]) <= +10, true) +where PipePress records the measured pipeline pressure +and response is a flag variable signifying a message to the +PLC. Here we consider the default pressure is 1000 psi and the +permitted pressure change per unit time is 10 psi (these can be +changed according to the demands of the system). Similarly +we have LOLA specifications monitoring other data injection +attacks such as Value Wave Injection, Setpoint Value Injection, +Single Data Injection, etc. and command injection attacks such +as Illegal Setpoint, Illegal PID Command, etc. +d) RACE Dataset: Runtime for Airspace Concept Eval- +uation (RACE) [12] is a framework developed by NASA that +is used to build an event based, reactive airspace simulation. +We use a dataset developed using this RACE framework. +This dataset contains three sets of data collected on three +different days. Each set was recorded at around 37 N Latitude +and 121 W Longitude. The dataset includes all 8 types of +messages being sent by the SBS unit by using a Telnet +14 + +application to listen to port 30003, but we only use the +messages with ID ‘MSG 3’ which is the Airborne Position +Message and includes a flight’s latitude, longitude and altitude +using which we verify the mutual separation of all pairs of +aircraft. Furthermore, calculating the distance between two +coordinates is computationally expensive, as we need to factor +in parameters such as curvature of the earth. In order to speed +up distance related calculations, we consider a constant latitude +distance of 111.2km and longitude distance of 87.62km, at the +cost of a negligible error margin. The corresponding LOLA +specification is mentioned below: +input flight1_alt : float +input flight1_lat : float +input flight1_lon : float +input flight2_alt : float +input flight2_lat : float +input flight2_lon : float +output distDiff := sqrt(pow(flight1_alt - +flight2_alt, 2) + pow((flight1_lon - +flight2_lon)*87620, 2) + pow((flight1_lat +- flight2_lat)*111200, 2)) +output check := distDiff > 500 +15 + diff --git a/ANFQT4oBgHgl3EQfMDYd/content/tmp_files/load_file.txt b/ANFQT4oBgHgl3EQfMDYd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4decf5ba6577baf2e3272f44cfc8e05bd8b4c5a3 --- /dev/null +++ b/ANFQT4oBgHgl3EQfMDYd/content/tmp_files/load_file.txt @@ -0,0 +1,869 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf,len=868 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='13266v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='DC] 30 Jan 2023 Stream-based Decentralized Runtime Verification Ritam Ganguly Department of Computer Science and Engineering Michigan State University East Lansing, USA gangulyr@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='edu Borzoo Bonakdarpour Department of Computer Science and Engineering Michigan State University East Lansing, USA borzoo@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='edu Abstract—Industrial Control Systems (ICS) are often built from geographically distributed components and often use pro- grammable logic controllers for localized processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Since ver- ification of such systems is challenging because of both time sensitivity of the system specifications and the inherent asyn- chrony in distributed components, developing runtime assurance that verifies not just the correctness of different components, but also generates aggregated statistics of the systems is of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In this paper, we first present a general technique for runtime monitoring of distributed applications whose behavior can be modeled as input/output streams with an internal compu- tation module in the partially synchronous semantics, where an imperfect clock synchronization algorithm is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Second, we propose a generalized stream-based decentralized runtime verification technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We also rigorously evaluate our algorithm on extensive synthetic experiments and several ICS and aircraft SBS message datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' INTRODUCTION Industrial Control Systems (ICS) are information systems to control industrial processes such as manufacturing, product handling, distribution, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' It includes supervisory control and data acquisition systems used to control geographically dispersed assets and distributed control systems using a pro- grammable logic controller for each of the localized processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' A typical programmable logic controller (PLC) receives data produced by a large number of sensors, fitted across the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The data produced by these components are often the target of cyber and ransom-ware attack putting the security of the system in jeopardy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Since these systems are linked to essential services, any attack on these facilities put the users life on the front line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The integrity of the data produced from these distributed components is very important as the PLC’s behavior is dictated by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Recent attacks have shown that an attack on a company’s ICS costs the company around $5 million and 50 days of system down time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Additionally, according to a recent report [1], it takes the effected company around 191 days to fully recover and around 54% of all organization are vulnerable to such attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In this paper, we advocate for a runtime verification (RV) approach, to monitor the behavior of a distributed system with respect to a formal specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Applying RV to multiple components of an ICS can be viewed as the general problem of distributed RV, where a centralized or decentralized monitor(s) observe the behavior of a distributed system in which the processes do not share a global clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Although RV deals with finite executions, the lack of a common global clock prohibits it from having a total ordering of events in a distributed setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' x y x + y 3 5 6 9 1 3 5 7 2(ǫ − 1) 2(ǫ − 1) 2(ǫ − 1) 1 3 4 6 9 11 {4} {4, 6, 8} {8} {8, 10, 9, 11} {11} {11, 13, 14, 16} {16} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 1: Partially Synchronous LOLA In other words, the monitor can only form a partial ordering of events which may yield different evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Enumerating all possible interleavings of the system at runtime incurs in an exponential blowup, making the approach not scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' To add to this already complex task, a PLC often requires time sensitive aggregation of data from multiple sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We propose an effective, sound and complete solution to distributed RV for the popular stream-based specification language LOLA [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Compared to other temporal logic, LOLA can describe both correctness/failure assertions along with statistical measures that can be used for system profiling and coverage analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' To present a high level of LOLA example, consider two input streams x and y and a output stream, sum as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Stream x has the value 3 until time instance 2 when it changes to 5 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' input x:int input y:int output sum := x+y We consider a fault proof decentralized set of monitors where each monitor only has a partial view of the system and has no access to a global clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In order to limit the blow- up of states posed by the absence of the global clock, we make a practical assumption about the presence of a bounded clock skew ǫ between all the local clocks, guaranteed by a clock synchronization algorithm (like NTP [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This setting is known to be partially synchronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 1, any two events less than ǫ = 2 time apart is considered to be concurrent and thus the non-determinism of the time of occurrence of each event is restricted to ǫ − 1 on either side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 1 When attempting to evaluate the output stream sum, we need to take into consideration all the possible time of occurrence of the values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For example, when evaluating the value of sum at time 1, we need to consider the value of x (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' y) as 3 and 5 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 1 and 3) which evaluates to 4, 6 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The same can be observed for evaluations across all time instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Our first contribution in this paper is introducing a partially synchronous semantics for LOLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In other words, we define LOLA which takes into consideration a clock-skew of ǫ when evaluating a stream expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Second, we introduce an SMT- based associated equation rewriting technique over a partially observable distributed system, which takes into consideration the values observed by the monitor and rewrites the associated equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The monitors are able to communicate within them- selves and are able to resolve the partially evaluated equations into completely evaluated ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We have proved the correctness of our approach and the upper and lower bound of the message complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Addi- tionally, we have completely implemented our technique and report the results of rigorous synthetic experiments, as well as monitoring correctness and aggregated results of several ICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' As identified in [4], most attacks on ICS components try to alter the value reported to the PLC in-order to make the PLC behave erroneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Through our approach, we were able to detect these attacks in-spite of the clock asynchrony among the different components with deterministic guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We also argue that our approach was able to evaluate system behavior aggregates that makes studying these system easier by the human operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Unlike machine learning approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=', [5]–[7]), our approach will never raise false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We put our monitoring technique to test, studying the effects of different parameters on the runtime and size of the message sent from one monitor to other and report on each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Organization: Section II presents the background con- cepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Partially synchronous LOLA and the formal problem statement are introduced in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Our RV technique is collectively presented in Sections Section IV – VII followed by the experimental results in Section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Related work is discussed in Section IX before we make concluding remarks in Section X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Details of syntax of LOLA, proofs of correctness and more details about the ICS case studies can be found in the Appendix XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' PRELIMINARIES – STREAM-BASED SPECIFICATION LANGUAGE (LOLA) [2] A LOLA [2] specification describes the computation of output streams given a set of input streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' A stream α of type T is a finite sequence of values, t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Let α(i), where i ≥ 0, denote the value of the stream at time stamp i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We denote a stream of finite length (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' infinite length) by T∗ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Tω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Definition 1: A LOLA specification is a set of equations over typed stream variables of the form: s1 = e1(t1, · · · , tm, s1, · · · , sn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' sn = en(t1, · · · , tm, s1, · · · , sn) where s1, s2, · · · , sn are called the dependent variables, t1, t2, · · · , tm are called the independent variables, and e1, e2, · · · , en are the stream expressions over s1, · · · , sn, t1, · · · , tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' ■ Typically, Input streams are referred to as independent variables, whereas output streams are referred as dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For example, consider the following LOLA specifi- cation, where t1 and t2 are independent stream variables of type boolean and t3 is an independent stream variable of type integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' s1 = true s2 = t1 ∨ (t3 ≤ 1) s3 = ite(s3, s4, s4 + 1) s4 = s9[−1, 0] + (t3 mod 2) where, ite is the abbreviated form of if-then-else and stream expressions s7 and s8 refers to the stream t1 with an offset of +1 and −1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Due to space constrains we present the full syntax of LOLA in Appendix XI-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The semantics of LOLA specifications is defined in terms of the evaluation model, which describes the relation between input and output streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Definition 2: Given a LOLA specification ϕ over indepen- dent variables, t1, · · · , tm, of type, T1, · · · , Tm, and depen- dent variables, s1, · · · , sn with type, Tm+1, · · · , Tm+n, let τ1, · · · , τm be the streams of length N + 1, with τi of type Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The tuple ⟨α1, · · · , αn⟩ of streams of length N + 1 is called the evaluation model, if for every equation in ϕ si = ei(t1, · · · , tm, s1, · · · , sn) ⟨α1, · · · , αn⟩ satisfies the following associated equations: αi(j) = val(ei)(j) for (1 ≤ i ≤ n) ∧ (0 ≤ j ≤ N) where val(ei)(j) is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For the base cases: val(c)(j) = c val(ti)(j) = τi(j) val(si)(j) = αi(j) For the inductive cases, where f is a function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=', arithmetic): val � f(e1, · · · , ek) � (j) = f � val(e1)(j), · · · , val(ek)(j) � val � ite(b, e1, e2) � (j) = if val(b)(j) then val(e1)(j) else val(e2)(j) val(e[k, c])(j) = � val(e)(j + k) if 0 ≤ j + k ≤ N c otherwise ■ The set of all equations associated with ϕ is noted by ϕα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 2 Definition 3: A dependency graph for a LOLA specification, ϕ is a weighted and directed graph G = ⟨V, E⟩, with vertex set V = {s1, · · · , sn, t1, · · · , tm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' An edge e : ⟨si, sk, w⟩ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' e : ⟨si, tk, w⟩) labeled with a weight w is in E iff the equation for αi(j) in ϕα contains αk(j +w) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' τk(j +w)) as a subexpression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Intuitively, an edge records that si at a particular position depends on the value of sk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' tk), offset by w positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Given a set of synchronous input streams {α1, α2, · · · , αm} of respective type T = {T1, T2, · · · , Tm} and a LOLA specification, ϕ, we evaluate the LOLA specification, given by: (α1, α2, · · · , αm) |=S ϕ given the above semantics, where |=S denotes the synchronous evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' PARTIALLY SYNCHRONOUS LOLA In this section, we extend the semantics of LOLA to one that can accommodate reasoning about distributed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Distributed Streams Here, we refer to a global clock which will act as the “real” timekeeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' It is to be noted that the presence of this global clock is just for theoretical reasons and it is not available to any of the individual streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We assume a partially synchronous system of n streams, denoted by A = {α1, α2, · · · , αn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For each stream αi, where i ∈ [1, |A|], the local clock can be represented as a monotonically increasing function ci : Z≥0 → Z≥0, where ci(G) is the value of the local clock at global time G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Since we are dealing with discrete-time systems, for simplicity and without loss of generality, we represent time with non-negative integers Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For any two streams αi and αj, where i ̸= j, we assume: ∀G ∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' | ci(G) − cj(G) |< ǫ, where ǫ > 0 is the maximum clock skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The value of ǫ is constant and is known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=', to a monitor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This assumption is met by the presence of an off-the-shelf clock synchronization algorithm, like NTP [3], to ensure bounded clock skew among all streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The local state of stream αi at time σ is given by αi(σ), where σ = ci(G), that is the local time of occurrence of the event at some global time G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Definition 4: A distributed stream consisting of A = {α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' , αn} streams of length N + 1 is represented by the pair (E, ⇝), where E is a set of all local states (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=', E = ∪i∈[1,n],j∈[0,N]αi(j)) partially ordered by Lamport’s happened-before (⇝) relation [8], subject to the partial syn- chrony assumption: For every stream αi, 1 ≤ i ≤ |A|, all the events happening on it are totally ordered, that is, ∀i, j, k ∈ Z≥0 : (j < k) → (αi(j) ⇝ αi(k)) For any two streams αi and αj and two corresponding events αi(k), αj(l) ∈ E, if k+ǫ < l then, αi(k) ⇝ αj(l), where ǫ is the maximum clock skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For events, e, f, and g, if e ⇝ f and f ⇝ g, then e ⇝ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' ■ Definition 5: Given a distributed stream (E, ⇝), a subset of events C ⊆ E is said to form a consistent cut if and only if when C contains an event e, then it should also contain all such events that happened before e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Formally, ∀e, f ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='(e ∈ C) ∧ (f ⇝ e) → f ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' ■ The frontier of a consistent cut C, denoted by front(C) is the set of all events that happened last in each stream in the cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' That is, front(C) is a set of αi(last) for each i ∈ [1, |A|] and αi(last) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We denote αi(last) as the last event in αi such that ∀αi(σ) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='(αi(σ) ̸= αi(last)) → (αi(σ) ⇝ αi(last)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Partially Synchronous LOLA We define the semantics of LOLA specifications for partially synchronous distributed streams in terms of the evaluation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The absence of a common global clock among the stream variables and the presence of the clock synchronization makes way for the output stream having multiple values at any given time instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Thus, we update the evaluation model, so that αi(j) and val(ti)(j) are now defined by sets rather than just a single value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This is due to nondeterminism caused by partial synchrony, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=', the bounded clock skew ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Definition 6: Given a LOLA [2] specification ϕ over in- dependent variables, t1, · · · , tm of type T1, · · · , Tm and de- pendent variables, s1, · · · , sn of type Tm+1, · · · , Tm+n and τ1, · · · , τm be the streams of length N + 1, with τi of type Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The tuple of streams ⟨α1, · · · , αn⟩ of length N + 1 with corresponding types is called the evaluation model in the partially synchronous setting, if for every equation in ϕ: si = ei(t1, · · · , tm, s1, · · · , sn), ⟨α1, · · · , αn⟩ satisfies the following associated equations: αi(j) = � val(ei)(k) | max{0, j−ǫ+1} ≤ k ≤ min{N, j+ǫ−1} � where val(ei)(j) is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For the base cases: val(c)(j) = {c} val(ti)(j) = � τi(k) | max{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' j − ǫ + 1} ≤ k ≤ min{N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' j + ǫ − 1} � val(si)(j) = αi(j) For the inductive cases: val � f(e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' ep) � (j) = � f(e′ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' e′ p) | e′ 1 ∈ val(e1)(j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' e′ p ∈ val(ep)(j) � val(ite(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' e1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' e2))(j) = � val(e1)(j) true ∈ val(b)(j) val(e2)(j) false ∈ val(b)(j) val(e[k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' c])(j) = � val(e)(j + k) if 0 ≤ j + k ≤ N c otherwise ■ Example 1: Consider the LOLA specification,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' over the independent boolean variables read and write: 3 read write count(read) count(write) check 1 2 3 4 5 6 7 {0} {0} {0} {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 1} {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 1} {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 2} {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 3} {2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 3} {0} {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 1} {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 2} {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 2} {2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 3} {2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 4} {3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 4} {4} {true} {true} {true} {true} {true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' false} {true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' false} {true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' false} {true} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 2: Partially Synchronous LOLA Example input read:bool input write:bool output countRead := ite(read, countRead[-1,0] + 1, countRead[-1,0]) output countWrite := ite(write, countWrite[-1,0] + 1, countWrite[-1,0]) output check := (countWrite - countRead) <= 2 In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 2, we have two input stream read and write which denotes the time instances where the corresponding events take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' It can be imagined that read and write are streams of type boolean with true values at time instances 4, 6, 7 and 2, 3, 5, 6 and false values at all other time instances respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We evaluate the above mentioned LOLA speci- fication considering a time synchronization constant, ǫ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The corresponding associated equations, ϕα, are: countRead(j) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ite(read, 1, 0) j = 0 ite � read, countRead(j− 1) + 1, countRead(j) � j ∈ [1, N) countWrite(j) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ite(write, 1, 0) j = 0 ite � write, countWrite(j− 1) + 1, countWrite(j) � j ∈ [1, N) check(j) = � countWrite(j) − countRead(j) � ≤ 2 Similar to the synchronous case, evaluation of the partially synchronous LOLA specification involves creating the depen- dency graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Definition 7: A dependency graph for a LOLA specification, ϕ is a weighted directed multi-graph G = ⟨V, E⟩, with vertex set V = {s1, · · · , sn, t1, · · · , tm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' An edge e : ⟨si, sk, w⟩ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' e : ⟨si, tk, w⟩) labeled with a weight w = {ω | p − ǫ < ω < p+ǫ} is in E iff the equation for αi(j) contains αk(j+p) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' τk(j + p)) as a sub-expression, for some j and offset p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' ■ a b1 b2 {0, 1, 2} {0, 1, 2}, {−2, −1, 0}, {−3, −2, −1} {−2, −1, 0} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 3: Dependency Graph Example Intuitively, the dependency graph records that evaluation of a si at a particular position depends on the value of sk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' tk), with an offset in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' It is to be noted that there can be more than one edge between a pair of vertex (si, sk) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' (si, tk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Vertices labeled by ti do not have any outgoing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Example 2: Consider the LOLA specification over the inde- pendent integer variable a: input a : uint output b1 := b2[1, 0] + ite(b2[-1,7] <= a[1, 0], b2[-2,0], 6) output b2 := b1[-1,8] Its dependency graph, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 3 for ǫ = 2, has 1 edge from b1 to a with a weight {0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Similarly, there are 3 edges from b1 to b2 with weights {0, 1, 2}, {−2, −1, 0} and {−3, −2, −1} and 1 edge from b2 to b1 with a weight of {−2, −1, 0} Given a set of partially synchronous input streams {α1, α2, · · · , α|A|} of respective type T = {T1, T2, · · · , T|A|} and a LOLA specification, ϕ, the evaluation of ϕ is given by (α1, α2, · · · , α|A|) |=P S ϕ where, |=P S denotes the partially synchronous evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' DECENTRALIZED MONITORING ARCHITECTURE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Overall Picture We consider a decentralized online monitoring system com- prising of a fixed number of |M| reliable monitor processes M = {M1, M2, · · · , M|M|} that can communicate with each other by sending and receiving messages through a complete point-to-point bidirectional communication links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Each com- munication link is also assumed to be reliable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=', there is no loss or alteration of messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Similar to the distributed system under observation, we assume the clock on the indi- vidual monitors are asynchronous, with clock synchronization constant = ǫM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Throughout this section we assume that the global dis- tributed stream consisting of complete observations of |A| streams is only partially visible to each monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Each monitor process locally executes an identical sequential algorithm which consists of the following steps (we will generalize this approach in Section VII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In other words, an evaluation iteration of each monitor consists of the following steps: 1) Reads the a subset of E events (visible to Mi) along with the corresponding time and valuation of the events, which results in the construction of a partial distributed stream;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 4 Algorithm 1 Behavior of a Monitor Mi, for i ∈ [1, |M|] 1: for j = 0 to N do 2: Let (Ei, ⇝i)j be the partial distributed stream view of Mi 3: LS j ← �(E, ⇝) |=P S ϕα � 4: Send: broadcasts symbolic view LSj 5: Receive: Πj ← {LS k j | 1 ≤ k ≤ M} 6: Compute: LSj+1 ← LC (Πj) 7: end for 2) Each monitor evaluates the LOLA specification ϕ given the partial distributed stream;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 3) Every monitor, broadcasts a message containing rewrit- ten associated equations of ϕ, denoted LS, and 4) Based on the message received containing associated equations, each monitor amalgamates the observations of all the monitors to compose a set of associated equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' After a evaluation iteration, each monitor will have the same set of associated equations to be evaluated on the upcoming distributed stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The message sent from monitor Mi at time π to another monitor Mj, for all i, j ∈ [1, |M|], during a evaluation iteration of the monitor is assumed to reach latest by time π + ǫM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Thus, the length of an evaluation iteration k can be adjusted to make sure the message from all other monitors reach before the start of the next evaluation iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Detailed Description We now explain in detail the computation model (see Algo- rithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Each monitor process Mi ∈ M, where i ∈ [1, |M|], attempts to read e ∈ E, given the distributed stream, (E, ⇝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' An event can either be observable, or not observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Due to distribution, this results in obtaining a partial distributed stream (Ei, ⇝) defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Definition 8: Let (E, ⇝) be a distributed stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We say that (E′, ⇝) is a partial distributed stream for (E, ⇝) and denote it by (E′, ⇝) ⊑ (E, ⇝) iff E′ ⊆ E (the happened before relation is obviously preserved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' ■ We now tie partial distributed streams to a set of decentral- ized monitors and the fact that decentralized monitors can only partially observe a distributed stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' First, all un-observed events is replaced by ♮, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=', for all αi(σ) ∈ E if αi(σ) ̸∈ Ei then Ei = Ei ∪ {αi(σ) = ♮}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Definition 9: Let (E, ⇝) be a distributed stream and M = {M1, M2, · · · , M|M|} be a set of monitors, where each monitor Mi, for i ∈ [1, |M|] is associated with a partial distributed stream (Ei, ⇝) ⊑ (E, ⇝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We say that these monitor observations are consistent if ∀e ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='∃i ∈ [1, |M|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='e ∈ Ei, and ∀e ∈ Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='∀e′ ∈ Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' (e = e′ ∧ e ̸= ♮) ⊕ � (e = ♮ ∨ e′ = ♮) � , where ⊕ denoted the exclusive-or operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In a partially synchronous system, there are different order- ing of events and each unique ordering of events might eval- uate to different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Given a distributed stream, (E, ⇝), a sequence of consistent cuts is of the form C0C1C2 · · · CN, where for all i ≥ 0: (1) Ci ⊆ E, and (2) Ci ⊆ Ci+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Given the semantics of partially-synchronous LOLA, evalu- ation of output stream variable si at time instance j requires events αi(k), where i ∈ [1, |A|] and k ∈ � π | max{0, j − ǫ + 1} ≤ π ≤ {N, j + ǫ − 1} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' To translate monitoring of a distributed stream to a synchronous stream, we make sure that the events in the frontier of a consistent cut, Cj are αi(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Let C denote the set of all valid sequences of consistent cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We define the set of all synchronous streams of (E, ⇝) as follows: Sr(E, ⇝) = � front(C0)front(C1) · · · | C0C1 · · · ∈ C � Intuitively, Sr(E, ⇝) can be interpreted as the set of all possi- ble “interleavings”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The evaluation of the LOLA specification, ϕ, with respect to (E, ⇝) is the following : � (E, ⇝) |=P S ϕ � = � (α1, · · · , αn) |=S ϕ | (α1, · · · , αn) ∈ Sr(E, ⇝) � This means that evaluating a partially synchronous distributed stream with respect to a LOLA specification results in a set of evaluated results, as the computation may involve several streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This also enables reducing the problem from evaluation of a partially synchronous distributed system to the evaluation of multiple synchronous streams, each evaluating to unique values for the output stream, with message complexity O � ǫ|A|N|M|2� Ω(N|M|2) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Problem Statement The overall problem statement requires that upon the termi- nation of the Algorithm 1, the verdict of all the monitors in the decentralized monitoring architecture is the same as that of a centralized monitor which has the global view of the system ∀i ∈ [1, m] : Resulti = � (E, ⇝) |=P S ϕ � where (E, ⇝) is the global distributed stream and ϕ is the LOLA specification with Resulti as the evaluated result by monitor Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' CALCULATING LS In this section, we introduce the rules of rewriting LOLA as- sociated equations given the evaluated results and observations of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In our distributed setting, evaluation of a LOLA specification involves generating a set of synchronous streams and evaluating the given LOLA specification on it (explained in Section VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Here, we make use of the evaluation of LOLA specification into forming our local observation to be shared with other monitors in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Given the set of synchronous streams, (α1, α2, · · · , α|A|), the symbolic locally computed result LS (see Algorithm 1) consists of associated LOLA equations, which either needs more information (data was unobserved) from other monitors to evaluate or the concerned monitor needs to wait (positive offset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In either case, the associated LOLA specification is shared with all other monitors in the system as the missing data 5 can be observed by either monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We divide the rewriting rules into three cases, depending upon the observability of the value of the independent variables required for evaluating the expression ei for all i ∈ [1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Each stream expression is categorized into three cases (1) completely unobserved, (2) completely observed or (3) partially observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This can be done easily by going over the dependency graph and checking with the partial distributed stream read by the corresponding monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Case 1 (Completely Observed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Formally, a completely observed stream expression si can be identified from the dependency graph, G = ⟨V, E⟩, as for all sk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' tk) ⟨si, sk, w⟩ ∈ E (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' ⟨si, tk, w⟩ ∈ E), sk(j + w) ̸= ♮ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' tk(j + w) ̸= ♮) are observed for time instance j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' If yes, this signifies, that all independent and dependent variables required to evaluate si(j), is observed by the monitor M, there by evaluating: si(j) = ei(s1, · · · , sn, t1, · · · , tm) and rewriting si(j) to LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Case 2 (Completely Unobserved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Formally, we present a completely unobserved stream expression, si from the dependency graph, G = ⟨V, E⟩, as for all sk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' tk), ⟨si, sk, w⟩ ∈ E (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' ⟨si, tk, w⟩ ∈ E), sk(j + w) = ♮ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' tk(j + w) = ♮) are unobserved, for time instance j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This signifies that the valuation of neither variables are known to the monitor M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Thus, we rewrite the following stream expressions s′ k(j) = � sk(j + w) 0 ≤ j + w ≤ N default otherwise t′ k(j) = � tk(j + w) 0 ≤ j + w ≤ N default otherwise for all ⟨si, sk, w⟩ ∈ E and ⟨si, tk, w⟩ ∈ E, and include the rewritten associated equation for evaluating si(j) as si(j) = ei(s′ 1, · · · , s′ n, t′ 1, · · · , t′ m) It is to be noted that the default value of a stream variable, sk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' tk), depends on the corresponding type Tk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Tm+k) of the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Case 3 (Partially Observed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Formally, we present a partially observed stream expression, si from the dependency graph, G = ⟨V, E⟩, as for all sk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' tk), they are either observed or unobserved, for time instance j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In other words, we can represent a set Vo = {sk | ∃sk(j + w) ̸= ♮} of all observed stream variable and a set Vu = {sk | sk(j + w) = ♮} of all unobserved dependent stream variable for all ⟨si, sk, w⟩ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The set can be expanded to include independent variables as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For all sk ∈ Vu (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' tk ∈ Vu) that are unobserved, are replaced by: su k(j) = � sk(j + w) 0 ≤ j + w ≤ N default otherwise tu k(j) = � tk(j + w) 0 ≤ j + w ≤ N default otherwise a b a b 1 2 3 4 5 6 1 7 5 4 4 7 3 5 9 3 5 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 4: Example of generating LS and for all sk ∈ Vo (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' tk ∈ Vo) that are observed, are replaced by: so k(j + w) = value to k(j + w) = value and there by partially evaluating si(j) as si(j) = ei(so 1, · · · , so n, to 1, · · · , to m, su 1, · · · , su n, tu 1, · · · , tu m) followed by adding the partially evaluated associated equation for si(j) to LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' It is to be noted, that a consistent partial distributed stream makes sure that for all sk (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' tk), can only be either observed or unobserved and not both or neither.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Example 3: Consider the LOLA specification mentioned below and the stream input of length N = 6 divided into two evaluation rounds and ǫ = 2 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 4 with the monitors M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' input a : uint input b : uint output c := ite(a[-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='0] <= b[1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' a[1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='0],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' b[-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0]) The associated equation for the output stream is: c = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ite(0 ≤ b(i + 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' a(i + 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0) i = 1 ite(a(i − 1) ≤ b(i + 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' a(i + 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' b(i − 1)) 2 ≤ i ≤ N − 1 ite(a(i − 1) ≤ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' b(i − 1)) i = N Let the partial distributed stream read by monitor M1 include {a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 5)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' {b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 9)} and the partial distributed stream read by monitor M2 include {a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 7)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' {b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Monitor M1 evaluates c(2) = 5 and partially evaluates c(1) and c(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Thus LS 1 1 = {c(1) = a(2), c(2) = 5, c(3) = ite(a(2) ≤ b(4), a(4), 5)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Monitor M2 partially evaluates all c(1), c(2) and c(3) and thus LS 2 1 = {c(1) = ite(0 ≤ b(2), a(2), 0), c(2) = a(3), c(3) = ite(7 ≤ b(4), a(4), b(2))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Let the partial distributed stream read by monitor M1 include {a, (4, 4), (5, 4)}, {b, (4, 3), (6, 1)} and the par- tial distributed stream read by monitor M2 include {a, (5, 4), (6, 7)}, {b, (4, 3), (5, 5)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Monitor M1 evaluates c(4) = 9 and c(5) = 3 and partially evaluates c(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Thus LS 1 2 = {c(4) = 9, c(5) = 3, c(6) = b(5)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Monitor M2 evaluates c(6) = 5 and partially evalues c(4) and c(5) and thus LS 2 2 = {c(4) = ite(a(3) ≤ 5, 4, 9), c(5) = ite(a(4) ≤ b(6), 7, 3), c(6) = 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' It is to be noted, the after the first round of evaluation, the corresponding local states, LS 1 1 and LS 2 1 will be shared which 6 will enable evaluating the output stream for few of the partially evaluated output stream (will be discussed in Section VII-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' These will be included in the local state of the following evaluation round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Note that generating LS takes into consideration an ordered stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' One where the time of occurrence of events and values are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' It can be imagined that generating the same for the distributed system involves generating it for all possible ordering of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This will be discussed in details in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' SMT-BASED SOLUTION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' SMT Entities SMT entities represent (1) LOLA equations, and (2) vari- ables used to represent the distributed stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Once we have generated a sequence of consistent cuts, we use the laws discussed in Section V, to construct the set of all locally computer or partially computed LOLA equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Distributed Stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In our SMT encoding, the set of events, E, is represented by a bit vector, where each bit corresponds to an individual event in the distributed stream, (E, ⇝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The length of the stream under observation is k, which makes |E| = k × |A| and the length of the entire stream is N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We conduct a pre-processing of the distributed stream where we create a E × E matrix, hbSet to incorporate the happen- before relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We populate hbSet as hbSet[e][f] = 1 iff e ⇝ f, else hbSet[e][f] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In order to map each event to its respective stream, we introduce a function, µ : E → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We introduce a valuation function, val : E → T (whatever the type is in the LOLA specification), in order to represent the values of the individual events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Due to the partially synchronous assumption of the system, the possible time of occurrence of an event is defined by a function δ : E → Z≥0, where ∀α(σ) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='∃σ′ ∈ [max{0, σ − ǫ + 1}, min{σ + ǫ − 1}, N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='δ � α(σ) � = σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We update the δ function when referring to events on output streams by updating the time synchroniza- tion constant to ǫM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This accounts for the clock skew between two monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Finally, we introduce an uninterpreted function ρ : Z≥0 → 2E that identifies a sequence of consistent cuts for computing all possible evaluations of the LOLA specification, while satisfying a number of given constrains explained in Section VI-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' SMT Constrains Once we have defined the necessary SMT entities, we move onto the SMT constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We first define the SMT constraints for generating a sequence of consistent cuts, followed by the ones for evaluating the given LOLA equations ϕα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Constrains for consistent cuts over ρ: In order to make sure that the uninterpreted function ρ identifies a sequence of consistent cuts, we enforce certain constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The first constraint enforces that each element in the range of ρ is in fact a consistent cut: ∀i ∈ [0, k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='∀e, e′ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' � (e ⇝ e′) ∧ (e′ ∈ ρ(i)) � → (e ∈ ρ(i)) Next, we enforce that each successive consistent cut consists of all events included in the previous consistent cut: ∀i ∈ [0, k − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='ρ(i) ⊆ ρ(i + 1) Next, we make sure that the front of each consistent cut constitutes of events with possible time of occurrence in accordance with the semantics of partially-synchronous LOLA: ∀i ∈ [0, k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='∀e ∈ front(ρ(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='δ(e) = i Finally, we make sure that every consistent cut consists of events from all streams: ∀i ∈ [0, k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='∀α ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='∃e ∈ front(ρ(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='µ(e) = α Constrains for LOLA specification: These constraints will evaluate the LOLA specifications and will make sure that ρ will not only represent a valid sequence of consistent cuts but also make sure that the sequence of consistent cuts evaluate the LOLA equations, given the stream expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' As is evident that a distributed system can often evaluate to multiple values at each instance of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Thus, we would need to check for both satisfaction and violation for logical expressions and evaluate all possible values for arithmetic expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Note that monitoring all LOLA specification can be reduce to evaluating expressions that are either logical or arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Below, we mention the SMT constraint for evaluating different LOLA equations at time instance j: ti[p, c] = � val(e) 0 ≤ j + p ≤ N c otherwise � ∃e ∈ front(ρ(j + p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' (µ(e) = αi) � si(j) = true front(ρ(j)) |= ϕα (Logical expression, satisfaction) si(j) = ei(∀e ∈ front(ρ(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='val(e)) (Arithmetic expression, evaluation) The previously evaluated result is included in the SMT in- stance as a entity and a additional constrain is added that only evaluates to unique value, in order to generate all possible evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The SMT instance returns a satisfiable result iff there exists at-least one unique evaluation of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This is repeated multiple times until we are unable to generate a sequence of consistent cut, given the constraints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=', generate unique values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' It is to be noted that stream expression of the form ite(si, sk, sj) can be reduced to a set of expressions where we first evaluate si as a logical expression followed by evaluating sj and sk accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' RUNTIME VERIFICATION OF LOLA SPECIFICATIONS Now that both the rules of generating rewritten LOLA equations (Section V) and the working of the SMT encoding (Section VI) have been discussed, we can finally bring them to- gether in order to solve the problem introduced in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 7 Algorithm 2 Computation on Monitor Mi 1: LS i 1[0] = ∅ 2: for r = 1 to ⌈N/k⌉ do 3: (Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' ⇝i)r ← r-th Consistent partial distributed stream 4: j = 0 5: do 6: j = j + 1 7: (α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' α|A|) ∈ Sr(Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' ⇝i) 8: LSi r[j] ← LSi r[j − 1] ∪ � (α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' α|A|) |=S ϕα � 9: while (LS i r[j] ̸= LS i r[j − 1]) 10: Send: broadcasts symbolic view LSi r[j] 11: Receive: Πi r ← {LS k r | 1 ≤ k ≤ M} 12: Compute: LSi r+1[0] ← LC(Πi r) ⊲ Section VII-A 13: end for 14: Resulti ← � r∈[1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='⌈N/k⌉+1] LSi r[0] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Computing LC Given a set of local states computed from the SMT encod- ing, each monitor process receives a set of rewritten LOLA associated equations, denoted by LS i j, where i ∈ [1, |M|] for j-th computation round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Our idea to compute LC from these sets is to simply take a prioritized union of all the associated equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' LC (Πi j) = � i∈[1,|M|] LS i j The intuition behind the priority is that an evaluated LOLA equation will take precedence over a partially evaluat- ed/unevaluated LOLA equation, and two partially-evaluated LOLA equation will be combined to form a evaluated or partially evaluated LOLA equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For example, taking the lo- cally computed LS 1 1 and LS 2 1 from Example 3, LC (LS 1 1, LS 2 1) is computed to be {c(1) = a(2), c(2) = 5, c(3) = ite(7 ≤ b(4), a(4), 5)} at Monitor M1 and {c(1) = 7, c(2) = 5, c(3) = ite(7 ≤ b(4), a(4), 5)} at Monitor M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Subse- quently, LC (LS 1 2, LS 2 2) is computed to be {c(4) = 9, c(5) = 3, c(6) = 5} at Monitor M1 and {c(4) = 9, c(5) = 3, c(6) = 5} at Monitor M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Bringing it all Together As stated in Section IV-A, the monitors are decentralized and online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Since, setting up of a SMT instance is costly (as seen in our evaluated results in Section VIII), we often find it more efficient to evaluate the LOLA specification after every k time instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This reduces the number of computation rounds to ⌈N/k⌉ as well as the number of messages being transmitted over the network as well with an increase to the size of the messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We update Algorithm 1 to reflect our solution more closely to Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Each evaluation round starts by reading the r-th partial distributed system which consists of events occurring between the time max{0, (r − 1) × ⌈N/k⌉} and min{N, r × ⌈N/k⌉} (line 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We assume that the partial distributed system is consistent in accordance with the assumption that each event has been read by atleast one monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' To account for any concurrency among the events in (r − 1)-th computation round with that in the r-th computation round, we expand the length by ǫ time, there-by making the length of the r-th computation round, max{0, (r − 1) × ⌈N/k⌉ − ǫ + 1} and min{N, r × ⌈N/k⌉}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Next, we reduce the evaluation of the distributed stream problem into an SMT problem (line 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We represent the distributed system using SMT entities and then by the help of SMT constraints, and we evaluate the LOLA specification on the generated sequence of consistent cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Each sequence of consistent cut presents a unique ordering of the events which evaluates to a unique value for the stream expression (line 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This is repeated until we no longer can generate a sequence of consistent cut that evaluates ϕα to unique values (line 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Both the evaluated as well as partially evaluated results are included in LS as associated LOLA equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This is followed by the communication phase where each monitor shares its locally computed LS i r, for all i ∈ [1, |M|] and r evaluation round (line 10-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Once, the local states of all the monitors are received, we take a prioritized union of all the associated equation and include them into LS i r+1 set of associated equations (line 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Following this, the computation shifts to next computation round and the above mentioned steps repeat again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Once we reach the end of the computation, all the evaluated values are contained in Resulti Lemma 1: Let A = {S1, S2, · · · , Sn} be a distributed sys- tem and ϕ be an LOLA specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Algorithm 1 terminates when monitoring a terminating distributed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Theorem 1: Algorithm 2 solves the problem stated in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Theorem 2: Let ϕ be a LOLA specification and (E, ⇝) be a distributed stream consisting of |A| streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The message complexity of Algorithm 2 with |M| monitors is O � ǫ|A|N|M|2� Ω(N|M|2) VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' CASE STUDY AND EVALUATION In this section, we analyze our SMT-based decentralized monitoring solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We note that we are not concerned about data collections, data transfer, etc, as given a distributed setting, the runtime of the actual SMT encoding will be the most dominating aspect of the monitoring process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We evaluate our proposed solution using traces collected from synthetic ex- periments (Section VIII-A) and case studies involving several industrial control systems and RACE dataset (Section VIII-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The implementation of our approach can be found on Google Drive(https://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='com/2p6ddjnr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Synthetic Experiments 1) Setup: Each experiment consists of two stages: (1) generation of the distributed stream and (2) verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For data generation, we develop a synthetic program that randomly generates a distributed stream (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=', the state of the local computation for a set of streams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We assume that streams are of the type Float, Integer or Boolean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For the streams of the type Float and Integer, the initial value is a random value s[0] and we generate the subsequent values by s[i-1] + N(0, 2), for all i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We also make sure that the value of a stream is always non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' On the other 8 hand, for streams of the type Boolean, we start with either true or false and then for the subsequent values, we stay at the same value or alter using a Bernoulli distribution of B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='8), where a true signifies the same value and a false denotes a change in value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For the monitor, we study the approach using Bernoulli distribution B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='2), B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='5) and B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='8) as the read distribution of the events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' A higher readability offers each event to be read by higher number of monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We also make sure that each event is read by at least one monitor in accordance with the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' To test the approach with respect to different types of stream expression, we use the following arithmetic and logical expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' input a1 : uint input a2 : uint output arithExp := a1 + a2 output logicExp := (a1 > 2) && (a2 < 8) 2) Result - Analysis: We study different parameters and analyze how it effects the runtime and the message size in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' All experiments were conducted on a 2017 Mac- Book Pro with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='5GHz Dual-Core Intel core i7 processor and 16GB, 2133 MHz LPDDR3 RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Unless specified otherwise all experiments consider number of streams, |A| = 3, time synchronization constant, ǫM = ǫ = 3s, number of monitors same as the number of streams, computation length, N = 100, with k = 3 with a read distribution B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Time Synchronization Constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Increasing the value of the time synchronization constant ǫ, increases the possible number of concurrent events that needs to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This increases the complexity of evaluating the LOLA specification and there-by increasing the runtime of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In addi- tion to this, higher number of ǫ corresponds to higher number of possible streams that needs to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We observe that the runtime increases exponentially with increasing the value of ǫ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 5a, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' An interesting observation is that with increasing the value of k, the runtime increases at a higher rate until it reaches the threshold where k = ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This is due to the fact, that the number of streams to be considered increases exponentially but ultimately gets bounded by the number of events present in the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Increasing the value of the time synchronization constant is also directly proportional to the number of evaluated re- sults at each instance of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This is because, each stream corresponds to a unique value being evaluated until it gets bounded by the total number of possible evaluations, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' However, comparing Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 5a and 6a, we see that the runtime increases at a faster rate to the size of the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This owes to the fact that initially a SMT instance evaluates unique values at all instance of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' However, as we start reaching all possible evaluations for certain instance of time, only a fraction of the total time instance evaluates to unique values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This is the reason behind the size of the message reaching its threshold faster than the runtime of the monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Type of Stream Expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Stream expressions can be divided into two major types, one consisting of arithmetic op- erations and the other involving logical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Arithmetic operations can evaluate to values in the order of O(|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='ǫ), where as logical operations can only evaluate to either true or false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' When the monitors have high readability of the distributed stream, it is mostly the case, that the monitor was able to evaluate the stream expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Thus, we observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 5c that the runtime grows exponentially for evaluating arithmetic expressions but is linear for logical expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' However, with low readability of the computation, irrespective of the type of expression, both takes exponential time since neither can completely evaluate the stream expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' So, each monitor has to generate all possible streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Similarly, for high readability and logical expressions, the message size is constant given the monitor was was able to evaluate the stream expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' However with low readability, message size for evaluating logical expressions matches with that of its arithmetic counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 6c and is due to the fact, that with low readability, complete evaluation of the expression is not possible at a monitor and thus needs to send the rewritten expression with the values observed to the other monitors where it will be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Number of Streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' As the number of streams increases, the number of events increase linearly and thereby making exponential increase in the number of possible synchronous streams (due to interleavings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 5b, where the runtime increases exponentially with increase in the number of streams in the distributed stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Similarly, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 6b, increase in the number of streams linearly effects the number of unique values that the LOLA expression can evaluate to and there-by increasing the size of the message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Case Studies: Decentralized ICS and Flight Control RV We put our runtime verification approach to the test with respect to several industrial control system datasets that in- cludes data generated by a (1) Secure Water Treatment plant (SWaT) [9], comprising of six processes, corresponding to different physical and control components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' (2) a Power Dis- tribution system [10] that includes readings from four phaser measurement unit (PMU) that measures the electric waves on an electric grid, and (3) a Gas Distribution system [11] that includes messages to and from the PLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In these ICS, we monitor for correctness of system properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Additionally we monitor for mutual separation between all pairs of aircraft in RACE [12] dataset, that consists of SBS messages from aircrafts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For more details about each of the systems along with the LOLA specifications refer to the Appendix XI-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For our setting we assume, each component has its own asynchronous local clock, with varying time synchronization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Next we discuss the results of verifying different ICS with respect to LOLA specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Result Analysis: We employed same number of monitors as the number of components for each of the ICS case-studies and divided the entire airspace into 9 different ones with one monitor responsible for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We observe that our approach 9 1 2 3 4 5 1 5 10 50 100 500 1,000 Time Synchronization Constant (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=') ε Runtime (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=') k = 5 k = 4 k = 3 k = 2 k = 1 (a) Epsilon 2 3 4 5 7 10 1 5 10 50 100 500 1,000 500 10,000 50,000 Number of Streams |A| Runtime (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=') k = 5 k = 4 k = 3 k = 2 k = 1 (b) Number of Streams 2 3 4 5 7 10 1 5 10 50 100 500 1,000 Number of Streams |A| Runtime (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=') arithExp, B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='8) logicExp, B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='8) arithExp, B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='5) logicExp, B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='5) arithExp, B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='2) logicExp, B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='2) (c) Different LOLA Specification Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 5: Impact of different parameters on runtime for synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 1 2 3 4 5 5 10 50 100 Time Synchronization Constant (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=') ε Size of Messages (bytes) k = 5 k = 4 k = 3 k = 2 k = 1 (a) Epsilon 2 3 4 5 7 10 5 10 50 100 500 1,000 Number of Streams |A| Size of Messages (bytes) k = 5 k = 4 k = 3 k = 2 k = 1 (b) Number of Streams 2 3 4 5 7 10 10 50 100 500 1,000 Number of Streams |A| Size of Messages (bytes) arithExp, B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='8) logicExp, B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='8) arithExp, B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='5) logicExp, B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='5) arithExp, B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='2) logicExp, B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='2) (c) Different LOLA Specification Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 6: Impact of different parameters on message size for synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='5 1 2 3 100 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='7 101 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='3 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='6 Time-Synchronization constant ǫ Average % of False-Positives SWaT Power Distribution Gas Distribution RACE Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 7: False-Positives for ICS Case-Studies does not report satisfaction of system property when there has been an attack on the system in reality (false-negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' However, due to the assumption of partial-synchrony among the components, our approach may report false positives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=', it reports a violation of the system property even when there was no attack on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 7, with decreasing time synchronization constant, the number of false- positives reduce as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This is due to the fact that with decreasing ǫ, less events are considered to be concurrent by the monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This makes the partial-ordering of events as observed by the monitor closer to the actual-ordering of events taking place in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We get significantly better result for aircraft monitoring with fewer false-positives compared to the other dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This can be attributed towards Air Traffic Controllers maintaining greater separation between two aircrafts than the minimum that is recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' As part of our monitoring of other ICS, we would like to report that our monitoring approach could successfully detect several attacks which includes underflow and overflow of tank and sudden change in quality of water in SWaT, differentiate between manual tripping of the breaker from the breaker being tripped due to a short-circuit in Power Distribution and Single-point data injection in Gas distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' RELATED WORK Online predicate detection for both centralized and de- centralized monitoring setting have been extensively studies in [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Extensions to more expressive temporal operators are introduced in [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Monitoring approaches introduced in [13], [15], [16] considers a fully asynchronous distributed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' An SMT-based predicate detection solution has been introduced in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Runtime Verification for synchronous dis- tributed system has been studied in [18]–[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The assumption of a common global clock shared among all the components act as a major shortcoming of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Finally, fault- tolerant monitoring, where monitors can crash, has been inves- tigated in [21] for asynchronous and in [22] for synchronized distributed processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 10 Runtime Verification of stream-based specification was in- troduced in [2], [23], where the occurrence of the events was assumed to be synchronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' To extend the stream-based runtime verification to more complex systems, one where the occurrence of events is asynchronous, a real-time based logic was introduced in [24]–[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' However, these methods fall short to verify large geographically separated distributed system, due to their assumption regarding the presence of a shared global clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' On the contrary, we assume the presence of a clock synchronization algorithm which limits the maximum clock skew among components to a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This is a realistic assumption since different components of a large industrial system have their own clock and it is certain to have a skew between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' A similar SMT-based solution was studied for LTL and MTL specifications in [27], [28] respectively, which we extend to include a more expressive stream-based specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' CONCLUSION In this paper, we studied distributed runtime verifica- tion w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' to the popular stream-based specification language LOLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We propose a online decentralized monitoring approach where each monitor takes a set of associated LOLA specifica- tion and a partial distributed stream as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' By assuming partial synchrony among all streams and by reducing the verification problem into an SMT problem, we were able to reduce the complexity of our approach where it is no longer dependent on the time synchronization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We also conducted extensive synthetic experiments, verified system properties of large Industrial Control Systems and airspace monitoring of SBS messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Comparing to machine learning- based approaches to verify the correctness of these system, our approach was able to produce sound and correct results with deterministic guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' As a better practice, one can also use our RV approach along with machine-learning based during training or as a safety net when detecting system violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' For future work, we plan to study monitoring of distributed systems where monitors themselves are vulnerable to faults such as crash and Byzantine faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This will let us design a technique with faults and vulnerabilities mimicking a real life monitoring system and thereby expanding the reach and application of runtime verification on more real-life safety critical systems.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Los Alamitos, CA, USA: IEEE Computer Society, jul 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 23–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Available: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='ieeecomputersociety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='1109/ICDCS54860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='00012 XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' LOLA Syntax A stream expression is constructed as follows: If c is a constant of type T, then c is an atomic stream expression of type T If s is a stream variable of type T, then s is an atomic stream expression of type T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' If f : T1 × T2 × · · · Tk → T is a k-ary operator and for 1 ≤ i ≤ k, ei is an expression of type Ti, then f(e1, e2, · · · , ek) is a stream expression of type T If b is a stream expression of type boolean and e1, e2 are stream expressions of type T, then ite(b, e1, e2) is a stream expression of type T, where ite is the abbreviated form of if-then-else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' If e is a stream expression of type T, c is a constant of type T and i is an integer, then e[i, c] is a stream expres- sion of type T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' e[i, c] refers to the value of the expression e offset by i positions from the current position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In case the offset takes it beyond the end or before the beginning of the stream, then the default value is c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Furthermore, LOLA can be used to compute incremental statistics, where a given a stream, α, a function, fα(v, u), computes a measure, where u represents the measure thus far and v, the current value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Given a sequence of values, v1, v2, · · · , vn, with a default value d, the measure over the data is given as u = fα(vn, fα(vn−1, · · · , fα(v1, d))) Example of such functions include count, fcount(v, u) = u+1, sum, fsum(v, u) = u + v, max, fmax(v, u) = max{v, u}, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Aggregate functions like average, can be de- fined using two incremental functions, count and sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Proofs Lemma 2: Let A = {S1, S2, · · · , Sn} be a distributed sys- tem and ϕ be an LOLA specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Algorithm 1 terminates when monitoring a terminating distributed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Proof 1: First, we note that our algorithm is designed for terminating system, also, note that a terminating program only produces a finite distributed computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In order to prove the lemma, let us assume that the system send out a stop signal to all monitor processes when it terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' When such a signal is received by a monitor, it starts evaluating the output stream expression using the terminal associated equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This might arise to two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' One where all the values required for the evaluation has been observed or one where the values required for the evaluation has not been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Although the termination of the monitor process for the first case is trivial, the termination of the monitor process for the second case is dependent upon replacing such unobserved stream value by the default value of the stream expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Thus, terminating the monitor process eventually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Theorem 3: Algorithm 2 solves the problem stated in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Proof 2: We prove the soundness and correctness of Al- gorithm 2, by dividing it into three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In the first step 12 we prove that given a LOLA specification, ϕ, the values of the output stream when computed over the distributed computation, (E, ⇝), of length N is the same as when the distributed computation is divided into N k computation rounds of length k each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Second, we prove that for all time instances the stream equation is eventually evaluated after the communication round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Finally we prove the set of all evaluated result is consistent over all monitors in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Step 1: From our approach, we see that the value of a output stream variable, is evaluated on the events present in the consistent cut with time j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Therefore, we can reduce the proof to: Sr(E, ⇝) = Sr(E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='E2 · · · E N k , ⇝) (⇒) Let Ck be a consistent cut such that Ck is in Sr(E, ⇝) , but not in Sr(E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='E2 · · · E N k , ⇝), for some k ∈ [0, |E|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This implies that the frontier of Ck, front(Ck) ̸⊆ E1 and front(Ck) ̸⊆ E2 and · · · and front(Ck) ̸⊆ E N k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' However, this is not possible, as according to the computation round construction in Section VII-B, there must be a Ei, where 1 ≤ i ≤ N k such that front(Ck) ⊆ Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Therefore, such Ck cannot exist, and (α1, α2, · · · , αn) ∈ Sr(E, ⇝) =⇒ (α1, α2, · · · , αn) ∈ Sr(E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='E2 · · · E N k , ⇝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' (⇐) Let Ck be a consistent cut such that Ck is in Sr(E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='E2 · · · E N k , ⇝) but not in Sr(E, ⇝) for some k ∈ [0, |E|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This implies, front(Ck) ⊆ Ei and front(Ck) ̸⊆ E for some i ∈ [1, N k ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' However, this is not possible due to the fact that ∀i ∈ [1, N k ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='Ei ⊂ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' There, such Ck cannot exist, and (α1, α2, · · · , αn) ∈ Sr(E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='E2 · · · E N k , ⇝) =⇒ (α1, α2, · · · , αn) ∈ Sr(E, ⇝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Therefore, Sr(E, ⇝) = Sr(E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='E2 · · · E N k , ⇝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Step 2: Given a output stream expression si and the dependency graph G = ⟨V, E⟩, for each ⟨si, sk, w⟩ ∈ E, evaluating the value at time instance j ∈ [1, N], αk(j+w) ̸= ♮ or αk(j + w) = ♮ or αk(w + j) not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' If αk(j +w) ̸= ♮, then we evaluate the stream expression If αk(j + w) = ♮, there exists at-least one other monitor where αk(j + w) ̸= ♮.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Thereby evaluating the stream expression, followed by sharing the the evaluated result with all other monitors If αk(w+j) not observed, then at some future evaluation round and at some monitor αk(j + w) ̸= ♮ and there-by evaluating the stream expression si Similarly, it can be proved for ⟨si, tk, w⟩ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Step 3: Each monitor in our approach is fault-proof with communication taking place between all pairs of monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We also assume, all messages are eventually received by the monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This guarantees all observations are either directly or indirectly read by each monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Together with Step 1 and 2, soundness and correctness of Algorithm 1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Theorem 4: Let ϕ be a LOLA specification and (E, ⇝) be a distributed stream consisting of |A| streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The message complexity of Algorithm 2 with |M| monitors is O � ǫ|A|N|M|2� Ω(N|M|2) Proof 3: We analyze the complexity of each part of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The algorithm has a nested loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The outer loop iterates for ⌈N/k⌉ times, that is O(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The inner loop is dependent on the number of unique evaluations of the stream expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Upper-bound Due to our assumption of partial- synchrony, each event’s time of occurrence can be off by ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This makes the maximum number of unique eval- uations in the order of O(ǫ|A|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Lower-bound The minimum number of unique evalua- tions is in the order of Ω(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In the communication phase, each monitor sends |M| messages to all other monitors and receives |M| messages from all other monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' That is |M|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Hence the message complexity is O � ǫ|A|N|M|2� Ω(N|M|2) As a side note, we would like to mention that in case of high readability of the monitors and evaluation of logical expres- sion, the complexity is closer to the lower-bound, whereas with low readability and arithmetic expressions, the complexity is closer to the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Industrial Control Systems a) SWaT Dataset: Secure Water Treatment (SWaT) [9] utilizes a fully operational scaled down water treatment plant with a small footprint, producing 5 gallons/minute of doubly filtered water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' It comprises of six main processes correspond- ing to the physical and control components of the water treatment facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' It starts from process P1 where it takes raw water and stores it in a tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' It is then passed through the pre-treatment process, P2, where the quality of the water is assessed and maintained through chemical dosing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The water then reaches P3 where undesirable materials are removed using fine filtration membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Any remaining chlorine is destroyed in the dechlorination process in P4 and the water is then pumped into the Reverse Osmosis system (P5) to reduce inorganic impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Finally in P6, water from the RO system is stored ready for distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The dataset classifies different attack on the system into four types, based on the point and stage of the attack: Single Stage-Single Point, Single Stage-Multi Point, Multi Stage- Single Point and Multi Stage-Multi Point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We for the scope of this paper are the most interested in the attacks either covering multiple stages or multiple points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Few of the LOLA specifications used are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' input FIT-101 : uint input MV-101 : bool input LIT-101 : uint input P-101 : bool input FIT-201 : uint output inflowCorr := ite(MV-101 == true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' FIT-101 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' FIT-101 == 0) output outflowCorr := ite(P-101 == true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' FIT-201 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' FIT-201 == 0) output tankCorr := ite(MV-101 == true || P-101 == true,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' LIT-101 = LIT-101[-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0] + FIT-101[-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0] - FIT-201[-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0]) 13 where FIT-101 is the flow meter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' measuring inflow into raw water tank,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' MV-101 is a motorized valve that controls water flow to the raw water tank,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' LIT-101 is the level transmitter of the raw water tank,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' P-101 is a pump that pumps water from raw water tank to the second stage and FIT-201 is the flow transmitter for the control dosing pumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The above LOLA specification checks the correctness of the inflow meter and valve pair (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' outflow meter and pump pair) in inflowCorr (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' outflowCorr) output expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' On the other hand, tankCorr checks if the water level in the tank adds up to the in-flow and out-flow meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' input AIT-201 : uint input AIT-202 : uint input AIT-203 : uint output numObv := numObv[-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0] + 1 output NaClAvg := (NaClAvg[-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0] * numObv[-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0] + AIT-201) / numObv output HClAvg := (HClAvg[-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0] * numObv[-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0] + AIT-202) / numObv output NaOClAvg := (NaOClAvg[-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0] * numObv[-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' 0] + AIT-202) / numObv where AIT-201,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' AIT-202 and AIT-203 represents the NaCl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' HCl and NaOCl levels in water respectively and NaClAvg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' HClAvg and NaOClAvg keeps a track of the average levels of the corresponding chemicals in the water,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' where as numObv keeps a track of the total number of observations read by the monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' b) Power System Attack Dataset: Power System Attack Dataset [10] consists of three datasets developed by Missis- sippi State University and Oak Ridge National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' It consists of readings from four phaser measurement unit (PMU) or synchrophasor that measures the electric waves on an electric grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Each PMU measures 29 features consisting of voltage phase angle, voltage phase magnitude, current phase angle, current phase magnitude for Phase A-C, Pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=', Neg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' and Zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' It also measures the frequency for relays, the frequency delta for relay, status flag for relays, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Apart from these 116 PMU measurements, the dataset also consists of 12 control panel logs, snort alerts and relay logs of the 4 PMU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The dataset classifies into either natural event/no event or an attack event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Few of the LOLA specifications used are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The first attempts to detect a single-line-to-ground (1LG) fault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' input R1-I : float input R2-I : float input R1-Relay : bool input R2-Relay : bool output R1-I-low := R1-I < 200 output R1-I-high := R1-I > 1000 output R2-I-low := R2-I < 200 output R2-I-high := R2-I > 1000 output 1LG := R1-I-high && R2-I-high && R1-Relay[+2, false] && R2-Relay[+2, false] && R1-I-low[+4, false] && R2-I-low[+4, false] where R1-I and R2-I represents the current measured at the R1 and R2 PMU respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Additionally, R1-Relay and R2-Relay keeps a track of the state of the corresponding relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' As a part of the 1LG attack detection, we first categorize the current measured as either low or high depending upon the amount of the current measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We categorize an attack as 1LG if both R1 and R2 detects high current flowing followed by the relay tripping followed by low current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' input R1-PA1-I : float input R1-PA2-I : float input R1-PA3-I : float output phaseBal := (R1-PA1-I - R1-PA2-I) <= 10 && (R1-PA2-I - R1-PA3-I) <= 10 && (R1-PA3-I - R1-PA1-I) <= 10 where R1-PA1-I, R1-PA2-I and R1-PA3-I are the amount of current measured by R1 PMU at Phase A, B and C respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The monitor helps us to check if the load on three phases are equally balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' c) Gas Distribution System: Gas Distributed System [11] is a collection of labeled Remote Terminal Unit (RTU) teleme- try streams from a Gas pipeline system in Mississippi State University’s Critical Infrastructure Protection Center with col- laboration from Oak Ridge National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The telemetry streams includes messages to and from the Programmable Logic Controller (PLC) under normal operations and attacks involving command injection and data injection attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The feature set includes the pipeline pressure, setpoint value, command data from the PLC, response to the PLC and the state of the solenoid, pump and the Remote Terminal Unit (RTU) auto-control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' One of the most common data injection attack is Fast Change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Here the reported pipeline pressure value is succes- sively varied to create a lack of confidence in the correct op- eration of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The corresponding LOLA specification monitoring against such attack is mentioned below: input PipePress : float input response : bool output fastChange := ite(response, mod(PipePress - PipePress[-1, 1000]) <= 10, true) where PipePress records the measured pipeline pressure and response is a flag variable signifying a message to the PLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Here we consider the default pressure is 1000 psi and the permitted pressure change per unit time is 10 psi (these can be changed according to the demands of the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Similarly we have LOLA specifications monitoring other data injection attacks such as Value Wave Injection, Setpoint Value Injection, Single Data Injection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' and command injection attacks such as Illegal Setpoint, Illegal PID Command, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' d) RACE Dataset: Runtime for Airspace Concept Eval- uation (RACE) [12] is a framework developed by NASA that is used to build an event based, reactive airspace simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' We use a dataset developed using this RACE framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' This dataset contains three sets of data collected on three different days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Each set was recorded at around 37 N Latitude and 121 W Longitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The dataset includes all 8 types of messages being sent by the SBS unit by using a Telnet 14 application to listen to port 30003, but we only use the messages with ID ‘MSG 3’ which is the Airborne Position Message and includes a flight’s latitude, longitude and altitude using which we verify the mutual separation of all pairs of aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' Furthermore, calculating the distance between two coordinates is computationally expensive, as we need to factor in parameters such as curvature of the earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' In order to speed up distance related calculations, we consider a constant latitude distance of 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='2km and longitude distance of 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content='62km, at the cost of a negligible error margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} +page_content=' The corresponding LOLA specification is mentioned below: input flight1_alt : float input flight1_lat : float input flight1_lon : float input flight2_alt : float input flight2_lat : float input flight2_lon : float output distDiff := sqrt(pow(flight1_alt - flight2_alt, 2) + pow((flight1_lon - flight2_lon)*87620, 2) + pow((flight1_lat flight2_lat)*111200, 2)) output check := distDiff > 500 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANFQT4oBgHgl3EQfMDYd/content/2301.13266v1.pdf'} diff --git a/C9A0T4oBgHgl3EQfAf_H/content/tmp_files/2301.01964v1.pdf.txt b/C9A0T4oBgHgl3EQfAf_H/content/tmp_files/2301.01964v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..31bf115e1485216025d38dff1b417a11c7ae3dcb --- /dev/null +++ b/C9A0T4oBgHgl3EQfAf_H/content/tmp_files/2301.01964v1.pdf.txt @@ -0,0 +1,1805 @@ +Nonlinear photoconductivities and quantum geometry of chiral multifold fermions +Hsiu-Chuan Hsu,1, 2, ∗ Jhih-Shih You,3, † Junyeong Ahn,4, ‡ and Guang-Yu Guo5, 6, § +1Graduate Institute of Applied Physics, National Chengchi University, Taipei 11605, Taiwan +2Department of Computer Science, National Chengchi University, Taipei 11605, Taiwan +3Department of Physics, National Taiwan Normal University, Taipei 11677, Taiwan +4Department of Physics, Harvard University, Cambridge, MA, USA +5Department of Physics, National Taiwan University, Taipei 10617, Taiwan +6Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan +(Dated: January 6, 2023) +Chiral multifold fermions are quasi-particles that appear only in chiral crystals such as transition metal sili- +cides in the cubic B20 structure (i.e., the CoSi family), and they may show exotic physical properties. Here +we study the injection and shift photoconductivities and also the related geometrical quantities for several types +of chiral multifold fermions, including spin-1/2 as well as pseudospin-1 and -3/2 fermions, dubbed as Kramers +Weyl, triple point and Rarita-Schwinger-Weyl (RSW) fermions, respectively. We utilize the minimal symmor- +phic model to describe the triple point fermions (TPF). We also consider the more realistic model Hamiltonian +for the CoSi family including both linear and quadratic terms. We find that circular injection currents are quan- +tized as a result of the Chern numbers carried by the multifold fermions within the linear models. Surprisingly, +we discover that in the TPF model, linear shift conductivities are proportional to the pseudo spin-orbit coupling +and independent of photon frequency. In contrast, for the RSW and Kramer Weyl fermions, the linear shift +conductivity is linearly proportional to photon frequency. The numerical results agree with the power-counting +analysis for quadratic Hamiltonians. The frequency independence of the linear shift conductivity could be +attributed to the strong resonant symplectic Christoffel symbols of the flat bands. Moreover, the calculated sym- +plectic Christoffel symbols show significant peaks at the nodes, suggesting that the shift currents are due to the +strong geometrical response near the topological nodes. +I. +INTRODUCTION +Multifold fermions are types of quasi-particles that only ap- +pear in solids with particular crystal symmetries. Their pesu- +dospin degrees of freedom are the degeneracies at the high- +symmetry points in the Brillouin zone. There is no counterpart +in the elementary particle model. Thus, the study of the phys- +ical properties and genuine signatures of multifold fermions +in solids is of great interest. +Recent advances in solid state physics show that the topo- +logical and geometrical properties of quantum states manifest +in several physical quantities, one of which is photovoltaic +effect. It is the generation of d.c. current in a noncentrosym- +metric solid under the irradiation of light without an external +bias. Thus, the photovoltaic effect plays an important role in +the search for green energy supplications [1, 2]. The photo- +voltaic response functions are closely related to the quantum +geometrical quantities, such as connections, quantum metric +and Berry curvature. The quantum geometric properties are +related to transport in semiclassical picture. For anomalous +Hall effect, Berry curvature gives rise to the anomalous veloc- +ity of carriers [3, 4]. More recently, the second-order response +of electrons to electromagnetic fields is shown to relate to the +quantum metric and Christoffel symbols, which give rise to +the gravity in momentum space [5, 6]. The possibility of the +quantization of quantum metric in topological semimetals has +∗ hcjhsu@nccu.edu.tw +† jhihshihyou@ntnu.edu.tw +‡ junyeongahn@fas.harvard.edu +§ gyguo@phys.ntu.edu.tw +been discussed [7, 8]. In another perspective, the photoelectric +response can be utilized to probe quantum geometry of Bloch +states [1, 9–11]. Therefore, the investigation of the seemingly +pure mathematical structure would deepen the theoretical and +experimental understanding of solids. +The photovoltaic effect in topological semimetals have +been widely studied [5, 12–18]. It has been found that the +Weyl semimetal possesses low frequency divergence which +makes it a promising candidate for terahertz photodetectors +[5, 13, 14, 17, 18]. +However, for chiral symmetric Weyl +semimetals, the photovoltaic response of the topological node +and antinode cancels out unless the Weyl nodes are tilted [13]. +In contrast, for chiral crystals, the Weyl points are separated +in energy, as a result of the chiral symmetry breaking. There +is an available energy window for nonvanishing photocurrent +even for upright cones. Therefore, the chiral Weyl semimet- +als are promising materials for realizing strong photovoltaic +response. +The relation between the second-order photoconductivity +tensors and topology has been investigated by several authors. +It has been theoretically shown that in chiral symmetry bro- +ken Weyl semimetals, the circular photogalvanic response is +quantized due to the Chern number of the Weyl node near the +Fermi level [2]. Moreover, the second-order photoconductiv- +ity is related to the connection and curvature, reflecting the +geometry of Bloch states involved in the transition [9]. +The photoconductivities in chiral multifold fermions have +been studied in real materials, especially in the CoSi fam- +ily of space group 198 [17–22]. +The material hosts sev- +eral types of topological semimetals, including, type-I, type-II +Weyl semimetal and chiral multifold fermions [17–23]. Thus, +it is a very suitable material for investigation of physical prop- +arXiv:2301.01964v1 [cond-mat.mes-hall] 5 Jan 2023 + +2 +erties of topological semimetals. +For second-order photoconducitvites, there are two contri- +butions, the injection and shift current. The injection current +is related to Berry curvature [5, 12] and quantum metric [6, 7], +while the shift current is related to Hermitian connections +[1, 9, 24]. However, an understanding of shift current and its +geometrical origin for multifold fermions have been lacking. +How momentum space quantum geometry contributes to op- +tical response via Christoffel symbols has not been carefully +examined. This paper aims at shedding light on this topic. +Two model Hamiltonians for multifold fermions are studied +in this paper. The first is a pseudospin-1 excitation, which +is dubbed as triple point fermion (TPF). The minimal sym- +morphic model for TPF, of which the degenerate nodal point +is protected by C4 and an anti-commuting mirror symmetry, +is used in this study. The second is the low-energy effective +Hamiltoinan for space group 198. When spin-orbit coupling +is switched off, the effective Hamiltoinian represents two de- +generate TPFs (DTPF). In contrast, when spin-orbit coupling +is included, the degenerate TPFs split into two sets of degener- +ate points, a spin-3/2 excitation, dubbed as Rarita-Schwinger- +Weyl (RSW) or a four-fold fermion, and a spin-1/2 Weyl +point. +In this paper, we give analytical expressions of the second- +order photoconductivities in terms of geometrical quantities +and report the numerical results for TPF, DTPF, RSW and +Kramer Weyl fermions. The injection conductivity is shown +to be related to quantum geometric tensors. The shift conduc- +tivity is not only contributed by Christoffel symbols, but also +the contorsion tensors. The numerical results show that the +shift conductivity can be merely given by the contorsion ten- +sors, whereas the corresponding Christoffel symbols vanish. +Our findings disclose the significance of contorsion tensors +which have been overlooked in previous studies [25]. More- +over, for chiral fermions described by the quadratic Hamil- +toinan, our results show that the lowest order of the second- +order photoconductivity scales as ω0. Particularly, the lowest +order of the shift conductivity is proportional to the pseudo +spin-orbit coupling. In contrast, the lowest order of the in- +jection conductivity is independent of model parameters, in +agreement with the quantization of circular injection conduc- +tivity. The remainder of this paper is organized as follows. In +Sec. II, the second-order photoconductivities and their rela- +tions to the quantum geometrical quantities are given. In Sec. +III, the model Hamiltonians and the power counting analysis +of the second-order photoconductivities for quadratic Hamil- +toinans are presented. The numerical results and discussions +are given in Sec. IV. Finally, the conclusion is given in Sec. +V. +II. +SECOND-ORDER PHOTOCONDUCTIVITIES AND +QUANTUM GEOMETRY +In this study, we consider two contributions to the d.c. re- +sponse of the second-order photoconductivies [26]. Accord- +ing to their mechanisms, they are characterized into two pro- +cesses, injection and shift current. The injection (shift) refers +to the change of group velocity (position) during the interband +transition. The topological and goemetrical aspects have been +discussed in literatures, some of them will be reviewed in this +section. +The shift photoconductivity is given by [5, 27] +σc,ab +shift = −πe3 +ℏ2 +� +ddk +(2π)d +� +n,m +fnmIc,ab +mn δ(ωmn − ω) (1) +where ℏωmn = Em − En is the energy difference between +two bands, d is the spatial dimension, fnm = fn − fm, where +fn,m is the Fermi-Dirac distribution. The electron charge is +−e and e > 0. The integrand for shift conductivity is +Ic,ab +mn = (Rc,a +mn − Rc,b +nm)rb +nmra +mn, +(2) +where Rc,a +mn is the shift vector +Rc,a +mn = rc +mm − rc +nn + i∂clog ra +mn +(3) +and ra +mn = ⟨m|i∂a|n⟩ is the Berry connection. The term +rb +nmra +mn is the real part of the band-resolved quantum geo- +metric tensor, defined as Qba = � +n∈occ +� +m∈unocc rb +nmra +mn +[28, 29], where (un)occ denotes the (un)occupied bands. The +real part of Qba is the quantum metric gba, while the imagi- +nary part is proportional to Berry curvature Ωba. The relation +is +Qba = gba − i +2Ωba. +(4) +Eq. 2 can also be written as i(rb +nmra +mn,c − rb +nm,cra +mn), where +ra +mn,c = ∂cra +mn − i(rc +mm − rc +nn)ra +mn. Notably, rb +nmra +mn,c = +Cbca +nm is a geometrical quantity for the quantum states [9]. +The non-abelian Berry connections form tangent vectors in +the manifold of the Bloch states. In the subpace of the tan- +gent vectors, Cbca +nm is the Hermitian connection that defines +the covariant derivative. +Note that the order of the index +for Hermitian connections is bca for the conductivity ten- +sor cab. Cbca +nm is in general complex and can be written as +Cbca +nm = M bca +nm − i ˜ +M bca +nm, where the real part is the metric con- +nection and the imaginary part is the symplectic connection +apart from a negative sign. Note that the metric connection +here is different from the Levi-Civita connection +Γbca +nm = 1 +2 +� +∂cgba +nm + ∂agbc +nm − ∂bgca +nm +� +(5) +when the number of bands in the system exceeds two. The dif- +ference is characterized by the contorsion tensors. We define a +generalized complex-valued contorsion tensor Kbca +nm such that +it satisfies +Γbca +nm = Re +� +Cbca +nm − Kbca +nm +� +(6) +and define the corresponding symplectic part by +˜Γbca +nm = −Im +� +Cbca +nm − Kbca +nm +� +. +(7) +The expression of the contorsion tensor is given in Appendix +A. The fully symmetric part with respect to the permutation +of b, c, and a of the Im +� +Kbca +nm +� +is chosen to be zero. Eq. 6 +(Eq.7) is the Levi-Civita connection part of the metric (sym- +plectic) connection. We refer to Γbca +nm (˜Γbca +nm) as (symplectic) +Christoffel symbols in this paper. +The linear shift conductivity can be written in terms of the +symplectic Christoffel symbols, + +3 +σc,ab +shift +L = −πe3 +ℏ2 +� +ddk +(2π)d +� +n,m +fnm +� +˜Γbca +nm + ˜Γacb +nm − Im +� +Kbca +nm + Kacb +nm +�� +δ(ωmn − ω). +(8) +The circular shift conductivity can be written in terms of the band resolved Christoffel symbol of the first kind, +σc,ab +shift +C = −πe3 +ℏ2 +� +ddk +(2π)d +� +n,m +fnm +� +Γbca +nm − Γacb +nm − Re +� +Kbca +nm − Kacb +nm +�� +δ(ωmn − ω). +(9) +For numerical calculations, Cbca +nm is written in terms of the ve- +locity operators and double derivatives of the Hamiltonian +Cbca +nm = vb +nm +ω2mn +� +wac +mn − vc +mn∆a +mn + va +mn∆c +mn +ωmn ++ +� +p̸=m,n +�vc +mpva +pn +ωmp +− va +mpvc +pn +ωpn +� � +, +(10) +where wac +mn = ℏ−1⟨m| +∂2H +∂ka∂kc |n⟩, va +mn = ℏ−1⟨m| ∂H +∂ka |n⟩, +∆a +mn = va +mm − va +nn. +The injection conductivity is given by +σc,ab +inj = −τ 2πe3 +ℏ2 +� +ddk +(2π)d +� +nm +fnm∆c +mnrb +nmra +mnδ(ωmn − ω), +(11) +where τ is the relaxation time. +For topological semimetal +that carries topological charges under the irraidation of cir- +cular polarized light, trace of the injection conductivity is +quantized, dubbed as quantized circular photogavanic effect +(CPGE) [12, 15]. +� +cycl σc,ab = β0Cτ, where � +cycl de- +notes the summation over c, a, b in cyclic permutation, C is +the topological charge of the semimetal, β0 = πe3 +h2 and h is +the Planck constant. +The explicit forms of the injection conductivity tensors in +terms of quantum geometrical tensor are given below. The +linear injection conductivity is +σc,ab +inj +L = −τ 2πe3 +ℏ2 +� +ddk +(2π)d +� +n,m +fnm∆c +mngba +nmδ(ωmn − ω). +(12) +The circular injection conductivity is +σc,ab +inj +C = τ 2πe3 +ℏ2 +� +ddk +(2π)d +� +n,m +fnm∆c +mnΩba +nmδ(ωmn − ω), +(13) +where gba +nm = Re +� +rb +nra +m +� +and Ωba +nm = 2iIm +� +rb +nra +m +� +are band +resolved quantum metric and Berry curvature, respectively. +In the numerical calculation, the Dirac delta function in the +equations is replaced with the Lorentzian function +L = 1 +π +Γ/2 +(ωmn − ω)2 + (Γ/2)2 , +(14) +where Γ is the broadening. +TABLE I. The sign change for the matrix elements under z-mirror +symmetry (Mz). +Omn(k) ⇒ ±Omn(k) +rz +mn,z(k) rz +mn,c̸=z(k) ra̸=z +mn,z(k) rz +mn(k) vz +mn(k) ra̸=z +mn (k) va̸=z +mn (k) ++1 +-1 +-1 +-1 +-1 ++1 ++1 +III. +MODEL HAMILTONIANS AND POWER COUNTING +ANALYSIS +The model Hamiltonians of the triple point fermion and the +multifold fermions in the CoSi family are introduced in this +section. +The first model Hamiltonian considered in this paper is the +minimal symmorphic model for TPF [32]. This model can +be viewed as stacked layers of Chern insulators along the z- +direction and thus the time-reversal symmetry is broken. The +topolgoical charge of the Weyl point is ±2. The band dis- +persion is a result of the coupling between the quadratic Weyl +point and a additional flat band via pseudo spin-orbit coupling. +The Hamiltonian for the quadratic Weyl fermions is +Hq(⃗k) = [s(2 − cos(kx) − cos(ky)) − 2t cos(kz)] σz ++ 2b sin(kx) sin(ky)σy ++ 2b [cos(kx) − cos(ky)] σx, +(15) +where b is the pseudo spin-orbit coupling strength, s is the on- +site hopping strength. The z-direction hopping term t, lattice +constant a and ℏ are taken to be 1 in this model. The Weyl +points are at (0, 0, ±π/2) and of opposite chirality. By intro- +ducing a flat band that couples to the quadratic Weyl fermions, +we obtain an effective Hamiltonian for the triple point fermion +[32] +Ht(⃗k) = +� +� +Hq +λ† ++ +λ† +− +λ+ λ− +0 +� +� , +(16) +where λ± = λei(φ±π/4)(sin kx ∓ i sin ky). Hereafter, we +choose λ = +√ +2 and φ = π/2 for isotropic dispersion (to +the lowest order). +The coupling between the flat band and Hq preseves the +symmetry of Hq. Both Hamiltonians obey C4 rotation sym- +metry and anticommute with mirror symmetry Rxy that maps +x ↔ y, preserving chiral symmetry, while time-reversal sym- +metry is broken. The two opposite topological nodes are re- +lated by the mirror symmetry along z-direction Mz. The sign + +4 +FIG. 1. The energy band along [110] direction for Ht [Eq. (6)] to +the quadratic order with b = 1, s = 1. (a) λ = 0. (b) λ = +√ +2. The +numbers annotated on the figure labels the band indexed from low to +high energy. The energy at which the TPF lies is denoted by WT P F . +change of the matrix elements for the photoconductivities un- +der Mz are shown in Table I. The conductivity tensor of which +components with odd numbers of z changes sign for opposite +nodes, leading to vanishing response for the lattice. To break +the chiral symmetry, an additional term that breaks the mirror +symmetry along z- direction, d sin(kz)I3×3, is added to the +Hamiltonian (Eq. 16), where I3×3 is the 3 × 3 identity matrix +[12]. Thus, the chiral symmetry is broken and the two TPFs +are separated in energy. In the following, we consider the re- +sponse near one of the nodes. We consider the low-energy +expansion of the Hamiltonian Eq. (16) up to quadratic order +of k near the node. Thus, λ± = λei(φ±π/4)(kx ∓iky) and Hq +becomes +Hq(⃗k) = +� +sk2 +x + k2 +y +2 ++ 2ckz +� +σz + +2bkxkyσy + b +� +k2 +y − k2 +x +� +σx, +(17) +where c = ∓1 is the chirality of the Weyl point for the node +at (0, 0, ±π/2). The quadratic term in the diagonal does not +change the Chern number of the bands. Thus, changing the +value of s can be treated as a smooth deformation to the +Hamiltonian. The eigenenergies are 0, and +±1 +2 +� +16k2 + (4b2 + s2)k4ρ + 8skzk2ρ +(18) +where k2 = k2 +x + k2 +y + k2 +z, k2 +ρ = k2 +x + k2 +y. The dispersion +relations for λ = 0, +√ +2 with b = 1, s = 1 are shown in Fig. +1. For λ = 0, the upper and lower bands are quadratic, while +for λ = +√ +2, the upper and lower bands disperse linearly. The +spin-excitation sits at zero energy, labeled by WT P F . +For a more realistic model, we take the effective Hamil- +tonian for transition metal silicides that belong to the space +group 198. There are one threefold rotation symmetry along +(111) axis and three twofold screw symmetries along the x, y +and z axis for this group [33, 34]. +FIG. 2. Energy bands along [111] direction for HΓ +198 [Eq. (9)] to the +quadratic order. (a) Without spin-orbit coupling. The bands are dou- +bly degenerate. The black dashed line denotes the energy level at the +double TPF node. (b) With spin-orbit coupling. The blue (red)dahsed +line indicates the energy levels of the RSW node (Kramer Weyl). The +blue numbers denote the band index of the RSW node. The zero en- +ergy denotes the Fermi level. +In order to isolate the multifold fermions at high symmetry +point, we expand the tight-binding Hamiltonian to the sec- +ond order of crystal momentum k. The effective low-energy +Hamiltonian for Γ point is [34] +HΓ198 = +� +i +(H(i) +o ++ H(i) +SOC), +(19) +where H(1,2) +o +is the spinless part, H(1,2) +SOC is the spin-orbit cou- +pled term and the superscripts (1, 2) denote the order in mo- +mentum k of the expansion. For the effective Hamiltonian to +the linear order, i = 1. For the quadratic order, the summation +runs over i = 1, 2. Each part of the Hamiltonian is given by + +(a) +(b) +0.2 +3 +0.1 +0 +E +2 +-0.1 +1 +-0.2 +-0.1 +0 +0.1 -0.1 +0 +0.1 +k/T +k/T(a) +(b) +4 +0.1 +0 +(eV) +WRSW +WDTPF +-0.1 +Wk +-0.2 +-0.1 +0 +0.1 -0.1 +0 +0.1 +k/ +k/5 +H(1) +o += 3v2 + v1 [τx + τxµx + µx] + vp +2 [µykx + τyµzky + τyµxkz] +(20) +H(1) +SOC = vr [τyσz + τxµyσx + τzµyσy] + vs +2 [τxσxkx + τxµxσyky + µxσzkz] +(21) +H(2) +o += −v2k2 +2 ++ −v1 +8 +� +τx(k2 +x + k2 +y) + τxµx(k2 +y + k2 +z) + µx(k2 +z + k2 +x) +� +(22) +H(2) +SOC = −vr +8 +� +τyσz(k2 +x + k2 +y) + τxµyσx(k2 +y + k2 +z) + τzµyσy(k2 +x + k2 +z) +� ++ v′ +r +4 [τyµzσxkxky + τyµxσykykz + µyσzkzkx] , +(23) +TABLE II. Notations for energy levels at each topological node for +HΓ198. +WDT P F -0.07 eV +WK +-0.131 eV +WRSW +-0.04 eV +where τ, µ, σ are Pauli matrices and lattice constant a has +taken to be 1. The parameters are obtained from fitting to +the first-principle calculations. For RhSi, the fitted parameters +are v1 = 0.55, v2 = 0.16, vp = −0.76, vr = −0.03, v′ +r = +0.01, vs = −0.04 (eV) [34]. The tight-binding model pre- +serves the screw and threefold rotation symmetry of the space +group 198. It was constructed with symmetry-allowed nearest +neighbor hoppings [34]. +When the spin-orbit coupling is turned off, i.e. vr, v′ +r, vs = +0 , there are two degenerate spin-1 excitation at Γ point in the +Brillouin zone. The energy band diagram for the quadratic +Hamiltoinan without spin-orbit coupling is shown in Fig. 2(a). +The bands show spin-1 excitation and are doubly degenerate, +dubbed as double TPF. The node locates at energy WDT P F = +−0.07eV. The low energy dispersion is similar to that of Ht, +although with different symmetry properties from Ht. There- +fore, the two models have different nonvanishing components +of the optical conductivities even though the pseudospin de- +grees of freedom are the same. +When SOC is turned on, the six-fold degenerate point splits +up into two sets [33], as denoted by dashed lines in the +band diagram in Fig. 2 (b). One is the fourfold degener- +ate point which is a pseudospin -3/2 excitation and named +as RSW fermion. +The other is the twofold crossing point +which is a spin-1/2 excitation. The energy of each node is +WRSW = −0.04eV for RSW and WK = −0.131eV for +Kramer Weyl. Because the degenerate point is at Γ point, +which is one of the time-reversal invariant momentum, the +two-fold degenerate point is called a Kramer Weyl [35]. The +effective Hamiltonian for the Kramer Weyl is +HK = ⃗k · ⃗σ, +(24) +where σ is the Pauli matrix for electron spin, not pseudospin +degrees of freedom. As a result, the real spin of a Kramer +Weyl align along the principal axis kx, ky, kz [35]. +Power counting analysis. The resonance effect of photo +response in topological semimetals is interesting, because it +suggests the potential application as terahertz photodetectors. +By dimension analysis, the dependence of the shift and injec- +tion conductiviy on photon frequency can be revealed. In pre- +vious studies [5, 30], the analysis was constrained for k-linear +Hamiltonian. Since in our study, the quadratic terms have sig- +nificant roles, we will include linear and quadratic terms in the +Hamiltoinan for dimension analysis. The following analysis +considers three dimensional case, i.e. d = 3. The dimension +of the Hamiltoinan is +H ∼ ℏvk + ℏv′k2 +(25) +and the eigenenergy is denoted by E. Thus, the dimension for +Berry connection is +r ∼ 1 +E +∂H +∂k = ℏv + ℏv′k +E +. +(26) +For E in the denominator, to the lowest order of k gives E ≈ +ℏvk, Thus, +r ∼ 1 +k + v′ +v +(27) +and +r3 ∼ +1 +k3 + 1 +k2 v′ +v + 1 +k +� +v′ +v +�2 ++ +� +v′ +v +�3 +, +(28) +to the lowest order ω ∼ vk. +The delta function, δ(ωmn −ω), has dimension ω−1. Thus, +the shift conductivity scales as +σsh ∼ e3 +ℏ2 +� +a−1 +ω + a0 v′ +v2 + a1 v′2 +v4 ω + H.O.T. +� +(29) +where a−1,0,1 are dimensionless coefficients given by the mo- +mentum space integration in Eq. 1. Note that the a−1ω−1 +diverging term is contributed only by the k-linear terms in +the Hamiltonian and vanishes for upright Weyl cones [5, 30], +which is the case for the multifold fermions considered in this +study. Thus, a−1 = 0. The second term shows that the shift +conductivity is independent of ω, but proportional to v′. The +similar result was found in a previous study that shows the +linear shift conductivity for Dirac surface state is linearly de- +pendent on the warping term and independent of photon fre- +quency [31]. +For injection conductivity, ∆ ∼ v + v′k and r2 ∼ k−2 + +v′ +vk + ( v′ +v )2, +σinj ∼ τe3 +ℏ2 +� +c0 + c1 v′ +v2 ω + H.O.T. +� +, +(30) + +6 +and c0,1 are dimensionaless coefficients given by the momen- +tum space integration in Eq. 11. The leading term, which is +independent of frequency, does not depend on the model pa- +rameters. This term corresponds to the quantization of circular +injection conductivity. The values of the coefficients a−1,0,1 +and c0,1 are determined by the details of the Hamiltonian. +IV. +NUMERICAL CALCULATIONS +In this section, we present the calculated second order pho- +toconductivity spectra and also related geometric quantities +for the model Hamiltonians discribed in the preceeding sec- +tion. +A. +Triple point fermions +The only symmetry for the effective triple point fermion +model Ht in the low-energy expansion is C4 symmetry along +the z axis. As a result of the lowest symmetry, Ht has more +nonzero components of second order photoconductivity than +the CoSi family. Furthermore, because of the broken time- +reversal symmetry, all four types of the photocurrents are +present, namely, linear and circular shift currents as well as +circular and linear injection currents [5]. +From symmetry +analysis, there are 11 nonvanishing linear and 10 nonvanish- +ing circular conductivity tensor elements [36]. Among them, +there are four (three) independent linear (circular) conductiv- +ity elements. For simplicity, we show the most prominant con- +ductivity elements in Fig. 3. +Linear injection current. The linear injection conductivity +spectra are shown in Fig. 3 (a). The xzx and zxx components +are both linear with photon frequency. The linear injection +conductivity is related to the quantum metric gba. In Fig. 4 +(a), gzx and gxx are plotted as a function of kz. The metric el- +ement gxx shows a more drastic change near the node kz = 0, +while gzx is zero along kz. As shown in Fig. 4 (b), gzx on +the kz = 0 plane is an odd function in kx. Therefore, the +integration over the plane is zero. For linear injection conduc- +tivity σxzx +inj +L, gzx is multiplied by ∆x +mn, which is also an odd +function, and the momentum space integration gives rise to +nonvanishing values, as shown in Fig. 3 (a). The distribution +of gxx on the kz = 0 plane is shown in Fig. 4 (c). For lin- +ear injection conductivity σzxx +inj +L, gxx is multiplied by ∆z +mn, +which is a constant because the Hamiltonian is linear in kz. +The values are all positive. Thus, σzxx +inj +L is proportional to the +momentum space integration of the quantum metric gxx. +Circular injection current. The circular injection conduc- +tivity is shown in Fig. 3 (b), which is related to the Berry +curvature. The Berry curvature is an antisymmetric tensor +and thus its diagonal elements Ωaa vanish. Therefore, only +the nondiagonal element Ωzx of Berry curvature is shown in +Fig. 4 (d). Clearly, Ωzx is odd in ky. Thus, when multi- +plied by ∆y +mn, the integral gives rise to nonvanishing circu- +lar injection current element in Fig. 3 (b). When the photon +frequency is larger than the chemical potential, the value satu- +rates at ∼0.65. This value is close to one-third of the topologi- +FIG. 3. Some of the nonvanishing components of the photoconduc- +tivity tensors for the TPF model. (a) Linear injection, (b) circular +injection, (c) linear shift and (d) circular shift conductivity. For all +the panels, the chemical potential is set to −0.1t. The vertical dashed +line denotes that ω = |µ|. +FIG. 4. +Quantum metric tensor elements related to the linear in- +jection conductivity (a, b, c) [components xzx and zxx] and circular +injection (d) conductivity [component xyz], respectively, for the TPF +model with µ = −0.1t. (b-d) are plotted on the kz = 0 plane for the +TPF model. + +0.03 +0.6 +(a) +(b) +0.02 +P +F +0.01 +0.4 +XZX +ZXX +0.00 +0.2 +-0.01 +-0.02 +0.0 +-0.0 +8.0 +(c) +(d) +-0.4 +6.0 +1 +XZX +-0.8 +4.0 +zyx +R +XZX +2.0 +-1.2 +ZXx +ZZZ +0.0 +-1.6 +0.0 +0.1 +0.2 +0.3 +0.0 +0.1 +0.2 +0.3 +w/t +↑/m(a) +4 +zX +0.2 +(b)gzx +2 +XX +E +g +0 +0 +ky +2 +-0.2 +2 +-0.2 +0 +0.2 +0- +kx() +-0.5 +0 +0.5 +kz(1/a) +0.2 +(c)gxx +100 +0.2 +xzU(p) +200 +E +E +0 +0 +0 +0 +-0.2 +-100 +-0.2 +200 +-0.2 +0 +0.2 +-0.2 +0 +0.2 +kx() +kx()7 +FIG. 5. +Circular injection conductivity for linear and quadratic +Hamiltonians of the TPF model. βcc = ϵcabσcab/τ, where is ϵcab +the Levi-Civita symbol. +cal charge for TPF. When taking the trace of the injection con- +ductivity tensor, the value is close to the Chern number, albeit, +with slight deviation. The deviation results from the nonzero +Chern number between each pair of bands for the quadratic +Hamiltonian. In Fig. 5, the � +cycl σc,ab spectrum for HΓ198 +to the linear order is shown. Clearly, with the linear order ex- +pansion of the Hamiltonian, the conductivity is quantized at 2, +the Chern number of the Weyl node. While with the second- +order expansion, the conductivity shifts away from the integer +at higher photon frequency. This is due to the nonzero Berry +curvature between a pair of bands. For injection and shift cur- +rent, only the interband transitions are considered. Thus, we +write the Chern number as combination of the Berry curvature +between pairs of band. Assume only the lowest band is occu- +pied and the bands are indexed from 1 to 3 starting from the +lowest energy band. The Chern number is decomposed into +C = C12 + C13, +(31) +where Cnm is obtained from the the surface integration of the +Berry curvature (Ωnm(θ, φ)), +Cnm = 1 +2π +� π +0 +dθ +� 2π +0 +dφΩnm(θ, φ), +(32) +Ωnm(θ, φ) = −2 Im +⟨m| ∂H +∂θ |n⟩⟨n| ∂H +∂φ |m⟩ +(En − Em)2 +. +(33) +The calculation is done in spherical coordinate and the Chern +number is obtained after intergrating the Berry cuvature on the +constant energy surface. For quadratic Hamiltonian, C13 be- +comes nonzero at higher energy, as shown in the inset of Fig. +5. As a result, the cyclic trace of the injection conductivity +between the optically active pair of bands is not quantized for +the quadratic Hamiltonian. +Linear shift current. Fig. 3(c) shows the linear shift con- +ductivities. zxx, xzx and zzz components are independent +of photon frequency after the photon frequency is larger than +chemical potential. The results is the lowest order in ω, as +FIG. 6. Momentum resolved symplectic Christoffel symbol for TPF. +(a) Along kz, (b) on the kz = 0 plane for the zzz component, (c) on +the kz = 0 plane for the xzx component. The chemical potential is +set slightly below the node, µ = WT P F − 0.1 eV. +suggested by Eq. 29. To understand the numerical results, +we resort to the analytical sulotions. For analytical calcula- +tion, we use Eq. 3 with Berry connetions. Below, the re- +sults for zzz and zxx components are presented. We define +Icab = � +nm fnmIcab +nm. To the lowet order of k, the analytical +form of Icab +nm for the isotropic cone is, +Izzz ≈ 3b +� +k2 +x + k2 +y +�2 +4k6 +(34) +and +Izxx ≈ b +� +2k4 +x + 5k2 +x +� +k2 +y + k2 +z +� ++ 3k2 +y +� +k2 +y + k2 +z +�� +4k6 +. (35) +It shows that the linear shift current depends linearly on the +pseudo spin-orbit coupling b. After inserting the integrand +to Eq. +1 and converting to spherical coordinate, d3k be- +comes dΩkk2dk, where Ωk is the solid angle in k-space, and +δ(ωmn − ω) is replaced with δ(k − k(ω))/|dE/dk|, where +|dE/dk| ∼ 2 in the linear order of k and k(ω) = ω/2. The +integral becomes +σc,ab = −e3 +2h2 +� +dΩk +� +k2dk δ(k − k(ω)) +2 +Icab. +(36) +One +obtains +σzzz +≈ +7.07 b µA/V 2 +and +σzxx +≈ +8.99 b µA/V 2. In Fig. 3, b = 1 is used for the numerical +calculation. The analytical values are close to the numerical +values with less than 4% error. Thus, the plateau corresponds +to a model dependent value. Similarly, the xzx component for +both the linear and circular shift conductivity is independent +of ω, as shown in Fig. 3 (c,d). +The relevant momentum resolved symplectic Christoffel +symbols for linear shift conductivity zzz and zxx are shown +in Fig. 6. The zzz and xzx components both show a peak near +the node in the kz resovled plot. On the kz = 0 plane, the zzz + +2.0 +1.5 + number +linear C12 + linear C13 +1 +Trβ/βo +quadratic C12 +Chern +quadratic Ci3 +1.0 +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +E/t +0.5 +linear +quadratic +0.8 +8.00 +0.25 +0.50 +0.75 +1.00 +1.25 +↑/m1 +>50 +(a) +-0.5 +(b) +ZZZ +0 +XZX +ky 0 +0 +-1 +(c) +0.5 +<-50 +-0.5 +0 +0.5 + -3 +>50 +-0.5 +(c) +-4 +ky 0 +0 +p +-5 +0.5 +-6 +<-50 +-0.5 +0 +0.5 +-0.5 +0 +0.5 +kz +kx8 +component is circularly symmetric, while the xzx component +shows mirror symmetry about the kx = ky plane. +Circular shift current. +Components of the circular shift +conductivity are shown in Fig. 3(d) for s = 1 and b = 1. +Interestingly, both xzx and zyx components of the circular +shift conductivity vanish when s = 0, showing that this term +is vulnerable to the deformation of the on-site hopping. The +corresponding Christoffel symbol of the first kind for σxzx +is shown in Fig. +7. +The kz resolved Γ in Fig. +7 (a) in- +dicate that both xxz and zxx components change sign near +the node kz = 0. For the circular shift zyx component, the +Christoffel symbols are zero along kz. Equation 9 shows that +the contribution to the zyx circular shift conductivity is the +contorsion tensor, not the Christoffel symbol. +Fig. 8 shows +the contorsion tensor Re [Kxzy] in the momentum space. Be- +cause Re [Kxzy] = −Re [Kyzx], only the xzy component is +shown. For other components of conductivities, the numeri- +cal values of the contorsion tensor is negligible compared to +the Christoffel symbols; thus their contorsion tensors are not +shown. +If we further simplify the Hamiltonian to the linear order of +k, the Berry curvature for each band of the TPF has a simple +form ∓ sin θ, 0 for the valence, conduction and flat band, re- +spectively, thereby giving rise to the Chern number of ∓2, 0. +Nevertheless, the shift current vanishes after integration, since +the integrands are either 0 or odd functions. +B. +Multifold fermions in the CoSi family +In this subsection, we present the numerical results of the +model Hamiltonians for multifold fermions in the CoSi fam- +ily. Since these model Hamiltonians have the time-reversal +symmetry, only the linear shift current and circular injection +FIG. 7. Momentum resolved Christoffel symbol of the first kind +(Γxxz and Γzxx) for the TPF model. (a) Along kz, (b) on the kz = 0 +plane for the zzz component, (c) on the kz = 0 plane for the xzx +component. The chemical potential is set slightly below the node, +µ = WT P F − 0.1 eV. +FIG. 8. Momentum resolved contorsion tensor Re [Kxzy] for the +TPF model. This tensor contributes to the circular shift conductivity +zyx. (a) Along kz, (b) on the kz = 0 plane. +FIG. 9. +(a) Circular injection and (b) linear shift conductivity +for HΓ198 without spin-orbit coupling with chemical potential set +slightly below the DTPF node (µ = −0.075 eV = WDT P F −0.005 +eV). In (b), the curve for chemical potential at the DTPF node is also +displayed. + +25 +>10 +XXZ +-0.5 (b) +(a) +ZXX +ky 0 +15 +0 +0.5 +(ge)1 +(b) +<-10 +5 +-0.5 +0 +0.5 +>10 +-0.5 +(c) +(t) +.5 +ky 0 +0 +-15 +0.5 +<-10 +-0.5 +0 +0.5 +-0.5 +0 +0.5 +kz +kx2 +(a) +(b) +1.0 +0.5 +0.0 +1 +-0.5 +0 +V +0 +-1 +0.5 +-0.5 +0 +0.5 +kx +-2 +-0.5 +0 +0.5 +kz5 +(a) +4 +Trβ/(iβo) +3 +2 +WDTPF - 0.005 +0 +20 +(b) +Re[oxyz]() +10 +WDTPF - 0.005 +0 +WDTPF +-10 +-20 +0.000 +0.005 +0.010 +0.015 +hw(eV)9 +FIG. 10. Momentum resolved symplectic Christoffel symbol for the +DTPF node at µ = WDT P F . (a) Along kz, (b) on the kz = 0 +plane for the zxy component and (c) on the kz = 0 plane for the +yxz component. At the Γ point, both zxy and yxz components di- +verge negatively, and thus, after the integration of kx, ky, the value +is negative, as shown by the peak at kz = 0 in (a). +current would occur [5]. +Double triple point fermions. HΓ198 without SOC is a de- +generate TPF, dubbed as double TPF (DTPF). An symmetry +analysis indicates that the xyz element is the only nonvan- +ishing independent component for both circular injection and +linear shift current, which is plotted as a function of photon +energy (ℏω) in Fig. 9(a) and Fig. 9(b), respectively. Figure +9(a) shows that for chemical potential being set slightly be- +low the DTPF node (µ = WDT P F − 0.005 eV), the circular +injection current is nearly zero when ℏω is smaller than the +energy difference between µ and WDT P F (0.005 eV). Nev- +ertheless, it increases sharply when ℏω approaches to 0.005 +eV and quickly becomes saturated as ℏω further increases. As +a result of the topological charge carried by the degenerate +point, the circular injection response would show quantiza- +tion. Figure 9(a) indicates that the circular injection conduc- +tivity is quantized at 4 when ℏω > 0.005 eV because of the +double degeneracy of the Weyl point with chiral charge 2. +In Fig. 9 (b), the linear shift conductivity is shown. In- +terestingly, the shift conductivity for the chemical potential at +the node and below the node are opposite in sign. The ma- +jor contribution comes from the quantum geometry of the flat +band. At low photon frequency, the flat band changes from +being unoccupied at µ = WDT P F − 0.005 to occupied at +µ = WDT P F . This change is approximately equivalent to +taking the complex conjugate of the Hermitian connection. +Since the linear shift conductivity is given by the imaginary +part of the Hermitian connection, the shift of chemical po- +tential leads to the sign change. This is similar to the sign +change of the Berry curvature when chemical potential shifts +across the node. In both cases, the magnitude of the shift con- +ductivity increases monotonically as ℏω increases from zero. +Similar to the circular injection current, the shift conductivity +becomes saturated when ℏω is well above 0.005 eV. Neverthe- +less, in contrast to the circular injection current, the saturation +of the shift conductivity apparently does not result from its +quantization behavior. The saturation can be understood from +power counting analysis, which shows that the lowest order of +the shift conductivity is proportional to a0v′/v. +Since the linear shift conductivity comes from the divergent +behavior of the symplectic Christoffel symbols near the topo- +logical nodes, we show in Figure 10 the symplectic Christof- +fel symbols for the DTPF node. Figure 10 indicates that the +yxz component is more than one order of magnitude stronger +than the zxy component, the linear shift conductivity reveals +mainly the yxz component of the symplectic Christoffel sym- +bols. +RSW fermions. When the spin-orbit coupling is included in +HΓ198, the DTPF nodal point [see Fig. 2(a)] splits into the +RSW and Kramers nodes [see Fig. 2(b)]. The calculated pho- +toconductivity spectra for the RSW node are displayed in Fig. +11. Figure 11(a) shows that the circular injection conductivity +for the RSW fermions increases when ℏω approaches to 0.002 +eV and becomes nearly saturated at ∼3 β0 between 0.0025 +and 0.005 eV. As ℏω further increases, it first dips slightly +and then increases rapidly to the saturated value of 4 [see Fig. +11(a)]. This interesting behavior of the circular injection con- +ductivity for the RSW node can be understood by the band +dispersion of the RSW Hamiltonian displayed in Fig. 2(b) +where the RSW bands of RSW are labeled with blue num- +bers 1-4. When only the transition from the lowest band is +active, the circular injection conductivity reveals the Chern +number of the lowest band, which is 3, and this explains the +first plateau of ∼3. At higher photon frequencies, the transi- +tion between the second and the third band also occurs, giving +rise to a quantization of 1. The saturated value of the circu- +lar injection conductivity thus reveals the sum of the Chern +numbers of the lowest two bands, which is 4. +The linear shift conductivity for the RSW node is displayed +in Fig. +11 (b). +Interestingly, the conductivity in the low +light frequency region below ∼0.007 eV changes sign when +chemical potential is slightly lowered from the RSW node +to WRSW − 0.003 eV. Specifically, when µ = WRSW (red +curve), the linear shift conductivity is negatively proportional +to ω. When µ = WRSW −0.003 eV (blue curve) (i.e. slightly +below the RSW node), the conductivity shows a pronounced +positive peak at the low frequencies. In this low frequency +region, it can be seen from the band structure [Fig. 2(b)] that +the optically active bands are the second and third (first and +second) for µ = WRSW (WRSW − 0.003) eV. Thus, the lin- +ear shift conductivity reveals that the symplectic Chirstoffel +symbols are opposite in sign between different pairs of bands. +Moreover, for the xyz componenet of the linear shift conduc- +tivity, the related components of the Christoffel symbols are +zxy and yxz. We calculate both components of the symplec- +tic Christoffel symbols for the RSW fermions, as shown in +Fig. 12. ˜Γyxz shows a very strong peak near kz = 0. As for +the DTPF node (Figure 10), in contrast, ˜Γzxy is much weaker +and no resonance is found at kz = 0. Thus, the major con- +tribution to the linear shift conductivity is the yxz component +of the symplectic Christoffel symbol. The distribution of ˜Γ on +the kz = 0 plane is also shown in Fig. 12 (b,c). There is a + +>300 +zxy × 10 +-0.2 +(b) +200 +yxz +Ky 0 +0 +(a) +() +0.2 +<-300 +3 +-0.2 +0 +0.2 +e +100 +>300 +((b) +-0.2 +(c) +ky 0 +0 +0.2 +0 +<-300 +-0.2 +0 +0.2 +-0.2 +0 +0.2 +Kz +kx10 +FIG. 11. (a) Circular injection and (b) linear shift conductivity for +HΓ198 with spin-orbit coupling (i.e., the RSW and Kramers nodes). +drastic change near the node. +After turning on the spin-orbit coupling in HΓ198, the band +structure changes drastically. The flat band in DTPF no longer +exists in RSW node. The existence and absence of the flat +band would alter the quantum geometry. The difference can +be observed in comparing the symplectic Christoffel symbols +[Fig. 10 and Fig. 12]. As a consequence, the linear shift +conductivity would have different behavivors. Comparing the +linear shift conductivity [Fig. [9(b) and Fig. 11(b)], the de- +pendence on the photon frequency changes to be linear. The +difference is likely to be the result of the large Christoffel sym- +bols of the flat band. +Kramers Weyl fermions. The calculated photoconductivity +spectra for the Kramers Weyl fermions are also shown in Fig. +11. In this case, µ = WK = −0.131 eV, and the circular in- +jection current probes the Chern number of the Kramer Weyl +node. Thus, the circular injection conductivity is quantized at +1 for ℏω > 0.005 eV. +Interestingly, the linear shift conductivity for the Kramers +Weyl node is proportional to ω, thus exhibiting the same trend +as the type-I Weyl points [30]. In Fig. 13, the symplectic +Christoffel symbols for the Kramer Weyl node are displayed, +which is the source of the linear shift current. Figure 13 thus +indicates that the linear shift conductivity σx,yz is dominated +by the yxz component of the symplectic Christoffel symbol, +similar to that of the DTPF and RSW nodes shown above. +V. +DISCUSSION AND CONCLUSION +The second-order photoconductivities and geometrical +properties of chiral multifold fermions are studied in this pa- +per. The analytical expressions for the injection and shift con- +ductivities in terms of geometrical objects are given. As a +result of the chiral symmetry breaking, the topological node +and antinode are separated in energy. +Thus, we study the +second-order optical response of a single node. Our dimen- +sion analysis reveals that the lowest order of second-order +photoconductivity is ∝ ω0 and the second to the lowest or- +der is ∝ ω1. The quantities are calculated for the minimal +symmorphic TPF model and the effective Hamiltonian for the +CoSi family. Whether the ω0 term survives depends on the de- +tails of the Berry connections. For the TPF, RSW and Kramer +Weyl nodes, the circular injection conductivity shows quanti- +zations, as a result of the Chern number carried by the node. +The linear shift conductivity for the RSW and Kramer Weyl +node is ∝ ω. This behavior is similar to the type-I Weyl node. +In contrast, the linear shift conductivity for the TPF node is +independent of ω, but proportional to pseudo spin-orbit cou- +pling. This relation has not been found in other Weyl semimet- +als, to the best of our knowledge. Furthermore, by analyzing +the momentum-resolved geometrical objects, it is found that +the quantum metric and Christoffel symbols are strongest near +the nodes. The shift conductivities are related to contorsion +tensors. The numerical results show that the contorsion ten- +sors in general are at least one order of magnitude smaller +than Christoffel symbols and symplectic Christoffel symbols +for both model Hamiltonians. However, the contorsion ten- +sors could be dominant. +It is found that the circular shift +conductivity σzyx for the symmorphic TPF model is solely +FIG. 12. Momentum resolved symplectic Christoffel symbol for the +RSW node, µ = WRSW . (a) Along kz, (b) on the kz = 0 plane for +zxy component, (c) on the kz = 0 plane for the yxz component. + +5 +(a) +4 +Trβ/(iβo) +WRsW-0.003 +Wk-0.002 +0 +20 +(b) +WRSW +WRSW -0.003 +10 +Wk +0 +-10 +0.000 +0.005 +0.010 +0.015 +hw(eV)>300 +-0.2 (b) +zxy × 10 +100 +a +yxz +ky 0 +0 +0 +(b) +0.2 +-() +<-300 +3 +-0.2 +0 +0.2 +a +-100 +>300 +-0.2 +(c) +-200 +ky 0 +0 +0.2 +-300 +-0.2 +0.2 +<-300 +-0.2 +0 +0.2 +0 +Kz +kx11 +FIG. 13. +Momentum resolved symplectic Christoffel symbol for +Kramer Weyl of HΓ198. µ = WK − 0.002. (a) Along kz, (b) on the +kz = 0 plane for the zxy component, (c) on the kz = 0 plane for the +yxzthe component. +contributed by contorsion tensors, whereas the corresponding +Christoffel symbols are zero. The study of these geometrical +objects sheds light on the optical probe of the Hilbert space of +lattices. +ACKNOWLEDGMENTS +H.-C.H., J.-S. Y. and G.-Y. G. acknowledge the support +from the National Science and Technology Counsil (NSTC) +and the National Center for Theoretical Sciences (NCTS) in +Taiwan. J.A. was supported by the Center for Advancement of +Topological Semimetals, an Energy Frontier Research Center +funded by the U.S. Department of Energy Office of Science, +Office of Basic Energy Sciences, through the Ames Labora- +tory under contract No. DE-AC02-07CH11358. +Appendix A: Second order photoconductivities in terms of +quantum geometrical quantities +The second-order conductivity tensors are expressed in +term of geometrical quantities in Eq. 8 and 9 of which the +contorsion tensor Kbca +nm is defined as +Kbca +nm = i +2 +� +rb +nm +� +p̸=m,n +(rc +mpra +pn − ra +mprc +pn) + ra +nm +� +p̸=m,n +(rb +mprc +pn − rc +mprb +pn) − ra +nm +� +p̸=m,n +(ra +mprb +pn − rb +mpra +pn) +� +− i +3Re +� +ra +nm +� +p̸=m,n +(rb +mprc +pn − rc +mprb +pn) − rc +nm +� +p̸=m,n +(ra +mprb +pn − rb +mpra +pn) +� ++ Snm +bca . +(A1) +Here, Snm +bca is imaginary and fully symmetric with respect to +the permutation of b, c, and a. 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(Aca- +demic Press, Inc., USA, 2008). + diff --git a/C9A0T4oBgHgl3EQfAf_H/content/tmp_files/load_file.txt b/C9A0T4oBgHgl3EQfAf_H/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6102534474646158c97c1926e9975d32e0a3eb4b --- /dev/null +++ b/C9A0T4oBgHgl3EQfAf_H/content/tmp_files/load_file.txt @@ -0,0 +1,1074 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf,len=1073 +page_content='Nonlinear photoconductivities and quantum geometry of chiral multifold fermions Hsiu-Chuan Hsu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' ∗ Jhih-Shih You,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' † Junyeong Ahn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' ‡ and Guang-Yu Guo5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' § 1Graduate Institute of Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' National Chengchi University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Taipei 11605,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Taiwan 2Department of Computer Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' National Chengchi University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Taipei 11605,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Taiwan 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' National Taiwan Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Taipei 11677,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Taiwan 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Harvard University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' USA 5Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' National Taiwan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Taipei 10617,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Taiwan 6Physics Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' National Center for Theoretical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Taipei 10617,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Taiwan (Dated: January 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 2023) Chiral multifold fermions are quasi-particles that appear only in chiral crystals such as transition metal sili- cides in the cubic B20 structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=', the CoSi family), and they may show exotic physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Here we study the injection and shift photoconductivities and also the related geometrical quantities for several types of chiral multifold fermions, including spin-1/2 as well as pseudospin-1 and -3/2 fermions, dubbed as Kramers Weyl, triple point and Rarita-Schwinger-Weyl (RSW) fermions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' We utilize the minimal symmor- phic model to describe the triple point fermions (TPF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' We also consider the more realistic model Hamiltonian for the CoSi family including both linear and quadratic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' We find that circular injection currents are quan- tized as a result of the Chern numbers carried by the multifold fermions within the linear models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Surprisingly, we discover that in the TPF model, linear shift conductivities are proportional to the pseudo spin-orbit coupling and independent of photon frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In contrast, for the RSW and Kramer Weyl fermions, the linear shift conductivity is linearly proportional to photon frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The numerical results agree with the power-counting analysis for quadratic Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The frequency independence of the linear shift conductivity could be attributed to the strong resonant symplectic Christoffel symbols of the flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Moreover, the calculated sym- plectic Christoffel symbols show significant peaks at the nodes, suggesting that the shift currents are due to the strong geometrical response near the topological nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' INTRODUCTION Multifold fermions are types of quasi-particles that only ap- pear in solids with particular crystal symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Their pesu- dospin degrees of freedom are the degeneracies at the high- symmetry points in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' There is no counterpart in the elementary particle model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, the study of the phys- ical properties and genuine signatures of multifold fermions in solids is of great interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Recent advances in solid state physics show that the topo- logical and geometrical properties of quantum states manifest in several physical quantities, one of which is photovoltaic effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' It is the generation of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' current in a noncentrosym- metric solid under the irradiation of light without an external bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, the photovoltaic effect plays an important role in the search for green energy supplications [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The photo- voltaic response functions are closely related to the quantum geometrical quantities, such as connections, quantum metric and Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The quantum geometric properties are related to transport in semiclassical picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For anomalous Hall effect, Berry curvature gives rise to the anomalous veloc- ity of carriers [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' More recently, the second-order response of electrons to electromagnetic fields is shown to relate to the quantum metric and Christoffel symbols, which give rise to the gravity in momentum space [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The possibility of the quantization of quantum metric in topological semimetals has ∗ hcjhsu@nccu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='tw † jhihshihyou@ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='tw ‡ junyeongahn@fas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='edu § gyguo@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='tw been discussed [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In another perspective, the photoelectric response can be utilized to probe quantum geometry of Bloch states [1, 9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Therefore, the investigation of the seemingly pure mathematical structure would deepen the theoretical and experimental understanding of solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The photovoltaic effect in topological semimetals have been widely studied [5, 12–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' It has been found that the Weyl semimetal possesses low frequency divergence which makes it a promising candidate for terahertz photodetectors [5, 13, 14, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' However, for chiral symmetric Weyl semimetals, the photovoltaic response of the topological node and antinode cancels out unless the Weyl nodes are tilted [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In contrast, for chiral crystals, the Weyl points are separated in energy, as a result of the chiral symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' There is an available energy window for nonvanishing photocurrent even for upright cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Therefore, the chiral Weyl semimet- als are promising materials for realizing strong photovoltaic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The relation between the second-order photoconductivity tensors and topology has been investigated by several authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' It has been theoretically shown that in chiral symmetry bro- ken Weyl semimetals, the circular photogalvanic response is quantized due to the Chern number of the Weyl node near the Fermi level [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Moreover, the second-order photoconductiv- ity is related to the connection and curvature, reflecting the geometry of Bloch states involved in the transition [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The photoconductivities in chiral multifold fermions have been studied in real materials, especially in the CoSi fam- ily of space group 198 [17–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The material hosts sev- eral types of topological semimetals, including, type-I, type-II Weyl semimetal and chiral multifold fermions [17–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, it is a very suitable material for investigation of physical prop- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='01964v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='mes-hall] 5 Jan 2023 2 erties of topological semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For second-order photoconducitvites, there are two contri- butions, the injection and shift current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The injection current is related to Berry curvature [5, 12] and quantum metric [6, 7], while the shift current is related to Hermitian connections [1, 9, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' However, an understanding of shift current and its geometrical origin for multifold fermions have been lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' How momentum space quantum geometry contributes to op- tical response via Christoffel symbols has not been carefully examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' This paper aims at shedding light on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Two model Hamiltonians for multifold fermions are studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The first is a pseudospin-1 excitation, which is dubbed as triple point fermion (TPF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The minimal sym- morphic model for TPF, of which the degenerate nodal point is protected by C4 and an anti-commuting mirror symmetry, is used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The second is the low-energy effective Hamiltoinan for space group 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' When spin-orbit coupling is switched off, the effective Hamiltoinian represents two de- generate TPFs (DTPF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In contrast, when spin-orbit coupling is included, the degenerate TPFs split into two sets of degener- ate points, a spin-3/2 excitation, dubbed as Rarita-Schwinger- Weyl (RSW) or a four-fold fermion, and a spin-1/2 Weyl point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In this paper, we give analytical expressions of the second- order photoconductivities in terms of geometrical quantities and report the numerical results for TPF, DTPF, RSW and Kramer Weyl fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The injection conductivity is shown to be related to quantum geometric tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The shift conduc- tivity is not only contributed by Christoffel symbols, but also the contorsion tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The numerical results show that the shift conductivity can be merely given by the contorsion ten- sors, whereas the corresponding Christoffel symbols vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Our findings disclose the significance of contorsion tensors which have been overlooked in previous studies [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' More- over, for chiral fermions described by the quadratic Hamil- toinan, our results show that the lowest order of the second- order photoconductivity scales as ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Particularly, the lowest order of the shift conductivity is proportional to the pseudo spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In contrast, the lowest order of the in- jection conductivity is independent of model parameters, in agreement with the quantization of circular injection conduc- tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' II, the second-order photoconductivities and their rela- tions to the quantum geometrical quantities are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' III, the model Hamiltonians and the power counting analysis of the second-order photoconductivities for quadratic Hamil- toinans are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The numerical results and discussions are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Finally, the conclusion is given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' SECOND-ORDER PHOTOCONDUCTIVITIES AND QUANTUM GEOMETRY In this study, we consider two contributions to the d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' re- sponse of the second-order photoconductivies [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Accord- ing to their mechanisms, they are characterized into two pro- cesses, injection and shift current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The injection (shift) refers to the change of group velocity (position) during the interband transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The topological and goemetrical aspects have been discussed in literatures, some of them will be reviewed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The shift photoconductivity is given by [5, 27] σc,ab shift = −πe3 ℏ2 � ddk (2π)d � n,m fnmIc,ab mn δ(ωmn − ω) (1) where ℏωmn = Em − En is the energy difference between two bands, d is the spatial dimension, fnm = fn − fm, where fn,m is the Fermi-Dirac distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The electron charge is −e and e > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The integrand for shift conductivity is Ic,ab mn = (Rc,a mn − Rc,b nm)rb nmra mn, (2) where Rc,a mn is the shift vector Rc,a mn = rc mm − rc nn + i∂clog ra mn (3) and ra mn = ⟨m|i∂a|n⟩ is the Berry connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The term rb nmra mn is the real part of the band-resolved quantum geo- metric tensor, defined as Qba = � n∈occ � m∈unocc rb nmra mn [28, 29], where (un)occ denotes the (un)occupied bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The real part of Qba is the quantum metric gba, while the imagi- nary part is proportional to Berry curvature Ωba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The relation is Qba = gba − i 2Ωba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (4) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 2 can also be written as i(rb nmra mn,c − rb nm,cra mn), where ra mn,c = ∂cra mn − i(rc mm − rc nn)ra mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Notably, rb nmra mn,c = Cbca nm is a geometrical quantity for the quantum states [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The non-abelian Berry connections form tangent vectors in the manifold of the Bloch states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In the subpace of the tan- gent vectors, Cbca nm is the Hermitian connection that defines the covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Note that the order of the index for Hermitian connections is bca for the conductivity ten- sor cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Cbca nm is in general complex and can be written as Cbca nm = M bca nm − i ˜ M bca nm, where the real part is the metric con- nection and the imaginary part is the symplectic connection apart from a negative sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Note that the metric connection here is different from the Levi-Civita connection Γbca nm = 1 2 � ∂cgba nm + ∂agbc nm − ∂bgca nm � (5) when the number of bands in the system exceeds two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The dif- ference is characterized by the contorsion tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' We define a generalized complex-valued contorsion tensor Kbca nm such that it satisfies Γbca nm = Re � Cbca nm − Kbca nm � (6) and define the corresponding symplectic part by ˜Γbca nm = −Im � Cbca nm − Kbca nm � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (7) The expression of the contorsion tensor is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The fully symmetric part with respect to the permutation of b, c, and a of the Im � Kbca nm � is chosen to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 6 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='7) is the Levi-Civita connection part of the metric (sym- plectic) connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' We refer to Γbca nm (˜Γbca nm) as (symplectic) Christoffel symbols in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The linear shift conductivity can be written in terms of the symplectic Christoffel symbols, 3 σc,ab shift L = −πe3 ℏ2 � ddk (2π)d � n,m fnm � ˜Γbca nm + ˜Γacb nm − Im � Kbca nm + Kacb nm �� δ(ωmn − ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (8) The circular shift conductivity can be written in terms of the band resolved Christoffel symbol of the first kind, σc,ab shift C = −πe3 ℏ2 � ddk (2π)d � n,m fnm � Γbca nm − Γacb nm − Re � Kbca nm − Kacb nm �� δ(ωmn − ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (9) For numerical calculations, Cbca nm is written in terms of the ve- locity operators and double derivatives of the Hamiltonian Cbca nm = vb nm ω2mn � wac mn − vc mn∆a mn + va mn∆c mn ωmn + � p̸=m,n �vc mpva pn ωmp − va mpvc pn ωpn � � , (10) where wac mn = ℏ−1⟨m| ∂2H ∂ka∂kc |n⟩, va mn = ℏ−1⟨m| ∂H ∂ka |n⟩, ∆a mn = va mm − va nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The injection conductivity is given by σc,ab inj = −τ 2πe3 ℏ2 � ddk (2π)d � nm fnm∆c mnrb nmra mnδ(ωmn − ω), (11) where τ is the relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For topological semimetal that carries topological charges under the irraidation of cir- cular polarized light, trace of the injection conductivity is quantized, dubbed as quantized circular photogavanic effect (CPGE) [12, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' � cycl σc,ab = β0Cτ, where � cycl de- notes the summation over c, a, b in cyclic permutation, C is the topological charge of the semimetal, β0 = πe3 h2 and h is the Planck constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The explicit forms of the injection conductivity tensors in terms of quantum geometrical tensor are given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The linear injection conductivity is σc,ab inj L = −τ 2πe3 ℏ2 � ddk (2π)d � n,m fnm∆c mngba nmδ(ωmn − ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (12) The circular injection conductivity is σc,ab inj C = τ 2πe3 ℏ2 � ddk (2π)d � n,m fnm∆c mnΩba nmδ(ωmn − ω), (13) where gba nm = Re � rb nra m � and Ωba nm = 2iIm � rb nra m � are band resolved quantum metric and Berry curvature, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In the numerical calculation, the Dirac delta function in the equations is replaced with the Lorentzian function L = 1 π Γ/2 (ωmn − ω)2 + (Γ/2)2 , (14) where Γ is the broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The sign change for the matrix elements under z-mirror symmetry (Mz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Omn(k) ⇒ ±Omn(k) rz mn,z(k) rz mn,c̸=z(k) ra̸=z mn,z(k) rz mn(k) vz mn(k) ra̸=z mn (k) va̸=z mn (k) +1 1 1 1 1 +1 +1 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' MODEL HAMILTONIANS AND POWER COUNTING ANALYSIS The model Hamiltonians of the triple point fermion and the multifold fermions in the CoSi family are introduced in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The first model Hamiltonian considered in this paper is the minimal symmorphic model for TPF [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' This model can be viewed as stacked layers of Chern insulators along the z- direction and thus the time-reversal symmetry is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The topolgoical charge of the Weyl point is ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The band dis- persion is a result of the coupling between the quadratic Weyl point and a additional flat band via pseudo spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The Hamiltonian for the quadratic Weyl fermions is Hq(⃗k) = [s(2 − cos(kx) − cos(ky)) − 2t cos(kz)] σz + 2b sin(kx) sin(ky)σy + 2b [cos(kx) − cos(ky)] σx, (15) where b is the pseudo spin-orbit coupling strength, s is the on- site hopping strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The z-direction hopping term t, lattice constant a and ℏ are taken to be 1 in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The Weyl points are at (0, 0, ±π/2) and of opposite chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' By intro- ducing a flat band that couples to the quadratic Weyl fermions, we obtain an effective Hamiltonian for the triple point fermion [32] Ht(⃗k) = � � Hq λ† + λ† − λ+ λ− 0 � � , (16) where λ± = λei(φ±π/4)(sin kx ∓ i sin ky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Hereafter, we choose λ = √ 2 and φ = π/2 for isotropic dispersion (to the lowest order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The coupling between the flat band and Hq preseves the symmetry of Hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Both Hamiltonians obey C4 rotation sym- metry and anticommute with mirror symmetry Rxy that maps x ↔ y, preserving chiral symmetry, while time-reversal sym- metry is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The two opposite topological nodes are re- lated by the mirror symmetry along z-direction Mz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The sign 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The energy band along [110] direction for Ht [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (6)] to the quadratic order with b = 1, s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (a) λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (b) λ = √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The numbers annotated on the figure labels the band indexed from low to high energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The energy at which the TPF lies is denoted by WT P F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' change of the matrix elements for the photoconductivities un- der Mz are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The conductivity tensor of which components with odd numbers of z changes sign for opposite nodes, leading to vanishing response for the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' To break the chiral symmetry, an additional term that breaks the mirror symmetry along z- direction, d sin(kz)I3×3, is added to the Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 16), where I3×3 is the 3 × 3 identity matrix [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, the chiral symmetry is broken and the two TPFs are separated in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In the following, we consider the re- sponse near one of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' We consider the low-energy expansion of the Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (16) up to quadratic order of k near the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, λ± = λei(φ±π/4)(kx ∓iky) and Hq becomes Hq(⃗k) = � sk2 x + k2 y 2 + 2ckz � σz + 2bkxkyσy + b � k2 y − k2 x � σx, (17) where c = ∓1 is the chirality of the Weyl point for the node at (0, 0, ±π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The quadratic term in the diagonal does not change the Chern number of the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, changing the value of s can be treated as a smooth deformation to the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The eigenenergies are 0, and ±1 2 � 16k2 + (4b2 + s2)k4ρ + 8skzk2ρ (18) where k2 = k2 x + k2 y + k2 z, k2 ρ = k2 x + k2 y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The dispersion relations for λ = 0, √ 2 with b = 1, s = 1 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For λ = 0, the upper and lower bands are quadratic, while for λ = √ 2, the upper and lower bands disperse linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The spin-excitation sits at zero energy, labeled by WT P F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For a more realistic model, we take the effective Hamil- tonian for transition metal silicides that belong to the space group 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' There are one threefold rotation symmetry along (111) axis and three twofold screw symmetries along the x, y and z axis for this group [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Energy bands along [111] direction for HΓ 198 [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (9)] to the quadratic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (a) Without spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The bands are dou- bly degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The black dashed line denotes the energy level at the double TPF node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (b) With spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The blue (red)dahsed line indicates the energy levels of the RSW node (Kramer Weyl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The blue numbers denote the band index of the RSW node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The zero en- ergy denotes the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In order to isolate the multifold fermions at high symmetry point, we expand the tight-binding Hamiltonian to the sec- ond order of crystal momentum k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The effective low-energy Hamiltonian for Γ point is [34] HΓ198 = � i (H(i) o + H(i) SOC), (19) where H(1,2) o is the spinless part, H(1,2) SOC is the spin-orbit cou- pled term and the superscripts (1, 2) denote the order in mo- mentum k of the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For the effective Hamiltonian to the linear order, i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For the quadratic order, the summation runs over i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Each part of the Hamiltonian is given by (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 0 E 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 k/T k/T(a) (b) 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 0 (eV) WRSW WDTPF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 Wk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 k/ k/5 H(1) o = 3v2 + v1 [τx + τxµx + µx] + vp 2 [µykx + τyµzky + τyµxkz] (20) H(1) SOC = vr [τyσz + τxµyσx + τzµyσy] + vs 2 [τxσxkx + τxµxσyky + µxσzkz] (21) H(2) o = −v2k2 2 + −v1 8 � τx(k2 x + k2 y) + τxµx(k2 y + k2 z) + µx(k2 z + k2 x) � (22) H(2) SOC = −vr 8 � τyσz(k2 x + k2 y) + τxµyσx(k2 y + k2 z) + τzµyσy(k2 x + k2 z) � + v′ r 4 [τyµzσxkxky + τyµxσykykz + µyσzkzkx] , (23) TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Notations for energy levels at each topological node for HΓ198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' WDT P F -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='07 eV WK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='131 eV WRSW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='04 eV where τ, µ, σ are Pauli matrices and lattice constant a has taken to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The parameters are obtained from fitting to the first-principle calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For RhSi, the fitted parameters are v1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='55, v2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='16, vp = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='76, vr = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='03, v′ r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='01, vs = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='04 (eV) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The tight-binding model pre- serves the screw and threefold rotation symmetry of the space group 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' It was constructed with symmetry-allowed nearest neighbor hoppings [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' When the spin-orbit coupling is turned off, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' vr, v′ r, vs = 0 , there are two degenerate spin-1 excitation at Γ point in the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The energy band diagram for the quadratic Hamiltoinan without spin-orbit coupling is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The bands show spin-1 excitation and are doubly degenerate, dubbed as double TPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The node locates at energy WDT P F = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='07eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The low energy dispersion is similar to that of Ht, although with different symmetry properties from Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' There- fore, the two models have different nonvanishing components of the optical conductivities even though the pseudospin de- grees of freedom are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' When SOC is turned on, the six-fold degenerate point splits up into two sets [33], as denoted by dashed lines in the band diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' One is the fourfold degener- ate point which is a pseudospin -3/2 excitation and named as RSW fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The other is the twofold crossing point which is a spin-1/2 excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The energy of each node is WRSW = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='04eV for RSW and WK = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='131eV for Kramer Weyl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Because the degenerate point is at Γ point, which is one of the time-reversal invariant momentum, the two-fold degenerate point is called a Kramer Weyl [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The effective Hamiltonian for the Kramer Weyl is HK = ⃗k · ⃗σ, (24) where σ is the Pauli matrix for electron spin, not pseudospin degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' As a result, the real spin of a Kramer Weyl align along the principal axis kx, ky, kz [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Power counting analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The resonance effect of photo response in topological semimetals is interesting, because it suggests the potential application as terahertz photodetectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' By dimension analysis, the dependence of the shift and injec- tion conductiviy on photon frequency can be revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In pre- vious studies [5, 30], the analysis was constrained for k-linear Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Since in our study, the quadratic terms have sig- nificant roles, we will include linear and quadratic terms in the Hamiltoinan for dimension analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The following analysis considers three dimensional case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The dimension of the Hamiltoinan is H ∼ ℏvk + ℏv′k2 (25) and the eigenenergy is denoted by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, the dimension for Berry connection is r ∼ 1 E ∂H ∂k = ℏv + ℏv′k E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (26) For E in the denominator, to the lowest order of k gives E ≈ ℏvk, Thus, r ∼ 1 k + v′ v (27) and r3 ∼ 1 k3 + 1 k2 v′ v + 1 k � v′ v �2 + � v′ v �3 , (28) to the lowest order ω ∼ vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The delta function, δ(ωmn −ω), has dimension ω−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, the shift conductivity scales as σsh ∼ e3 ℏ2 � a−1 ω + a0 v′ v2 + a1 v′2 v4 ω + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' � (29) where a−1,0,1 are dimensionless coefficients given by the mo- mentum space integration in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Note that the a−1ω−1 diverging term is contributed only by the k-linear terms in the Hamiltonian and vanishes for upright Weyl cones [5, 30], which is the case for the multifold fermions considered in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, a−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The second term shows that the shift conductivity is independent of ω, but proportional to v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The similar result was found in a previous study that shows the linear shift conductivity for Dirac surface state is linearly de- pendent on the warping term and independent of photon fre- quency [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For injection conductivity, ∆ ∼ v + v′k and r2 ∼ k−2 + v′ vk + ( v′ v )2, σinj ∼ τe3 ℏ2 � c0 + c1 v′ v2 ω + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' � , (30) 6 and c0,1 are dimensionaless coefficients given by the momen- tum space integration in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The leading term, which is independent of frequency, does not depend on the model pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' This term corresponds to the quantization of circular injection conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The values of the coefficients a−1,0,1 and c0,1 are determined by the details of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' NUMERICAL CALCULATIONS In this section, we present the calculated second order pho- toconductivity spectra and also related geometric quantities for the model Hamiltonians discribed in the preceeding sec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Triple point fermions The only symmetry for the effective triple point fermion model Ht in the low-energy expansion is C4 symmetry along the z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' As a result of the lowest symmetry, Ht has more nonzero components of second order photoconductivity than the CoSi family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Furthermore, because of the broken time- reversal symmetry, all four types of the photocurrents are present, namely, linear and circular shift currents as well as circular and linear injection currents [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' From symmetry analysis, there are 11 nonvanishing linear and 10 nonvanish- ing circular conductivity tensor elements [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Among them, there are four (three) independent linear (circular) conductiv- ity elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For simplicity, we show the most prominant con- ductivity elements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Linear injection current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The linear injection conductivity spectra are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The xzx and zxx components are both linear with photon frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The linear injection conductivity is related to the quantum metric gba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 4 (a), gzx and gxx are plotted as a function of kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The metric el- ement gxx shows a more drastic change near the node kz = 0, while gzx is zero along kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 4 (b), gzx on the kz = 0 plane is an odd function in kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Therefore, the integration over the plane is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For linear injection conduc- tivity σxzx inj L, gzx is multiplied by ∆x mn, which is also an odd function, and the momentum space integration gives rise to nonvanishing values, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The distribution of gxx on the kz = 0 plane is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 4 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For lin- ear injection conductivity σzxx inj L, gxx is multiplied by ∆z mn, which is a constant because the Hamiltonian is linear in kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The values are all positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, σzxx inj L is proportional to the momentum space integration of the quantum metric gxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Circular injection current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The circular injection conduc- tivity is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 3 (b), which is related to the Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The Berry curvature is an antisymmetric tensor and thus its diagonal elements Ωaa vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Therefore, only the nondiagonal element Ωzx of Berry curvature is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 4 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Clearly, Ωzx is odd in ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, when multi- plied by ∆y mn, the integral gives rise to nonvanishing circu- lar injection current element in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' When the photon frequency is larger than the chemical potential, the value satu- rates at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' This value is close to one-third of the topologi- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Some of the nonvanishing components of the photoconduc- tivity tensors for the TPF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (a) Linear injection, (b) circular injection, (c) linear shift and (d) circular shift conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For all the panels, the chemical potential is set to −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The vertical dashed line denotes that ω = |µ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Quantum metric tensor elements related to the linear in- jection conductivity (a, b, c) [components xzx and zxx] and circular injection (d) conductivity [component xyz], respectively, for the TPF model with µ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (b-d) are plotted on the kz = 0 plane for the TPF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='6 (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='02 P F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='4 XZX ZXX 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 (c) (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 1 XZX 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 zyx R XZX 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 ZXx ZZZ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='3 w/t ↑/m(a) 4 zX 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 (b)gzx 2 XX E g 0 0 ky 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0- kx() 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 kz(1/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 (c)gxx 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 xzU(p) 200 E E 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 kx() kx()7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Circular injection conductivity for linear and quadratic Hamiltonians of the TPF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' βcc = ϵcabσcab/τ, where is ϵcab the Levi-Civita symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' cal charge for TPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' When taking the trace of the injection con- ductivity tensor, the value is close to the Chern number, albeit, with slight deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The deviation results from the nonzero Chern number between each pair of bands for the quadratic Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 5, the � cycl σc,ab spectrum for HΓ198 to the linear order is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Clearly, with the linear order ex- pansion of the Hamiltonian, the conductivity is quantized at 2, the Chern number of the Weyl node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' While with the second- order expansion, the conductivity shifts away from the integer at higher photon frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' This is due to the nonzero Berry curvature between a pair of bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For injection and shift cur- rent, only the interband transitions are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, we write the Chern number as combination of the Berry curvature between pairs of band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Assume only the lowest band is occu- pied and the bands are indexed from 1 to 3 starting from the lowest energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The Chern number is decomposed into C = C12 + C13, (31) where Cnm is obtained from the the surface integration of the Berry curvature (Ωnm(θ, φ)), Cnm = 1 2π � π 0 dθ � 2π 0 dφΩnm(θ, φ), (32) Ωnm(θ, φ) = −2 Im ⟨m| ∂H ∂θ |n⟩⟨n| ∂H ∂φ |m⟩ (En − Em)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (33) The calculation is done in spherical coordinate and the Chern number is obtained after intergrating the Berry cuvature on the constant energy surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For quadratic Hamiltonian, C13 be- comes nonzero at higher energy, as shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' As a result, the cyclic trace of the injection conductivity between the optically active pair of bands is not quantized for the quadratic Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Linear shift current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 3(c) shows the linear shift con- ductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' zxx, xzx and zzz components are independent of photon frequency after the photon frequency is larger than chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The results is the lowest order in ω, as FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Momentum resolved symplectic Christoffel symbol for TPF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (a) Along kz, (b) on the kz = 0 plane for the zzz component, (c) on the kz = 0 plane for the xzx component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The chemical potential is set slightly below the node, µ = WT P F − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' suggested by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' To understand the numerical results, we resort to the analytical sulotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For analytical calcula- tion, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 3 with Berry connetions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Below, the re- sults for zzz and zxx components are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' We define Icab = � nm fnmIcab nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' To the lowet order of k, the analytical form of Icab nm for the isotropic cone is, Izzz ≈ 3b � k2 x + k2 y �2 4k6 (34) and Izxx ≈ b � 2k4 x + 5k2 x � k2 y + k2 z � + 3k2 y � k2 y + k2 z �� 4k6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (35) It shows that the linear shift current depends linearly on the pseudo spin-orbit coupling b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' After inserting the integrand to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 1 and converting to spherical coordinate, d3k be- comes dΩkk2dk, where Ωk is the solid angle in k-space, and δ(ωmn − ω) is replaced with δ(k − k(ω))/|dE/dk|, where |dE/dk| ∼ 2 in the linear order of k and k(ω) = ω/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The integral becomes σc,ab = −e3 2h2 � dΩk � k2dk δ(k − k(ω)) 2 Icab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (36) One obtains σzzz ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='07 b µA/V 2 and σzxx ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='99 b µA/V 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 3, b = 1 is used for the numerical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The analytical values are close to the numerical values with less than 4% error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, the plateau corresponds to a model dependent value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Similarly, the xzx component for both the linear and circular shift conductivity is independent of ω, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 3 (c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The relevant momentum resolved symplectic Christoffel symbols for linear shift conductivity zzz and zxx are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The zzz and xzx components both show a peak near the node in the kz resovled plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' On the kz = 0 plane, the zzz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 number linear C12 linear C13 1 Trβ/βo quadratic C12 Chern quadratic Ci3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 E/t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 linear quadratic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='25 ↑/m1 >50 (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 (b) ZZZ 0 XZX ky 0 0 1 (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 <-50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 3 >50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 (c) 4 ky 0 0 p 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 6 <-50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 kz kx8 component is circularly symmetric, while the xzx component shows mirror symmetry about the kx = ky plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Circular shift current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Components of the circular shift conductivity are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 3(d) for s = 1 and b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Interestingly, both xzx and zyx components of the circular shift conductivity vanish when s = 0, showing that this term is vulnerable to the deformation of the on-site hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The corresponding Christoffel symbol of the first kind for σxzx is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The kz resolved Γ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 7 (a) in- dicate that both xxz and zxx components change sign near the node kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For the circular shift zyx component, the Christoffel symbols are zero along kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Equation 9 shows that the contribution to the zyx circular shift conductivity is the contorsion tensor, not the Christoffel symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 8 shows the contorsion tensor Re [Kxzy] in the momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Be- cause Re [Kxzy] = −Re [Kyzx], only the xzy component is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For other components of conductivities, the numeri- cal values of the contorsion tensor is negligible compared to the Christoffel symbols;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' thus their contorsion tensors are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' If we further simplify the Hamiltonian to the linear order of k, the Berry curvature for each band of the TPF has a simple form ∓ sin θ, 0 for the valence, conduction and flat band, re- spectively, thereby giving rise to the Chern number of ∓2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Nevertheless, the shift current vanishes after integration, since the integrands are either 0 or odd functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Multifold fermions in the CoSi family In this subsection, we present the numerical results of the model Hamiltonians for multifold fermions in the CoSi fam- ily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Since these model Hamiltonians have the time-reversal symmetry, only the linear shift current and circular injection FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Momentum resolved Christoffel symbol of the first kind (Γxxz and Γzxx) for the TPF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (a) Along kz, (b) on the kz = 0 plane for the zzz component, (c) on the kz = 0 plane for the xzx component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The chemical potential is set slightly below the node, µ = WT P F − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Momentum resolved contorsion tensor Re [Kxzy] for the TPF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' This tensor contributes to the circular shift conductivity zyx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (a) Along kz, (b) on the kz = 0 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (a) Circular injection and (b) linear shift conductivity for HΓ198 without spin-orbit coupling with chemical potential set slightly below the DTPF node (µ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='075 eV = WDT P F −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='005 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In (b), the curve for chemical potential at the DTPF node is also displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 25 >10 XXZ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 (b) (a) ZXX ky 0 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 (ge)1 (b) <-10 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 >10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 (c) (t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 ky 0 0 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 <-10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 kz kx2 (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 0 V 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 kx 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='5 kz5 (a) 4 Trβ/(iβo) 3 2 WDTPF - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='005 0 20 (b) Re[oxyz]() 10 WDTPF - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='005 0 WDTPF 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='015 hw(eV)9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Momentum resolved symplectic Christoffel symbol for the DTPF node at µ = WDT P F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (a) Along kz, (b) on the kz = 0 plane for the zxy component and (c) on the kz = 0 plane for the yxz component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' At the Γ point, both zxy and yxz components di- verge negatively, and thus, after the integration of kx, ky, the value is negative, as shown by the peak at kz = 0 in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' current would occur [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Double triple point fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' HΓ198 without SOC is a de- generate TPF, dubbed as double TPF (DTPF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' An symmetry analysis indicates that the xyz element is the only nonvan- ishing independent component for both circular injection and linear shift current, which is plotted as a function of photon energy (ℏω) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 9(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 9(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Figure 9(a) shows that for chemical potential being set slightly be- low the DTPF node (µ = WDT P F − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='005 eV), the circular injection current is nearly zero when ℏω is smaller than the energy difference between µ and WDT P F (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='005 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Nev- ertheless, it increases sharply when ℏω approaches to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='005 eV and quickly becomes saturated as ℏω further increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' As a result of the topological charge carried by the degenerate point, the circular injection response would show quantiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Figure 9(a) indicates that the circular injection conduc- tivity is quantized at 4 when ℏω > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='005 eV because of the double degeneracy of the Weyl point with chiral charge 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 9 (b), the linear shift conductivity is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In- terestingly, the shift conductivity for the chemical potential at the node and below the node are opposite in sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The ma- jor contribution comes from the quantum geometry of the flat band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' At low photon frequency, the flat band changes from being unoccupied at µ = WDT P F − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='005 to occupied at µ = WDT P F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' This change is approximately equivalent to taking the complex conjugate of the Hermitian connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Since the linear shift conductivity is given by the imaginary part of the Hermitian connection, the shift of chemical po- tential leads to the sign change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' This is similar to the sign change of the Berry curvature when chemical potential shifts across the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In both cases, the magnitude of the shift con- ductivity increases monotonically as ℏω increases from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Similar to the circular injection current, the shift conductivity becomes saturated when ℏω is well above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='005 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Neverthe- less, in contrast to the circular injection current, the saturation of the shift conductivity apparently does not result from its quantization behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The saturation can be understood from power counting analysis, which shows that the lowest order of the shift conductivity is proportional to a0v′/v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Since the linear shift conductivity comes from the divergent behavior of the symplectic Christoffel symbols near the topo- logical nodes, we show in Figure 10 the symplectic Christof- fel symbols for the DTPF node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Figure 10 indicates that the yxz component is more than one order of magnitude stronger than the zxy component, the linear shift conductivity reveals mainly the yxz component of the symplectic Christoffel sym- bols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' RSW fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' When the spin-orbit coupling is included in HΓ198, the DTPF nodal point [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 2(a)] splits into the RSW and Kramers nodes [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 2(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The calculated pho- toconductivity spectra for the RSW node are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Figure 11(a) shows that the circular injection conductivity for the RSW fermions increases when ℏω approaches to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='002 eV and becomes nearly saturated at ∼3 β0 between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='0025 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='005 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' As ℏω further increases, it first dips slightly and then increases rapidly to the saturated value of 4 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 11(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' This interesting behavior of the circular injection con- ductivity for the RSW node can be understood by the band dispersion of the RSW Hamiltonian displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 2(b) where the RSW bands of RSW are labeled with blue num- bers 1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' When only the transition from the lowest band is active, the circular injection conductivity reveals the Chern number of the lowest band, which is 3, and this explains the first plateau of ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' At higher photon frequencies, the transi- tion between the second and the third band also occurs, giving rise to a quantization of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The saturated value of the circu- lar injection conductivity thus reveals the sum of the Chern numbers of the lowest two bands, which is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The linear shift conductivity for the RSW node is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 11 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Interestingly, the conductivity in the low light frequency region below ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='007 eV changes sign when chemical potential is slightly lowered from the RSW node to WRSW − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='003 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Specifically, when µ = WRSW (red curve), the linear shift conductivity is negatively proportional to ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' When µ = WRSW −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='003 eV (blue curve) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' slightly below the RSW node), the conductivity shows a pronounced positive peak at the low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In this low frequency region, it can be seen from the band structure [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 2(b)] that the optically active bands are the second and third (first and second) for µ = WRSW (WRSW − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='003) eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, the lin- ear shift conductivity reveals that the symplectic Chirstoffel symbols are opposite in sign between different pairs of bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Moreover, for the xyz componenet of the linear shift conduc- tivity, the related components of the Christoffel symbols are zxy and yxz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' We calculate both components of the symplec- tic Christoffel symbols for the RSW fermions, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' ˜Γyxz shows a very strong peak near kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' As for the DTPF node (Figure 10), in contrast, ˜Γzxy is much weaker and no resonance is found at kz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, the major con- tribution to the linear shift conductivity is the yxz component of the symplectic Christoffel symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The distribution of ˜Γ on the kz = 0 plane is also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 12 (b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' There is a >300 zxy × 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 (b) 200 yxz Ky 0 0 (a) () 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 <-300 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 e 100 >300 ((b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 (c) ky 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0 <-300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 Kz kx10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (a) Circular injection and (b) linear shift conductivity for HΓ198 with spin-orbit coupling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=', the RSW and Kramers nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' drastic change near the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' After turning on the spin-orbit coupling in HΓ198, the band structure changes drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The flat band in DTPF no longer exists in RSW node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The existence and absence of the flat band would alter the quantum geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The difference can be observed in comparing the symplectic Christoffel symbols [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 10 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' As a consequence, the linear shift conductivity would have different behavivors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Comparing the linear shift conductivity [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' [9(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 11(b)], the de- pendence on the photon frequency changes to be linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The difference is likely to be the result of the large Christoffel sym- bols of the flat band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Kramers Weyl fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The calculated photoconductivity spectra for the Kramers Weyl fermions are also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In this case, µ = WK = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='131 eV, and the circular in- jection current probes the Chern number of the Kramer Weyl node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, the circular injection conductivity is quantized at 1 for ℏω > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='005 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Interestingly, the linear shift conductivity for the Kramers Weyl node is proportional to ω, thus exhibiting the same trend as the type-I Weyl points [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 13, the symplectic Christoffel symbols for the Kramer Weyl node are displayed, which is the source of the linear shift current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Figure 13 thus indicates that the linear shift conductivity σx,yz is dominated by the yxz component of the symplectic Christoffel symbol, similar to that of the DTPF and RSW nodes shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' DISCUSSION AND CONCLUSION The second-order photoconductivities and geometrical properties of chiral multifold fermions are studied in this pa- per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The analytical expressions for the injection and shift con- ductivities in terms of geometrical objects are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' As a result of the chiral symmetry breaking, the topological node and antinode are separated in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Thus, we study the second-order optical response of a single node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Our dimen- sion analysis reveals that the lowest order of second-order photoconductivity is ∝ ω0 and the second to the lowest or- der is ∝ ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The quantities are calculated for the minimal symmorphic TPF model and the effective Hamiltonian for the CoSi family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Whether the ω0 term survives depends on the de- tails of the Berry connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' For the TPF, RSW and Kramer Weyl nodes, the circular injection conductivity shows quanti- zations, as a result of the Chern number carried by the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The linear shift conductivity for the RSW and Kramer Weyl node is ∝ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' This behavior is similar to the type-I Weyl node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' In contrast, the linear shift conductivity for the TPF node is independent of ω, but proportional to pseudo spin-orbit cou- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' This relation has not been found in other Weyl semimet- als, to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Furthermore, by analyzing the momentum-resolved geometrical objects, it is found that the quantum metric and Christoffel symbols are strongest near the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The shift conductivities are related to contorsion tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The numerical results show that the contorsion ten- sors in general are at least one order of magnitude smaller than Christoffel symbols and symplectic Christoffel symbols for both model Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' However, the contorsion ten- sors could be dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' It is found that the circular shift conductivity σzyx for the symmorphic TPF model is solely FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Momentum resolved symplectic Christoffel symbol for the RSW node, µ = WRSW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (a) Along kz, (b) on the kz = 0 plane for zxy component, (c) on the kz = 0 plane for the yxz component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 5 (a) 4 Trβ/(iβo) WRsW-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='003 Wk-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='002 0 20 (b) WRSW WRSW -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='003 10 Wk 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='015 hw(eV)>300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 (b) zxy × 10 100 a yxz ky 0 0 0 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 () <-300 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 a 100 >300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 (c) 200 ky 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 <-300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='2 0 Kz kx11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Momentum resolved symplectic Christoffel symbol for Kramer Weyl of HΓ198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' µ = WK − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' (a) Along kz, (b) on the kz = 0 plane for the zxy component, (c) on the kz = 0 plane for the yxzthe component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' contributed by contorsion tensors, whereas the corresponding Christoffel symbols are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' The study of these geometrical objects sheds light on the optical probe of the Hilbert space of lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' ACKNOWLEDGMENTS H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' acknowledge the support from the National Science and Technology Counsil (NSTC) and the National Center for Theoretical Sciences (NCTS) in Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' was supported by the Center for Advancement of Topological Semimetals, an Energy Frontier Research Center funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Department of Energy Office of Science, Office of Basic Energy Sciences, through the Ames Labora- tory under contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' DE-AC02-07CH11358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' Appendix A: Second order photoconductivities in terms of quantum geometrical quantities The second-order conductivity tensors are expressed in term of geometrical quantities in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9A0T4oBgHgl3EQfAf_H/content/2301.01964v1.pdf'} +page_content=' 8 and 9 of which the contorsion tensor Kbca nm is defined as Kbca nm = i 2 � rb nm � p̸=m,n (rc mpra pn − ra mprc pn) + ra nm � p̸=m,n (rb mprc pn − rc mprb pn) − 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degree variation +Ahmed IBNOULOUAFI1,a, Mohamed EL HAZITI2, Hocine +CHERIFI3 +1 LRIT Laboratory, Associated Unit to CNRST (URAC29) IT Rabat Center - +Faculty of Sciences In Rabat, MOHAMMED V UNIVERSITY IN RABAT, B.P.1014 +RP, Rabat, Morocco +2 Higher School of technology (E.S.T) in SALE +3 Laboratoire Electronique, Informatique et Image (Le2i) UMR 6306 CNRS, +University of Burgundy, Dijon, France. +a Corresponding author: +E-mail: ahmedibnoulouafi@gmail.com +Abstract. +Identifying influential nodes in a network is a major issue due to the +great deal of applications concerned, such as disease spreading and rumor dynamics. +That is why, a plethora of centrality measures has emerged over the years in order to +rank nodes according to their topological importance in the network. Local metrics +such as degree centrality make use of a very limited information and are easy to +compute. Global metrics such as betweenness centrality exploit the information of +the whole network structure at the cost of a very high computational complexity. +Recent works have shown that combining multiple metrics is a promising strategy to +quantify the node’s influential ability. +Our work is in this line. +In this paper, we +introduce a multi-attributes centrality measure called M-Centrality that combines the +information on the position of the node in the network with the local information +on its nearest neighborhood. The position is measured by the K-shell decomposition, +and the degree variation in the neighborhood of the node quantifies the influence of +the local context. In order to examine the performances of the proposed measure, we +conduct experiments on small and large scale real-world networks from the perspectives +of transmission dynamics and network connectivity. +According to the empirical +results, the M-Centrality outperforms its alternatives in identifying both influential +spreaders and nodes essential to maintain the network connectivity. In addition, its +low computational complexity makes it easily applied to large scale networks. +Keywords: Complex networks, Centrality measures, Influential nodes. +arXiv:2301.01256v1 [cs.SI] 3 Jan 2023 + +M-Centrality +2 +1. Introduction +The problem of identifying nodes that are ”central” or ”influential” has attracted wide +attention from researchers due to its many applications. In epidemic spreading, the +transmission of the disease depends on the contacts that the infected person has with +the susceptible population. Thus, being able to locate and vaccinate the most influential +individuals can prevent from a potential outbreak of the disease [1]. In viral marketing, +being able to locate influential individuals can help to optimize the sales of products [2]. +A straightforward approach towards detecting these central nodes is to quantify +their influence using centrality measures. The idea of centrality was initially introduced +in the context of sociology to look whether there is a relation between the location +of an individual in the network and its influence in group processes. +Since then, +various centrality measures have emerged over the years. +They are employed in a +multitude of contexts to rank nodes according to their topological importance. +As +there is no consensual definition of the centrality of a node within a network, the issue +is considered from the multiple concepts reflecting the notion of influence. Therefore, we +can classify centralities from different perspectives. They can be classified according to +the underlying approach (geometric, spectral or path-based) [3], the way they quantify +influence (locally, globally, position within the network, dynamical processes such as +random walks) [4, 5], the computing ideas (iterative refinement) [6], the number of +attributes they take into consideration [6] or by taking into account the collective +influence of the whole set of nodes, i.e., in this case, the problem of finding influential +spreaders is targeted as an influence maximization problem [7, 8, 9], where we look for +a minimum set of nodes that maximize the spread of information to the whole network. +In geometric measures, influence is related to distances. In other words, it depends +only on how many nodes exist at every distance. Degree [10], closeness [11] and K- +shell [12] (and its variations, including MDD [13] and INK [14]) are geometric measures +that evaluate influence from respectively a local, global and position point of view. +In [15], authors proposed a local ranking measure, ClusterRank, that quantifies the +influence of a node by taking into account not only its direct influence (measured by +the number of its followers) and influences of its neighbors, but also its clustering +coefficient. +ClusterRank can be applied to directed as well as undirected networks +where its superiority in term of locating influential spreaders is significant compared +with degree centrality, K-shell decomposition, PageRank and LeaderRank. In addition, +ClusterRank, only making use of local information, is much more efficient considering +computational complexity. Spectral measures work on the spectral properties of the +graph (eigenvalues and eigenvectors of the adjacency matrix). +Iterative refinement +methods identify influential nodes based on the influence of its neighbors. +This is +known as the mutual enhancement effect. PageRank [16], HITS [17], LeaderRank [4] +and Personalized PageRank [18] (where the probability of jumping to a node when +abandoning the random walk is not uniform, but it is given by a preference vector +that favorites nodes over others) are good examples of these methods. +Path-based + +M-Centrality +3 +measures exploit the existence of shortest paths passing through a node. Betweenness +[19] is the best known centrality of this type. +It evaluates influence from a global +perspective by considering influential nodes those through which transits the largest +flow of information. As an alternative to betweenness centrality, a centrality measure +called DIL, was proposed in [20]. Instead of working on a global level, DIL centrality +ranks nodes based on local information (degree value and the importance of lines) to +identify network bridges. This measure showed great performances and is adapted to +large scale network due to its low complexity. Note that global and spectral measures +are inapplicable to very large graphs due to their computational complexity, while local +ones are simple but generally less effective because they only take in consideration the +neighborhood of nodes. Another way of approaching the problem of locating influential +nodes is to optimize an objective function of influence. In this context, Morone and +Makse proposed a method called Collective Influence expressed as the product of the +reduced degree of a node and the total reduced degree of all nodes at a distance d from +the node [8]. According to their results, optimal results can be reached at a distance d=3. +The Collective Influence formula constitutes the main core of their percolation algorithm +used to find the minimal set of nodes which are crucial for the global connectivity of +the network (their removal cause the destruction of the giant component). The major +drawback of this method is its computational complexity that stands at O(nlog(n)) +[8, 21]. +Recent works have raised the fact that centrality cannot be apprehended from +a single point of view and that combining measures can enhance the performances +of ranking methods [22]. +Comprehensive evidence centrality (CEC) [23] is a multi- +attributes method that uses the Dempster-Shafer evidence theory to combine degree, +betweenness and closeness centralities in order to characterize the influence of a node. +Another multi-criteria method proposed in [24] uses the Technique for Order Preference +by Similarity to Ideal Solution (TOPSIS) to evaluate the influence of each node from +more than one perspective based on four indicators which are the degree, betweenness, +closeness and improved K-shell (IKs). Dynamic sensitive centrality (DS) [25] is also a +multi-attributes method that takes in consideration topological features (a dynamical +process represented by the calculation of the total number of walks of length t from node +i to all nodes in the network) and dynamical properties that depend on the spreading +rate β over the time. DS centrality can locate influential nodes accurately and performs +very well in the early stages of spreading compared to degree, K-shell and eigenvector +centralities. In [26], Ma et al. proposed a centrality (labeled Gravity) based on the Isaac +Newton classical gravity formula. Gravity centrality considers the K-core value of a node +as its mass, and the shortest path distance between two nodes in a network is viewed +as their distance. To reduce the complexity of their method, only neighbors located in +a distance less than or equal to 3 are considered. This method shows that combining +a position based measure (Coreness) and geodesic distance, gives better results in term +of locating influential spreaders. +The main lesson learned from these studies is that combining multiple attributes is + +M-Centrality +4 +much more accurate than using a single one in order to evaluate the influence capability +of a node. +However, this raises new questions such as which centralities should be +combined and how to combine them. +Previous studies show that position [12] and +neighborhood [27, 28] are key factors to quantify the node influence. Based on this +evidence, we propose a new centrality measure combining these complementary aspects. +The measure is called M-Centrality in homage to the M-Theory that unifies all consistent +versions of the super-string theory. It is a weighted combination of : +• A ”global” measure that characterizes the location importance of the node in +the all network (core or periphery). +We use the K-shell decomposition for its +low computational complexity. Note, however, that any of its variations can be +substituted. +• A local measure that characterizes its neighborhood. This measure we choose to +call ∆D calculates the degree variation in the neighborhood of the node. +It is +inspired from the preferential attachment process [29]. +Since it cannot be assumed that attributes have equal weights, the Shannon entropy +method is used in order to find the appropriate weights for the global and local measure. +It is one of the most famous approach for determining the objective attribute weights +in multiple-criteria decision problems. +To evaluate the proposed centrality measure, we report a series of experiments +on real-world networks. Extensive comparisons with the most influential alternative +measures are performed. Results clearly show that globally, M-Centrality provides more +accurate ranking list. In addition to its effectiveness, one main advantage of this measure +as compared to alternative global ranking methods is that it has a low computational +complexity, i.e. O(n) (as shown in Section 3.4), that allows it to be used with large +scale networks. +The remainder of this article is organized as follows. In section 2, we review the +necessary background on evaluation metrics used to asses different centralities and the +preferential attachment concept that inspired our approach. Section 3 introduces the +M-Centrality measure, and it shows how it is related to the preferential attachment +process. In section 4, the datasets, the experimental setup and experimental results +are presented. The proposed measure is analyzed and compared to the most recent +methods, including Gravity (Gr), DIL, Personalized PageRank (PPR), ClusterRank +(CR) and Collective influence (COI). Finally, section 5 concludes the paper. +2. Background +In this section, we recall the definition of the evaluation measures that are used to +compare centralities (Monotonicity, Kendall’s Tau correlation coefficient (τ) and decline +rate of network efficiency). +Additionally, the model used to simulate an epidemic +spreading process in order to evaluate the performances of the different methods in + +M-Centrality +5 +the context of transmission dynamics is presented. Finally, we present the preferential +attachment process. +2.1. Evaluation of nodes ranking methods +The main role of centrality measures is to provide a means to rank nodes relative to +each other. +Indeed, the numerical values may not be directly interpretable. +Nodes +are usually ranked in descending order of influence. The node with largest influence is +ranked first and the one with smallest influence value is ranked last. There are many +ways to evaluate the performances of centrality measures, more or less linked to the +underlying applications. In this paper, we adopt generic measures (Monotonicity and +Kendall’s Tau correlation coefficient) that are commonly employed in the literature to +quantify influence. Monotonicity measures the ability of a ranking method to assign a +different rank to each node, while Kendall tau correlation allows a statistical comparison +of the agreement between two rankings. In addition, we also evaluate the performances +of centrality measures from two perspectives, one is based on transmission dynamics +where we look for the spreading capabilities of nodes, and the other is based on the +network connectivity and the theory that the network damage caused by deleting a +node is equivalent to its importance. +Monotonicity: +To quantify the resolution of a ranking method, the monotonicity M [30] of a ranking +vector R is defined as follows: +M(R) = +� +1 − +� +r∈R nr(nr − 1) +n(n − 1) +�2 +, M ∈ [0, 1] +(1) +where n is the size of ranking vector R and nr is the number of ties with the same +rank r . This metric quantifies the fraction of ties in the ranking list. The monotonicity +M(R) is equal to one if the ranking vector R is perfectly monotonic, and it is equal to +zero if all nodes in R have the same rank. Note that monotonicity allows to quantify +the discrimination ability of a centrality measure, however, a ranking with no tie is not +necessarily accurate. +Kendall tau correlation: +The Kendall’s Tau (τ) correlation coefficient [31] is generally used to compare the +performance of different topology-based measures. It measures the ranking consistency +of two lists that rank the same set of objects. Consider X = (x1, x2, ..., xn) and Y= +(y1, y2, ..., yn) are two ranked lists that contain n elements, respectively. Any pair of +ranks (xi, yi) and (xj, yj) is said concordant if xi > xj and yi > yj , or if xi < xj and +yi < yj . If xi > xj and yi < yj, or if xi < xj and yi > yj, then the pair is said to be +discordant. In the case of xi = xj or yi = yj (tied pair), the pair is neither concordant +nor discordant, in this case X and Y are independent. The Kendall’s Tau (τ) correlation + +M-Centrality +6 +coefficient is defined by: +τX,Y = +nc − nd +� +(n0 − n1)(n0 − n2) +, τ ∈ [−1, 1] +(2) +where n0 = n(n − 1)/2, n1 = � +i ti(ti − 1)/2, n2 = � +j tj(tj − 1)/2, nc and nd are +respectively the number of concordant pairs and discordant pairs, ti and tj are the +number of tied values in the ith and jth group of ties respectively. We can consider the +following ranges in order to qualify the strength of the relation between two rankings: +• No correlation: τ = 0, +• Low correlation: τ ∈ ]0, 0.50[, +• Moderate correlation: τ ∈ [0.50, 0.70[, +• High correlation: τ ∈ [0.70, 0.90[, +• Very high correlation: τ ∈ [0.90, 1[, +• Perfect correlation: τ = 1. +SIR model: +In literature, the susceptible-infected (SI) and susceptible–infected–removed (SIR) +models [32, 33] are generally used to simulate an epidemic spreading process in real +networks. Compared to the SI model, the SIR model is widely used for information +dissemination and disease diffusion and in various fields. In this paper, we employ the +SIR model to estimate the spreading capabilities of the nodes. In this model, a node +has three possible states: S (susceptible), I (infected) and R (recovered). Initially we +start from a single infected node and the other nodes are susceptible. At each step, +the infected node can infect its susceptible neighbors with infection probability β, and +then it recovered from the diseases with probability γ (set to 1 in this paper). The +spreading process stops when there is no infected node in the network. At last, the +number of recovered nodes represents the influence of the node. In this article, for the +SIR epidemic model we consider: +• For each node we simulate 102 diffusion process. The spreading capability of the +node will be the average number of infected nodes. +• We chose values of infection rate β that vary from 20% to 160% of βth (the epidemic +threshold [34]) representing both cases where β < βth and β ≥ βth. +Network efficiency: +Network efficiency [35] reflects the network connectivity. +The better the network +efficiency is, the better the network connectivity is. +Network efficiency η is defined +thus: +η = +1 +n(n − 1) +� +i̸=j∈V +ηij, +(3) + +M-Centrality +7 +where ηij is the efficiency between i and j, ηij = +1 +dij , dij is the shortest way between +i and j, n is the number of network nodes. The decline rate of network efficiency ν is +defined as: +ν = 1 − η +η0 +, +(4) +where η is the efficiency of the network after removing nodes. +η0 is the initial +efficiency of the network. +The bigger the ν is, the worse the network connectivity +destroyed by removing nodes is and the more important the node removed is. +2.2. Preferential attachment +Growth and preferential attachment are the two mechanisms used in the most prominent +approach to reproduce complex networks formation. Growth is a network construction +process where at each time step, a new node with m links is added to the existing +network [29]. The preferential attachment rule specify that new nodes select old nodes +with which they will form links based on their degree. That is, the probability Πn→i +that a new node n makes a connection to an existing node i with degree ki is given by : +Πn→i(ki) = +ki +� +j∈all kj +, +(5) +where all is the set of nodes to which the new node n could connect. +Due to this preference, a ”Rich get Richer” process takes place where the nodes +with higher degrees will further increase their connexions leading to the emergence of +hubs. An important feature of this model introduced by Barab´asi and Albert [36] is that +the generated networks display a power-law degree distribution. The growth mechanism +combined with preferential attachment is therefore the most influential explanation for +the prevalence of the scale-free networks that are ubiquitous in nature and man-made +systems. +3. Proposed method +3.1. Motivation +One of the main application of centrality measures is the assessment of node’s spreading +capability in the context of epidemic spreading. Various approaches based on centralities +have been proposed so far, one of them is based on node degrees [37]. However, as shown +in [30, 38], it is not sufficient since it is just the number of neighboring nodes that is +considered. An improvement of the approach consists in taking into account the degrees +of the neighbors of the node; a node with neighbors that have high degrees has greater +spreading capability. It is worth pointing out, however, that the high degree of a node, +or its neighbors, is not sufficient for specification of its spreading capability [39]. On + +M-Centrality +8 +the basis of this observation, and driven by the fact that quantifying node influence +from multiple points of view, we decided to introduce a new multi-attributes centrality, +labeled M-Centrality, that takes in consideration global and local features of a node. +Of course there is as much combination possible as the number of centralities, but the +main reason behind the choice of K-shell and degree in our combination is their great +performances in detecting influential spreaders. Indeed the K-shell shows the best results +in the case of single origin spreading [12] while the degree operates the best in the case +of multiple origins spreading [40, 41]. +3.2. M-Centrality measure +The main idea of this approach is that both position and neighborhood attributes play +important roles in shaping node influence. Therefore, by combining these two attributes, +we can enhance the performance of the centrality evaluation process. +M-Centrality +expresses how influential is a node based on the combination of the local information +contained in its neighborhood and a more global information about its position in the +network. More precisely, the M-Centrality of the node i that we note Mi is the weighted +sum of a global measure Ksi and a local measure ∆Di: +Mi = µKsi + (1 − µ)∆Di, 0 ≤ µ ≤ 1 +(6) +• Ksi is the coreness index. It is computed using K-core decomposition. This global +measure characterizes the position (core or periphery) of the node i in the network. +Note that any of its variation can be substituted. +• ∆Di is a new measure that we introduce to quantify the degree variation at a +local level. It takes inspiration from the preferential attachment process. We first +calculate the degree variation (di,j) between node i and each of its neighbors as +follows: +di,j = |Ni|. +��� kj − ki +� +j∈Ni kj +���, j ∈ Ni +(7) +where ki and kj represent the degree of nodes i and j respectively, Ni is the +neighborhood of the node i and |Ni| the number of i neighbors. We use the absolute +value to eliminate negative values in case of kj > ki. The degree variation in the +neighborhood of node i is given by: +∆Di = +� +j +di,j, +(8) +• µ is a tailored weighting factor that is estimated from the data. +Unlike most +traditional multi-attribute ranking methods that consider all attributes as equally +important (equal weights), we propose to compute the weight µ by targeting the +problem in a multi attribute decision-making framework. +Among the various +solutions, we choose an entropy technique that is known for its great performances + +M-Centrality +9 +in attributes weights determination [42]. The weight computation process proceeds +as follows: +First, we normalize the global and local measure attributes of the M-Centrality: +r1j = +Ksj +�n +j=1 Ksj and r2j = +∆Dj +�n +j=1 ∆Dj . +where n is the size of the network +Second, we build the matrix R2,n defined by: +R2,n = +� +r11 +r12 +r13 +. . . +r1n +r21 +r22 +r23 +. . . +r2n +� +(9) +Third, we compute the entropy Ei of the ith attribute: +Ei = − +1 +ln(n) +n +� +j=1 +rij ln(rij), i = 1, 2 and 1 ≤ j ≤ n +(10) +Finally, the weight of the two attributes is computed: +wi = +1 − Ei +2 − �2 +i=1 Ei +, i = 1, 2 +(11) +According to the properties of entropy, 0 ≤ wi ≤ 1 and �2 +i=1 wi = 1, thus µ = w1 +and 1 - µ = w2. +3.3. Relation with the preferential attachment +Since the centrality we propose takes inspiration from the preferential attachment phe- +nomenon, we will now start by developing Eq.(5) to establish that link. But first, let us +agree on the following notations: +• Ni: The set of node i neighbors. +• N c +i : The complement of the set Ni. +• V={v1, v2,..., vn}: The set of network nodes. +• all = V - {vi}: The set of all network nodes except the node i. +• all= Ni ∪ N c +i and Ni ∩ N c +i = ∅ +• Ni = all - N c +i +By developing Eq.(5): +( +� +j∈all +kj)Πn→i(ki) = ki +( +� +j∈all kj +� +j∈all kj − � +j∈Nc +i kj +)Πn→i(ki) = +ki +� +j∈all kj − � +j∈Nc +i kj +( +� +j∈all kj +� +j∈Ni kj +)Πn→i(ki) = +ki +� +j∈Ni kj + +M-Centrality +10 +We note +ki +� +j∈Ni kj by Πlocal +n→i(ki) to obtain: +Πlocal +n→i(ki) = ( +� +j∈all kj +� +j∈Ni kj +)Πn→i(ki). +(12) +Eq.(12) expresses the preferential attachment at a local level. This result allows us to +construct the local measure of M-Centrality. +By developing Eq.(7) we obtain: +di,j = |Ni|. +��� +kj +� +j∈Ni kj +− +ki +� +j∈Ni kj +��� +di,j = |Ni|. +��� +kj +� +j∈Ni kj +− ( +� +j∈all kj +� +j∈Ni kj +)Πn→i(ki) +��� +Replacing ( +� +j∈all kj +� +j∈Ni kj )Πn→i(ki) by Πlocal +n→i(ki) and +kj +� +j∈Ni kj by ¯kNi +j +gives: +di,j = |Ni|.|¯kNi +j +− Πlocal +n→i(ki)|. +(13) +Finally, replacing ∆Di by its value in Eq.(6) we obtain: +Mi = µKsi + (1 − µ) +� +j +|Ni|.|¯kNi +j +− Πlocal +n→i(ki)|. +(14) +This expression shows that the M-Centrality is linked to the preferential attachment +phenomenon. +3.4. Computational complexity +• The calculation of Ks index for all nodes has a complexity of O(n), where n is the +size of the graph. +• The calculation of ∆D for all nodes has a complexity of O(n). +• The calculation of the entropy matrix and the weight µ has a complexity of O(2n). +• The calculation of M-Centrality for all nodes has a complexity of O(n) + O(n) + +O(2n)= O(n). +4. Experimental results +To evaluate the efficiency of the proposed method, eight real-world networks‡ (small +[43, 44, 45, 46] and large scale [47, 48, 49, 50]) are studied. All datasets are considered +undirected and unweighted; also, the largest connected component was used in the +spreading process using the SIR model. +The statistical properties relative to these +‡ The networks used can be found via this link: https://icon.colorado.edu/ and http://vlado.fmf.uni- +lj.si/pub/networks/doc/erdos/ for Paul Erd˝os collaborations network + +M-Centrality +11 +networks are listed in Table 1. +Concerning the evaluation process, it comprises the +ability to detect key nodes§, monotonicity, rank correlation, impact of nodes removal +on network efficiency and finally the spreading capabilities of nodes. The results of +the proposed centrality are compared to Gravity (Gr), DIL, Personalized PageRank +(PPR)∥, ClusterRank (CR) and Collective influence (COI) centralities. +Table 1: The statistical properties of the networks under study, where < k > is the +average degree of the network, kmax the highest degree, Ksmax the highest coreness, σ +the size of the giant component and βth the epidemic threshold. +Network +Type +Number of nodes +Number of edges +< k > +kmax +Ksmax +βth +σ +Dolphins +Social +62 +159 +5.12 +12 +4 +0.139 +62 +Les Mis´erables +Co-appearance +77 +254 +6.59 +36 +9 +0.083 +77 +Game Of Thrones +107 +352 +6.57 +36 +7 +0.075 +107 +Paul Erd˝os collaborations +Collaboration +492 +1417 +5.76 +42 +9 +0.058 +446 +Netscience +1589 +2742 +3.45 +34 +19 +0.052 +379 +US Political blogs +Web-graph +1490 +16715 +22.43 +351 +36 +0.013 +1222 +E-mail +Communication +1133 +5451 +9.62 +71 +11 +0.048 +1133 +US airport 2010 +Traffic +1574 +17215 +21.87 +314 +64 +0.008 +1572 +4.1. Evaluation of nodes ranking +4.1.1. Small scale networks +Since there is no consensus on the concept of centrality, +real networks with known information about the node’s importance are commonly +used as benchmarks to assess the effectiveness of centrality measures. We present the +results of the experimental evaluation on three well documented networks (Dolphins, Les +Mis´erables and Game Of Thrones) and a network with no information about central +nodes (Paul Erd˝os collaborations). +We choose to concentrate on these small scale +networks in order to better understand the behavior of the centrality measures. +First, let’s study the influence of the weight variation µ on the M-Centrality measure +for the four networks. Tables.A.1 and A.2 illustrate the M-Centrality evolution of the +nodes ranking for various values of the weight. We can clearly distinguish three major +behaviors corresponding to the following cases µ = 0, 0 < µ <1, and µ = 1. At the +extremes, M-Centrality reduces to ∆D and Coreness while in between these two it +tends to adapt to both local and global topological properties of the network. +The +major shortcoming of this is the fact that µ can vary in a wide range without impacting +the nodes rankings. This is a very interesting result because it shows that a node needs +to be strategically positioned in the network along with a high degree variation in its +neighborhood in order to be considered influential. For more details about ranking, we +suggest to refer to Appendix A. +§ For all the Tables that follow, the nodes marked in color (blue, red, yellow, green, violet, orange +and olive) are the most central according to literature. For networks with unknown information about +central nodes, we mark in gray are the nodes that appear frequently in the ranking lists +∥ The parameters used are: (1) a damping factor α = 0.15 as experiments suggest that small changes +in α have little effect in practice[51, 52]. (2) for the preference vector, we choose to favor the Hubs. + +M-Centrality +12 +The next step is to determine the optimal weight value of the M-Centrality measure +using the entropy weighted technique. The values obtained for the four networks are +reported in Further analysis section, Table 11. For Dolphins (resp. Les Mis´erables and +Game of Thrones) the weight µ = 0.44 (resp. µ = 0.33 and µ = 0.36) puts more emphasis +on the local measure ∆D. The main reason for this is that Coreness assigns the same +rank to many nodes. Therefore, its entropy is smaller than the one associated to the local +measure ∆D, and so is its weight in the M-centrality. This result makes sense. Indeed, +∆D is more efficient at identifying key characters of the novel. However, combining +Coreness with degree variation allows to distinguish the characters that are grouped +in the same category by the Coreness centrality. For Paul Erd˝os collaborations, both +global and local information are given the same importance by the entropy weighted +technique. +After we study the behavior of M-Centrality based on µ values, we focus now on +its capability of detecting key nodes and then compare the results with the alternative +measures presented previously. For Dolphins, the information provided in [43] reports +that the female denoted SN100 plays an important role holding the community together. +Table 2 shows that DIL is the only centrality measure that puts SN100 at the top of +the list. It is understandable given the fact that its disappearance split the network +into two groups. ClusterRank, Personalized PageRank and Collective Influence are the +only ones that fail in identifying SN100 in the top 15. In fact the first one ranks it +at the 46th position while the second and third ones rank it at the 19th position. For +M-Centrality, it is in the top 15. However, this result suggests that the measures do +not exploit adequately the information about bridges that is well encoded in the DIL. +Further improvement for M-Centrality needs to be made in this direction. +For Les Mis´erables, The ex-convict, Jean Valjean, is a central character of the novel. +He spends a great deal of time running away from Inspector Javert. He is also closely tied +to his adopted daughter, Cosette, and her future husband, Marius. Fantine, Cosette’s +mother, Mr and Mme Thenardier and their son Gavroche are also important characters +of the novel. We report on Table 2 the top 15 nodes sorted by relevance. M-Centrality +is the only measure that succeeds in ranking Valjean as the most influential character of +the novel. Additionally, it identifies all the key characters of the novel in the top 15. For +Gravity, DIL and Personalized PageRank, they all rank Gavroche, Valjean and Marius +in the top 4. Thenardier and Javert are always in the top 15 but with very various +rankings. We may also notice that Collective Influence and ClusterRank give the worst +ranking as they clearly fail to detect main characters. The major shortcoming of these +rankings is that (except for M-Centrality) Cosette does not appear in the top 15 major +nodes and that the central hub (Valjean) is not necessarily the main bridge. +Moving on to the third network, Game Of Thrones, no one can deny the important +role played by Tyrion Lannister in the story. +Indeed, all the centralities (except +ClusterRank) identify this character as the most important one. We also can see that +M-Centrality performs the best as it identifies, along with Tyrion, a majority of other +key characters, including his brother and sister Jamie and Cersei Lannister, Jon Snow, + +M-Centrality +13 +Table 2: First 15 nodes sorted by relevance according to the centrality measures in the +networks Dolphins and Les Mis´erables. M (µ = 0.44) and M (µ = 0.33) refer to the +M-Centrality ranking corresponding to the weight obtained by the entropy weighted +technique. +Networks +Les Mis´erables +Dolphins +Rank +M (µ = 0.44) +Gr +DIL +CR +PPR +COI +M (µ = 0.33) +Gr +DIL +CR +PPR +COI +1 +Valjean +Gavroche +Gavroche +Bossuet +Gavroche +Myriel +Grin +Grin +SN100 +Hook +Grin +Beescratch +2 +Gavroche +Valjean +Enjolras +Combeferre +Valjean +Listolier +Trigger +SN4 +Grin +MN105 +SN4 +PL +3 +Myriel +Marius +Marius +Feuilly +Enjolras +Fameuil +Topless +Topless +Topless +Jonah +Topless +SN4 +4 +Marius +Enjolras +Valjean +Bahorel +Marius +Blacheville +Jet +Kringel +Web +MN83 +Scabs +Trigger +5 +Javert +Bossuet +Bossuet +Joly +Bossuet +Favourite +Web +SN9 +Gallatin +Gallatin +Trigger +TR77 +6 +Fantine +Courfeyrac +Courfeyrac +Enjolras +Courfeyrac +Dahlia +SN4 +Scabs +Scabs +Topless +Patchback +Upbang +7 +Thenardier +Thenardier +Bahorel +Courfeyrac +Bahorel +Zephine +Scabs +SN100 +SN4 +Scabs +TR99 +Oscar +8 +Enjolras +Javert +Joly +Prouvaire +Joly +Bahorel +Patchback +TR99 +Feather +Feather +SN9 +Scabs +9 +Cosette +Bahorel +Combeferre +Gueulmer +Combeferre +Joly +Kringel +Patchback +Trigger +SN90 +Hook +DN63 +10 +Bossuet +Joly +Feuilly +Babet +Feuilly +Fantine +SN63 +Beescratch +MN105 +DN21 +MN105 +SN96 +11 +Mme Thenardier +Combeferre +Grantaire +Grantaire +Mabeuf +Courfeyrac +Beescratch +Jonah +Jonah +Grin +Jonah +SN63 +12 +Montparnasse +Feuilly +Mabeuf +Montparnasse +Thenardier +Combeferre +Stripes +Trigger +Patchback +SN4 +SN63 +SN9 +13 +Gueulemer +Mabeuf +Thenardier +Claquesous +Javert +Feuilly +SN100 +Double +DN21 +Web +Kringel +Grin +14 +Babet +Grantaire +Prouvaire +Mabeuf +Grantaire +Mabeuf +Gallatin +Break +MN83 +Upbang +MN83 +Zap +15 +Courfeyrac +Gueulemer +Javert +MmeHucheloup +Prouvaire +Grantaire +SN9 +SN63 +Upbang +SN9 +Stripes +Topless +Catelyn stark and her three children Robb, Sansa and Arya and the mother of dragons +Daenerys Targaryen. Collective influence manages to identify some major characters, +with Daenerys ranked at the top of the list. However, we may notice the absence of +Tyrion in the top 15. Again, ClusterRank gives the poorest ranking results. Overall, +two important observations need to be made. First, the fact that only the proposed +method succeeds in ranking Daenerys in the top 15 (3rd position). Second, for Jon +Snow it is only M-Centrality that managed to give him importance by ranking him 2nd +most influential character. +Paul Erd˝os collaborations network is a collaboration graph of mathematicians where +two mathematicians are joined by an edge whenever they co-authored a paper together. +As the concept of ”central” nodes is relative, one may consider influence in term of the +number of co-authors (degree), others may consider nodes located in the core, or those +without the network will split in two or more sub-graphs, etc... This can intuitively seem +a good approach, but in reality it tends to be biased since nodes that are considered +central in term of connections (Hubs) in the topological context, can be less influential +when talking about spreading capabilities where nodes located in the core are more +important. +Being said, at this stage of the paper and especially for networks with unknown +information about central nodes, we make the assumption that all the rankings provided +by the centralities under study are correct. Later, these rankings will be exploited in +studying the impact of deleting keys nodes on network structure and efficiency and +thus we can conclude which measure(s) provide(s) the most suitable ranking. From +Table 3, the only observation we can make is that despite all centralities have different +conceptions of influence, some nodes (marked in gray) appear frequently in all ranking +results. There are also some concordances between the different centralities on ranking +certain nodes in the top 15. +For example M-Centrality and Gravity (resp. +DIL, +ClusterRank, Personnalized PageRank and Collective Influence) agree on 11 (resp. 7, +3, 8 and 1) out of 15 rankings. + +M-Centrality +14 +Table 3: First 15 nodes sorted by relevance according to the centrality measures in the +networks Game Of Thrones and Paul Erd˝os collaborations network. M (µ = 0.36) and +M (µ = 0.50) refer to the M-Centrality ranking corresponding to the weight obtained +by the entropy weighted technique. +Networks +Game Of thrones +Paul Erd˝os collaborations network +Rank +M (µ = 0.36) +Gr +DIL +CR +PPR +COI +M (µ = 0.50) +Gr +DIL +CR +PPR +COI +1 +Tyrion +Tyrion +Tyrion +Eddard +Tyrion +Daenerys +HARARY, FRANK +GRAHAM, RONALD L. +RODL, VOJTECH +SZEMEREDI, ENDRE +GRAHAM, RONALD L. +POMERANCE, CARL +2 +Jon +Sansa +Sansa +Meryn +Sansa +Mance +GRAHAM, RONALD L. +RODL, VOJTECH +GRAHAM, RONALD L. +TROTTER, WILLIAM T., JR. +RODL, VOJTECH +HAJNAL, ANDRAS +3 +Daenerys +Robb +Jaime +Ilyn +Jaime +Samwell +POMERANCE, CARL +ALON, NOGA +ALON, NOGA +LEHEL, JENO +ALON, NOGA +CHARTRAND, GARY +4 +Sansa +Jaime +Robb +Joffrey +Robb +Cersei +TUZA, ZSOLT +TUZA, ZSOLT +TUZA, ZSOLT +FRANKL, PETER +SPENCER, JOEL H. +HARARY, FRANK +5 +Robb +Tywin +Cersei +Balon +Cersei +Tyrion +RODL, VOJTECH +SPENCER, JOEL H. +SPENCER, JOEL H. +JACOBSON, MICHAEL S. +TUZA, ZSOLT +STRAUS, ERNST G. +6 +Tywin +Arya +Arya +Petyr +Tywin +Jon +SOS, VERA T. +HARARY, FRANK +FUREDI, ZOLTAN +RODL, VOJTECH +FUREDI, ZOLTAN +TUZA, ZSOLT +7 +Jaime +Cersei +Joffrey +Gregor +Joffrey +Jorah +ALON, NOGA +FUREDI, ZOLTAN +CHUNG, FAN RONG K. +GOULD, RONALD J. +CHUNG, FAN RONG K. +GYARFAS, ANDRAS +8 +Samwell +Robert +Tywin +Cersei +Arya +Sandor +SPENCER, JOEL H. +CHUNG, FAN RONG K. +GYARFAS, ANDRAS +FUREDI, ZOLTAN +SZEMEREDI, ENDRE +RODL, VOJTECH +9 +Catelyn +Joffrey +Robert +Aerys +Robert +Rhaegar +HAJNAL, ANDRAS +BOLLOBAS, BELA +SZEMEREDI, ENDRE +SAKS, MICHAEL E. +LOVASZ, LASZLO +SOS, VERA T. +10 +Cersei +Catelyn +Catelyn +Sandor +Catelyn +Joffrey +BOLLOBAS, BELA +FAUDREE, RALPH J. +FAUDREE, RALPH J. +ALON, NOGA +PACH, JANOS +BABAI, LASZLO +11 +Mance +Stannis +Sandor +Arya +Sandor +Beric +PACH, JANOS +SOS, VERA T. +PACH, JANOS +GRAHAM, RONALD L. +HARARY, FRANK +TURAN, PAL +12 +Arya +Eddard +Eddard +Jaime +Stannis +Davos +STRAUS, ERNST G. +PACH, JANOS +LOVASZ, LASZLO +KUBICKA, EWA MARIE +BABAI, LASZLO +SPENCER, JOEL H. +13 +Robert +Jon +Jon +Pycelle +Eddard +Jaime +KLEITMAN, DANIEL J. +SZEMEREDI, ENDRE +BABAI, LASZLO +KUBICKI, GRZEGORZ +GYARFAS, ANDRAS +SARKOZY, ANDRAS +14 +Joffrey +Sandor +Gregor +Sansa +Gregor +Tywin +CHARTRAND, GARY +LOVASZ, LASZLO +FRANKL, PETER +GYARFAS, ANDRAS +FAUDREE, RALPH J. +LOVASZ, LASZLO +15 +Bran +Gregor +Stannis +Stannis +Jon +Edmure +CHUNG, FAN RONG K. +HAJNAL, ANDRAS +JACOBSON, MICHAEL S. +SCHELP, RICHARD H. +FRANKL, PETER +RENYI, ALFRED A. +Now we turn to the comparative evaluation of the centrality measures. +Table +4 shows the monotonicity of the ranking methods. Results show that M-Centrality, +Gravity and Personalized PageRank clearly outperform the other measures. In other +words, few nodes are assigned the same rank. DIL is more competitive than ClusterRank +centrality except in the case of Game Of Thrones network. Collective Influence also gives +good performances in distinguishing between nodes importance. +Table 5 reports the rank correlation of the various centrality measures. The main +result is that M-Centrality is highly and positively correlated with Gravity, with τ values +ranging from 0.71 to 0.85. This is reasonable since they are both a variant of Coreness +centrality. Globally, the proposed method is moderately correlated with ClusterRank. +The lowest correlation value is registered between M-Centrality and Collective Influence +(τM,COI = 0.46) in the case of Les Mis´erables network. +In other words, Collective +Influence centrality behaves very differently than the proposed method for this network. +Note that it is evident if we refer to the top fifteen nodes. Further investigations about +the nature of the relation between the ranking produced by the M-Centrality and the +one of its alternatives are reported in Appendix B. +Table 4: Monotonicity M of centrality measures for the small scale real-world networks. +The weight of the M-Centrality is computed by the entropy weighted technique. +Network +M(M) +M(Gr) +M(DIL) +M(CR) +M(PPR) +M(COI) +Dolphins +0.989 +0.997 +0.958 +0.873 +0.997 +0.960 +Les Mis´erables +0.958 +0.958 +0.876 +0.854 +0.958 +.864 +Game of Thrones +0.993 +0.994 +0.937 +0.948 +0.995 +0.953 +Paul Erd˝os collaborations +0.982 +0.987 +0.922 +0.762 +0.987 +0.879 +4.1.2. Large scale networks +After we study the behavior of M-Centrality on small scale +networks, we move on to present the results of the experimental evaluation on four large +scale networks. It includes E-mail, Netscience, US airport and US Political blogs. As +the performances of the proposed measure have been established previously, it will be +interesting to test its effectiveness on much larger graphs that are not necessarily well + +M-Centrality +15 +Table 5: Kendall’s tau (τ) rank correlation coefficient for the small scale real-world +networks. +Network +τ(M, Gr) +τ(M, DIL) +τ(M, CR) +τ(M, PPR) +τ(M, COI) +Dolphins +0.716 +0.699 +0.589 +0.504 +0.517 +Les Mis´erables +0.824 +0.703 +0.632 +0.736 +0.467 +Game of Thrones +0.857 +0.820 +0.612 +0.748 +0.566 +Paul Erd˝os collaborations +0.856 +0.623 +0.648 +0.744 +0.857 +documented. Details concerning the impact of the weight variation on M-Centrality are +reported in Appendix A, Tables A.3 and A.4. +First, we determine the key nodes identified by the different centralities. Tables 6 +and 7 present our results. The main observation is that each centrality identifies various +key nodes. +However, some nodes appear more frequently in the top 15 of multiple +centralities, this can possibly suggest their potential importance. The first network, E- +mail, represents the exchange of emails among members of the Rovira i Virgili University +in Spain, in 2003. The nodes marked in gray appear in 4 out of 5 ranking results except +ClusterRank. However, an important thing to notice about these nodes is the fact that +none of them is strategically located in the network. In other words, they are not in the +core of the network. +Netscience is a network of co-authorships between scientists whose research centers +on the properties of networks. Edges join every pair of individuals whose names appear +together as authors of a paper. The nodes marked in gray are the ones who appear in +all ranking results except M-Centrality and Collective Influence. In addition, they all +belong to the core of the network. This result suggests that in this network, the proposed +measure and Collective Influence quantify node influence differently from its alternatives. +Note that these two methods agree both on ranking BARABASI, A, NEWMAN, M and +JEONG,H in the top 3. +Table 6: First 15 nodes sorted by relevance according to the centrality measures in the +networks E-mail and Netscience. +Networks +E-mail +Netscience +Rank +M (µ = 0.40) +Gr +DIL +CR +PPR +COI +M (µ = 0.50) +Gr +DIL +CR +PPR +COI +1 +105 +105 +105 +886 +105 +16 +BARABASI, A +UETZ, P +UETZ, P +GIOT, L +UETZ, P +NEWMAN, M +2 +23 +333 +16 +888 +16 +105 +NEWMAN, M +CAGNEY, G +CAGNEY, G +JUDSON, R +CAGNEY, G +BARABASI, A +3 +333 +42 +299 +887 +196 +24 +JEONG, H +MANSFIELD, T +MANSFIELD, T +KNIGHT, J +MANSFIELD, T +JEONG, H +4 +41 +23 +196 +788 +204 +564 +YOUNG, M +GIOT, L +GIOT, L +LOCKSHON, D +GIOT, L +SOLE, R +5 +16 +76 +3 +571 +42 +14 +BOCCALETTI, S +JUDSON, R +JUDSON, R +NARAYAN, V +JUDSON, R +HOLME, P +6 +42 +41 +42 +885 +49 +434 +OLTVAI, Z +KNIGHT, J +KNIGHT, J +SRINIVASAN, M +KNIGHT, J +MORENO, Y +7 +233 +233 +204 +299 +56 +196 +SOLE, R +LOCKSHON, D +LOCKSHON, D +POCHART, P +LOCKSHON, D +OLTVAI, Z +8 +24 +52 +205 +426 +116 +72 +ALON, U +NARAYAN, V +NARAYAN, V +QURESHIEMILI, A +NARAYAN, V +LATORA, V +9 +14 +3 +389 +3 +333 +204 +KURTHS, J +SRINIVASAN, M +SRINIVASAN, M +LI, Y +SRINIVASAN, M +PASTORSATORRAS, R +10 +196 +196 +552 +9 +3 +354 +DIAZGUILERA, A +POCHART, P +POCHART, P +GODWIN, B +POCHART, P +CALDARELLI, G +11 +21 +135 +434 +205 +23 +396 +LATORA, V +QURESHIEMILI, A +QURESHIEMILI, A +CONOVER, D +QURESHIEMILI, A +VAZQUEZ, A +12 +355 +332 +332 +389 +128 +116 +HU, G +LI, Y +LI, Y +KALBFLEISCH, T +LI, Y +VESPIGNANI, A +13 +578 +16 +726 +210 +21 +69 +MUTH, S +GODWIN, B +GODWIN, B +VIJAYADAMODAR, G +GODWIN, B +DIAZGUILERA, A +14 +135 +299 +56 +386 +206 +21 +VICSEK, T +CONOVER, D +CONOVER, D +YANG, M +CONOVER, D +STAUFFER, D +15 +76 +21 +49 +215 +41 +341 +KAHNG, B +KALBFLEISCH, T +KALBFLEISCH, T +JOHNSTON, M +KALBFLEISCH, T +BOCCALETTI, S +US airport is a complete network of flights among all commercial airports in the +United States, in 2010, derived from the U.S. Bureau of Transportation Statistics (BTS) +Transtats site. Weights represent the number of seats available on the flights between a +pair of airports. In this network, all centralities (except ClusterRank) agree on ranking + +M-Centrality +16 +the nodes marked in gray in the top 15. Again, they are all located in the core of the +network. +The last network, US Political blogs, is a network of hyperlinks between web blogs +on US politics, recorded in 2005 by Adamic and Glance. As for US airport and E-mail +networks, all centralities (except ClusterRank) agree on ranking the nodes marked in +gray in the top 15, with Gravity and DIL ranking them in the top 5. These nodes are +located in the core of the network except for instapundit.com that has a coreness value +of Ks = 32. +Table 7: First 15 nodes sorted by relevance according to the centrality measures in the +networks US airport and Political blogs. +Networks +US airport +US Political blogs +Rank +M (µ = 0.67) +Gr +DIL +CR +PPR +COI +M (µ = 0.50) +Gr +DIL +CR +PPR +COI +1 +766 +1200 +1200 +1201 +1200 +152 +blogsforbush.com +dailykos.com +dailykos.com +nielsenhayden.com/electrolite +dailykos.com +blogsforbush.com +2 +114 +114 +114 +1391 +114 +567 +dailykos.com +atrios.blogspot.com +atrios.blogspot.com +michaelberube.com +atrios.blogspot.com +gevkaffeegal.typepad.com/the alliance +3 +877 +435 +709 +683 +709 +977 +drudgereport.com +talkingpointsmemo.com +talkingpointsmemo.com +tbogg.blogspot.com +talkingpointsmemo.com +liberaloasis.com +4 +709 +1068 +1068 +759 +435 +844 +instapundit.com +instapundit.com +washingtonmonthly.com +roadtosurfdom.com +washingtonmonthly.com +corrente.blogspot.com +5 +1200 +709 +435 +1011 +1068 +402 +talkingpointsmemo.com +washingtonmonthly.com +instapundit.com +aintnobaddude.com +liberaloasis.com +pacificviews.org +6 +1016 +391 +391 +239 +711 +1660 +atrios.blogspot.com +digbysblog.blogspot.com +liberaloasis.com +busybusybusy.com +instapundit.com +bodyandsoul.typepad.com +7 +500 +1252 +711 +1820 +391 +47 +powerlineblog.com +juancole.com +digbysblog.blogspot.com +bodyandsoul.typepad.com +digbysblog.blogspot.com +busybusybusy.com +8 +389 +711 +1252 +169 +500 +1186 +michellemalkin.com +talkleft.com +bodyandsoul.typepad.com +atrios.blogspot.com/ +bodyandsoul.typepad.com +seetheforest.blogspot.com +9 +711 +389 +389 +75 +1252 +91 +truthlaidbear.com +liberaloasis.com +talkleft.com +nomoremister.blogspot.com +pandagon.net +wampum.wabanaki.net +10 +1068 +500 +500 +760 +389 +1677 +washingtonmonthly.com +madkane.com/notable.html +pandagon.net +wampum.wabanaki.net +talkleft.com +atrios.blogspot.com/ +11 +391 +982 +206 +1320 +877 +726 +littlegreenfootballs.com/weblog +powerlineblog.com +politicalstrategy.org +thetalkingdog.com +juancole.com +blogsagainsthillary.com +12 +215 +206 +982 +1628 +982 +482 +wizbangblog.com +pandagon.net +corrente.blogspot.com +nathannewman.org/log +politicalstrategy.org +coxandforkum.com +13 +435 +311 +877 +1376 +206 +1190 +hughhewitt.com +yglesias.typepad.com/matthew +tbogg.blogspot.com +elayneriggs.blogspot.com +corrente.blogspot.com +homespunbloggers.blogspot.com +14 +875 +1353 +311 +478 +215 +1742 +juancole.com +tbogg.blogspot.com +prospect.org/weblog +billmon.org +tbogg.blogspot.com +patriotboy.blogspot.com +15 +685 +877 +215 +997 +311 +836 +lashawnbarber.com +michellemalkin.com +dneiwert.blogspot.com +leanleft.com +dneiwert.blogspot.com +techievampire.net/wppol +Next we examine the monotonicity of the different methods. +The results are +reported in Table 8. Again M-Centrality and Gravity clearly offer the best performances +in all networks with a very high monotonicity score outperforming the one of the other +centralities. We also notice the poor results of Personalized PageRank and Collective +Influence in Netscience network, which can be interpreted as a sign of incapability to +distinguish between nodes influence. +Table 9 shows the correlation between M-Centrality and other centralities. One +can see that in the four networks, M-Centrality is very correlated with Gravity. We also +notice that the proposed measure exhibits poor relation with Personalized PageRank, +especially in Netscience network (τ(M, PPR) = 0.29). For the results concerning the +concordance in high ranks between the ranking list produced by M-Centrality and the +one of its alternatives, we suggest the reader to refer to Appendix B +Table 8: Monotonicity M of the centrality measures for the large scale networks. +Network +M(M) +M(Gr) +M(DIL) +M(CR) +M(PPR) +M(COI) +Netscience +0.910 +0.915 +0.803 +0.805 +0.016 +0.275 +US airport +0.997 +0.998 +0.897 +0.882 +0.998 +0.915 +E-mail +0.998 +0.999 +0.961 +0.870 +0.999 +0.964 +US Political blogs +0.936 +0.936 +0.918 +0.794 +0.937 +0.859 +4.2. Impact of nodes removal on network efficiency +A high monotonicity score alone does not mean necessarily the performance of a ranking +method, that is why in many studies about ranking influential nodes, the ranking list + +M-Centrality +17 +Table 9: +Correlation between M-Centrality and its alternatives for the large scale +networks. +Network +τ(M, Gr) +τ(M, DIL) +τ(M, CR) +τ(M, PPR) +τ(M, COI) +Netscience +0.883 +0.841 +0.806 +0.297 +0.500 +US airport +0.867 +0.811 +0.711 +0.617 +0.742 +E-mail +0.881 +0.527 +0.569 +0.766 +0.906 +US Political blogs +0.962 +0.750 +0.760 +0.885 +0.756 +of different centrality measures is evaluated in the context of network vulnerability and +transmission dynamics. In this work we explore the two ways. +First, we evaluate the efficiency of the proposed method by examining the impact +of removing the top most important nodes on network structure. Table 10 shows the +rest graph obtained after deleting the key nodes identified by each centrality. The main +observation is that M-Centrality outperforms the other centralities, with the exception +of the two collaborations networks where Collective Influence seems to be more effective. +Indeed the removal of the most important nodes identified by the proposed measure has +high impact on network structure. These results suggest that the ranking list of the +proposed measure is more consistent and accurate than the one of its alternatives. +Table 10: +Number of connected components C obtained after removing the most +important nodes according to ranking lists produced by the various centrality measures. +Network +Number of removed nodes +C(M) +C(Gr) +C(DIL) +C(CR) +C(PPR) +C(COI) +Dolphins +15 +12 +11 +10 +5 +7 +6 +Les Mis´erables +15 +26 +15 +14 +2 +14 +9 +Game Of Thrones +20 +29 +22 +19 +5 +22 +23 +Paul Erd˝os collaborations +30 +63 +58 +58 +48 +61 +70 +Netscience +50 +462 +411 +406 +395 +418 +490 +US airport 2010 +50 +286 +203 +203 +17 +203 +94 +E-mail +50 +42 +21 +17 +12 +22 +36 +US Political blogs +50 +362 +327 +292 +274 +296 +314 +After we study the structural damage caused by the removal of important nodes, we +move on to study the impact of deleting important nodes on network efficiency. Figure +1 shows the relationship between the decline rate of network efficiency and the number +of nodes removed from the network. Two main observations can be made. First, we can +see that the decline rate of network efficiency is rising with the increase of the number +of nodes removed. Second, the proposed measure clearly performs the best compared +to its alternatives, with ClusterRank giving the worst results in all the networks under +study. +For Dolphins network, the removal of node SN100 (identified by DIL) seems to +cause more damage to the network than nodes Grin and Hook. This comforts the fact +that SN100 is a key member in holding the group together. Gravity and DIL give good +performances in all stages of the removal process, while Personalized PageRank fails in +the last stages (after the removal of the 8th node). In Les Mis´erables, removing Valjean +(identified by M-Centrality) causes more damage than Gavroche. In the stages 2, 3 +and 4 of the removal process, we notice that Marius and Enjolras are more important + +M-Centrality +18 +than Myriel. The fifth stage of the removal process rises the fact that Javert is clearly +more central than Bossuet. +This suggests that the most suitable ranking would be +Valjean, Gavroche, Marius, Enjolras and Javert. For Game Of Thrones, the top 10 +nodes identified by M-Centrality are clearly the most important ones. Gravity, DIL +and Personalized PageRank give competitive results. Globally all centralities are very +competitive with a slight advantage of the proposed method due to its consistent ranking +as previously shown in Table 3. +For the remaining networks, the performance gap +between M-Centrality and its alternatives is consequent and more visible especially in +the cases of Political blogs, US airport and E-mail networks. In Netscience and Paul +Erd˝os, Collective Influence gives better results in the first network and is as competitive +as M-Centrality in the second one. This is concordant with the results presented in Table +10 and can be explained by the fact that in this type of networks, the importance of an +author is closely related to the importance of authors with whom he collaborates. This +topological property is captured perfectly by the definition of the Collective Influence +centrality. +Being said, given the computational complexity, monotonicity, structural damage +caused by the removal of key nodes and network efficiency scores, the proposed centrality +gives better results than its alternatives, which comforts the previous ranking results +presented in Section 4.1. This suggests a more consistent and accurate ranking. We +also notice that the highest vulnerability to the removal of important nodes identified by +M-Centrality is registered for Game Of Thrones (90%) and Les Mis´erable (87%) while +the lower one is for E-mail network (18%). This reflects that some networks are more +resilient to targeted attacks than others. +4.3. Evaluation of nodes spreading capability with SIR model +In addition to network vulnerability, we test the effectiveness of the methods by +evaluating their Kendall tau correlation with the SIR model for different values of β. +A greater correlation indicates the greater accuracy of the centrality in quantifying +the spreading capability of nodes. +Figure 2 shows the rank correlation results of +different ranking methods on the previously studied networks. Note that M-Centrality +is compared with the most recent and effective methods in detecting influential +spreaders. For the real-world networks investigated, the first observation to make is +that the correlation of different methods with the SIR epidemic model increases as +the transmission rate increases. Second, for small values of β, the correlation is low. +This is due to the fact that the epidemic is limited to the neighborhood of nodes and +thus couldn’t spread all over the network. Third, when β>βth, we can observe that +the variations in the transmission rate have in general little effect on the correlation +values. This can be explained by the fact that when the transmission rate is larger +than the epidemic threshold, the epidemic starting from any node can reach all the +network nodes and the role of topology in the spreading will be overridden. +Thus, +distinguishing between nodes spreading capabilities become much harder. This is the + +M-Centrality +19 +Figure 1: Decline rate of network efficiency for the eight real-world networks under +study. + +(a) Dolphins network +decline rate of network efficiency +5 +0 +■ +4 +0 +MC +Gr +Y +DIL +H +CR +PPR +2 +COI +0 +12 +8 +10 +14 +6 +top nodes removed(b) Les Misérables network +decline rate of network efficiency +8 +9 +MC +0 +Gr +DIL +4 +CR +0 +PPR +COI +2 +f+ +0 ++....+....+ +0 +2 +12 +14 +6 +8 +10 +4 +top nodes removed(c) Game Of Thrones network +0 +decline rate of network efficiency +8 +0 +IN +9 +MC +0 +Gr +DIL +CR +4 +PPR +O +COI +中 +X口 +2 +0 +0 +0 +5 +10 +15 +20 +top nodes removed(d) Paul Erdos collaborations network +decline rate of network efficiency +XX +0 +M +MC +A +2 +Gr +10000466 +DIL +0 +KX +CR +PPR +COI +0 +8 +0 +25 +5 +10 +15 +20 +30 +0 +top nodes removed(e) Netscience network +8 +0 +decline rate of network efficiency +6 +0 +口 +MC +Gr +11 +DIL +4 +t +0 +CR +PPR +COI +2 +XX +口 +0 +20 +0 +10 +30 +40 +50 +top nodes removed(f) Political blogs network +decline rate of network efficiency +0 +0 +乙 +8 +MC +0 +Gr +DIL +o +CR +PPR +0 +COI +二 +0 +8 +0 +20 +40 +0 +10 +30 +50 +top nodes removed(g) US airport network +5 +0 +decline rate of network efficiency +10 +4 +MC +3 +Gr +0 +O +DIL +CR +2 +PPR +0 +COI +0 +20 +10 +30 +40 +50 +top nodes removed(h) E-mail network +0 +decline rate of network efficiency +5 +0 +MC +Gr +O +0 +DIL +0 +CR +PPR +COI +5 +0 +8 +0 +20 +0 +10 +30 +40 +50 +top nodes removedM-Centrality +20 +Figure 2: The accuracy of five centrality measures in evaluating the spreading influence +of nodes according to the SIR model in the eight real-world networks, quantified by the +Kendall’s Tau coefficients. The dot line correspond to the epidemic threshold βth. + +(b) Les Misérables network +0 +5 +0 +MC +中 +Gr +O +公 +DIL +0 +0 +CR +PPR +COI +? +5 +0 +0.05 +0.10 +0.15 +0.20 +B(c) Game Of Thrones network +0 +-一+一 +5 +0 +MC +Gr +DIL +0 +0 +CR +PPR +COI +5 +0 +0.05 +0.10 +0.15 +0.20 +B(d) Paul Erdos collaborations network +0 +5 +0 +MC +Gr +DIL +0 +0 +CR +PPR +COI +5 +0 +0.05 +0.10 +0.15 +0.20 +B(e) Netscience network +0 +XXXXXXxX +5 +0 +MC +中 +Gr +O +DIL +0 +0 +CR +PPR +COI +5 +0 +0.05 +0.10 +0.15 +0.20 +B(f) Political blogs network +0 +5 +0 +MC +Gr +公 +DIL +0 +0 +CR +PPR +COI +5 +0 +0.05 +0.10 +0.15 +0.20 +B(g) US airport network +0 +5 +0 +MC +Gr +ol +DIL +0 +0 +CR +PPR +COI +5 +0 +0.05 +0.10 +0.15 +0.20 +B(h) E-mail network +0 +5 +0 +中 +MC +Gr +DIL +0 +A +0 +CR +PPR +COI +5 +0 +0.05 +0.10 +0.15 +0.20 +B(a) Dolphins network +0 +5 +0 +MC +Gr +DIL +0 +0 +CR +PPR +COI +? +5 +0.05 +0.10 +0.15 +0.20 +BM-Centrality +21 +case for all centralities except M-Centrality that seems to have the ability to capture +these small variations in the transmission rate due to its local component ∆D. This is +an interesting feature of the proposed method that needs to be perfected in the future. +We also can see that despite all centralities being very competitive, better results are +achieved by M-Centrality and Gravity with a slight advantage for the first one, see Table +C.1 for the numerical results corresponding to Figure 2. This is reasonable since they +are both the improved methods of k-shell method. Although the M-Centrality evaluates +the node influence from multiple attributes, the proposed method weight the position +and neighborhood attributes by using the entropy method, leading to a better rank +correlation and thus better performances than its alternatives. Overall, in the cases +of the three small networks and Political blogs, the less competitive and inconsistent +centralities are Collective Influence and ClusterRank. For Netscience, E-mail and Paul +Erd˝os collaborations, it is DIL centrality that fails to deliver suitable results. These +results confirm that the nodes located in the core of the network are more important +when considering epidemic spreading. +The main problem of centralities is that they characterize the importance of +nodes related to the topological features, which is perfect when we are considering the +structural importance of nodes. But they may not perform well in processes involving +transmission dynamics which consider more about the dynamics of nodes [25]. This can +explain why in Figure 2 the correlation between the different centralities and the SIR +model is not perfect. +4.4. Further Analysis +We conclude the analysis by highlighting the common features of the M-Centrality +revealed by the investigations on real-world networks. +In order to measure a node +influence, the M-Centrality combines a global information about its position in the +network with the local information of the degree variation in its neighborhood. +According to the weight value, more or less importance is given to these complementary +aspects of a node influence, see Table 11. In any case, varying the value of the mixing +proportion µ allows to tune the influence of the node. For µ = 0, its influence depends +only on the topology of its neighborhood, while for µ = 1, it is related to its global +position in the network. Between these two extremes, M-Centrality tends to adapt to +both local and global topological properties of the network. So far, the empirical results +demonstrate the effectiveness of the entropy weighted technique to estimate the optimum +mixing proportion of the M-Centrality components. Moreover, small variations around +the optimal weight value does not significantly alter the ranking of the nodes. +Table 11: Entropy associated with the global (EKs) and local measure (E∆D) of M- +Centrality for the real-world networks. +Network +Dolphins +Les Mis´erables +Game Of Thrones +Paul Erd˝os collaborations +Netscience +US airport +E-mail +Political blogs +EKs +0.44 +0.33 +0.36 +0.5 +0.5 +0.67 +0.40 +0.5 +E∆D +0.56 +0.67 +0.64 +0.5 +0.5 +0.33 +0.60 +0.5 + +M-Centrality +22 +These results show that M-Centrality is able to extract the most relevant +information from the network topology at both the global and local level in order to +measure the influence of the nodes. +The local influence as measured by the degree +variation in the neighborhood, allows to distinguish between the multiple nodes that +share the same position in the network according to the Coreness centrality. For the +real-world networks under study, it appears that M-Centrality shows a high and positive +correlation, especially with Gravity centrality. This is due to the fact that they are both +a variant of Coreness centrality. However, the results show that the proposed measure +performs better in term of quantifying node influence topologically and in transmission +dynamics. +The proposed measure, M-Centrality, is more subtle in ranking nodes than Coreness +centrality. This subtlety can be summarized in four major points: +(i) Its hybrid nature (combine global and local information) that leads to a more +accurate ranking of nodes. To illustrate this behavior, let’s consider nodes having +the same degree and Coreness in these networks, and let’s see if M-Centrality +can discern differences between them in term of influence. +Table 12 reports +centrality informations about different nodes that share the same degree centrality +and Coreness values in the three well documented networks under study. Results +shows that these nodes can be always distinguished by the M-centrality because the +information about the degree variation in their neighborhood is always different. For +example, node Topless and node SN4 of the Dolphins network are ranked similarly +by the Coreness and degree centralities, while they have different influence according +to M-Centrality (Topless is considered more central than SN4). This is due to the +fact that the degree variation in the neighborhood of Topless (∆DTopless = 6.28) +is higher than the one of SN4 (∆DSN4 = 5.70). So the distinction between nodes +having the same Coreness and degree is made subtly thanks to ∆D. We also notice +that M-Centrality is higher when the nodes are strategically positioned in the core +of network. +(ii) Invariance to change of attribute weights as shown by the wide range of µ values +for which ranking accuracy is conserved. +(iii) The very high monotonicity scores achieved on all the networks studied. In fact +the hybrid nature of the method and the entropy weighting strategy play a major +role. +(iv) Presents more correlation with the SIR model in the case of epidemic spreading, +which results in a better detection of influential spreaders. +5. conclusion +In this paper, a new centrality measure combining global information related to the +position of the node in the network and local information linked to its neighborhood +is proposed. The node position in the network is measured by its Coreness and the + +M-Centrality +23 +Table 12: M-Centrality, degree variation in the neighborhood (∆D), degree and Coreness +centralities of some nodes sharing the same degree and Coreness values in the Dolphins, +Les Mis´erables and Game Of Thrones networks. +Network +Nodes ID +M-Centrality +∆D +Degree +Coreness +Dolphins +Topless, SN4 +5.25, 4.93 +6.28, 5.70 +11 +4 +Mus, Notch +2.12, 2.07 +1.41, 1.31 +3 +3 +TR120, TR88 +1.51, 1.37 +1.11, 0.85 +2 +2 +Whitetip, Zig +0.93, 0.81 +0.87, 0.66 +1 +1 +Les Mis´erables +Brujon, Dahlia +4.40, 3.12 +3.10, 1.18 +7 +7 +Mme Magloire, Anzelma +2.63, 2.52 +2.44, 2.28 +3 +3 +Perpetue, Magnon +1.71, 1.70 +1.57, 1.55 +2 +2 +Gribier, Jondrette +0.83, 0.66 +0.75,0.50 +1 +1 +Game Of Thrones +Bronn, Theon +3.44, 3.42 +3.12, 3.09 +4 +4 +Viserys, Eddison +2.69, 2.61 +2.51, 2.40 +3 +3 +Jeyne, Chataya +1.88, 1.87 +1.81, 1.80 +2 +2 +Doran, Orell +0.98, 0.97 +0.97,0.96 +1 +1 +local information is quantified by the degree variation in its neighborhood. Rather than +assigning the same importance to both components of the M-Centrality, a weighting +strategy based on their respective entropy is applied in order to tune their relative +importance to the network topology. An extensive comparative evaluation has been +performed on eight real-world networks of different scales, and with existing as well +as unknown knowledge about influential nodes. +The experimental analysis provides +strong evidence of the effectiveness of the M-Centrality as compared to recent and +robust centrality measures such as Gravity, DIL, Personalized PageRank, ClusterRank +and Collective Influence. +Despite the fact that it shows strong correlation with +Gravity centrality, M-Centrality gives better results in term of capturing topological and +dynamical influence of nodes. It is therefore more subtle in measuring node influence. +Moreover the proposed method exhibits a low complexity (O(n)) which makes it suitable +for large scale networks. This work can be extended in various directions. First of all, +other variants of K-shell, including MDD and INK can be substituted to the position +component of the measure. Further progress can be obtained by a finer evaluation of +the degree variation in the neighborhood. Indeed, all the neighbors are given the same +weight independently of their context. +Finally, extensions to weighted and directed +networks must be addressed. +Appendix A. Impact of the weight µ on M-Centrality +Table A.1 reports for Dolphins and Les Mis´erables networks the top 15 nodes sorted +according to their M-Centrality values computed using various weights ranging from +µ = 0 to µ = 1. In Dolphins, for µ = 0, the M-Centrality reduces M = ∆D. The +proposed measure identify nodes Grin, Trigger, Topless, Jet and Web as the 5 most +important (See Table A.1). It also succeed in identifying SN100 in the top 15. When +the weight µ increases, the rank of SN100 remains constant. For µ = 1, there is four +groups of nodes. The biggest group with the highest coreness value (Ks = 4) contains 36 +individuals including node SN100, Grin, Jet and Kringel that are considered among the + +M-Centrality +24 +most influential by ∆D). The second one with Ks = 3 contains 9 individuals. The third +one with Ks = 2 contains 8 individuals. The last one (Ks = 1) is made of 9 individuals. +These results confirm the inability of Coreness to distinguish accurately the nodes in +a networks. For Les Mis´erables network, when µ = 0, the M-Centrality measures the +degree variation in the neighborhood of the nodes (M = ∆D). It ranks Valjean as the +most central, followed by Gavroche, Marius, Javert, Fantine, Mr and Mme Thenardier +and Cosette. Those are main characters of the novel. All the other nodes are not so +central that the measure may suggest. At the other extreme, (µ = 1) the M-Centrality +reduces to the Coreness. Among the top fifteen nodes, 12 share a Coreness value of 9 and +the 3 remaining have also a common coreness of 8. Note that there is some characters +in this list such as Gavroche and Marius (Ks = 9) that are ranked more important than +characters such as Valjean, Javert and Thenardier (Ks = 8). When the weight µ ranges +between 0.25 to 0.75 the top 15 characters are always the same with the exception of +Cosette that disappear from the list when µ = 0.75. Note that a high variation of µ is +needed in order to observe some evolutions of the ranking. +Table A.1: First 15 nodes sorted by relevance according to M-Centrality, Coreness +and ∆D in Dolphins and Les Mis´erables networks. The numbers between parenthesis +correspond to node centrality values. +Network +Rank +∆D(µ = 0) +M(µ = 0.25) +M(µ = 0.50) +M(µ = 0.75) +Ks(µ = 1) +Dolphins +1 +Grin +Grin +Grin +Grin +Beak (4) +2 +Trigger +Trigger +Trigger +Trigger +Beescratch (4) +3 +Topless +Topless +Topless +Topless +DN21 (4) +4 +Jet +Jet +Jet +Jet +DN63 (4) +5 +Web +Web +Web +Web +Double (4) +6 +SN4 +SN4 +SN4 +SN4 +Feather (4) +7 +Scabs +Scabs +Scabs +Scabs +Fish (4) +8 +Patchback +Patchback +Patchback +Patchback +Gallatin (4) +9 +Kringel +Kringel +Kringel +Kringel +Grin (4) +10 +SN63 +SN63 +SN63 +SN63 +Haecksel (4) +11 +Beescratch +Beescratch +Beescratch +Beescratch +Hook (4) +12 +Stripes +Stripes +Stripes +Stripes +Jet (4) +13 +SN100 +SN100 +SN100 +SN100 +SN100 (4) +14 +Gallatin +Gallatin +Gallatin +Gallatin +Knit (4) +15 +Shmuddel +SN9 +SN9 +SN9 +Kringel (4) +Les Mis´erables +1 +Valjean +Valjean +Valjean +Valjean +Gavroche (9) +2 +Gavroche +Gavroche +Gavroche +Gavroche +Marius (9) +3 +Myriel +Myriel +Myriel +Marius +Mabeuf (9) +4 +Marius +Marius +Marius +Javert +Enjolras (9) +5 +Javert +J avert +Javert +Thenardier +Combeferre (9) +6 +Fantine +Fantine +Fantine +Fantine +Prouvaire (9) +7 +Thenardier +Thenardier +Thenardier +Enjolras +Feuilly (9) +8 +Cosette +Enjolras +Enjolras +Bossuet +Courfeyrac (9) +9 +Enjolras +Cosette +Bossuet +Courfeyrac +Bahorel (9) +10 +MlleGillenormand +Bossuet +Cosette +Mabeuf +Bossuet (9) +11 +Gillenormand +MmeThenardier +Courfeyrac +Myriel +Joly (9) +12 +MmeThenardier +Montparnasse +Mabeuf +Prouvaire +Grantaire (9) +13 +Bamatabois +MlleGillenormand +Montparnasse +Bahorel +Valjean (8) +14 +Bossuet +Gueulemer +Prouvaire +Joly +Thenardier (8) +15 +Montparnasse +Babet +Gueulemer +Grantaire +Javert (8) +For the two remaining small scale networks, Game Of Thrones and Paul Erd˝os +collaborations, the same previous observations can be made. TableA.2 shows that µ +can vary in a wide range without changing the top nodes positions. This is interesting +because the estimate of µ can be very approximative without affecting too much the +M-Centrality behavior. +For the large scale networks under study, Tables A.3 and A.4 illustrate the M- + +M-Centrality +25 +Table A.2: First 15 nodes sorted by relevance according to M-Centrality, Coreness and +∆D in Game Of Thrones and Paul Erd˝os collaborations networks. The numbers between +parenthesis correspond to node centrality values. +Network +Rank +∆D(µ = 0) +M(µ = 0.25) +M(µ = 0.50) +M(µ = 0.75) +Ks(µ = 1) +Game Of Thrones +1 +Tyrion +Tyrion +Tyrion +Tyrion +Jaime (7) +2 +Jon +Jon +Jon +Jon +Robert (7) +3 +Daenerys +Daenerys +Sansa +Sansa +Tyrion (7) +4 +Sansa +Sansa +Robb +Robb +Tywin (7) +5 +Robb +Robb +Daenerys +Tywin +Arya (7) +6 +Tywin +Tywin +Tywin +Jaime +Cersei (7) +7 +Jaime +Jaime +Jaime +Daenerys +Gregor (7) +8 +Samwell +Samwell +Samwell +Catelyn +Joffrey (7) +9 +Mance +Catelyn +Catelyn +Cersei +Sandor (7) +10 +Catelyn +Cersei +Cersei +Arya +Catelyn (7) +11 +Cersei +Mance +Arya +Robert +Robb (7) +12 +Arya +Arya +Mance +Joffrey +Sansa (7) +13 +Robert +Robert +Robert +Samwell +Stannis (7) +14 +Joffrey +Joffrey +Joffrey +Stannis +Eddard (7) +15 +Bran +Bran +Bran +Bran +Bran (6) +Paul Erd˝os collaborations +1 +HARARY, FRANK +HARARY, FRANK +HARARY, FRANK +HARARY, FRANK +ALON, NOGA (9) +2 +POMERANCE, CARL +GRAHAM, RONALD L. +GRAHAM, RONALD L. +GRAHAM, RONALD L. +BABAI, LASZLO (9) +3 +GRAHAM, RONALD L. +POMERANCE, CARL +POMERANCE, CARL +TUZA, ZSOLT +BOLLOBAS, BELA (9) +4 +TUZA, ZSOLT +TUZA, ZSOLT +TUZA, ZSOLT +RODL, VOJTECH +BURR, STEFAN ANDRUS (9) +5 +RODL, VOJTECH +RODL, VOJTECH +RODL, VOJTECH +POMERANCE, CARL +CHUNG, FAN RONG K. (9) +6 +SOS, VERA T. +SOS, VERA T. +SOS, VERA T. +SOS, VERA T. +FAUDREE, RALPH J. (9) +7 +ALON, NOGA +ALON, NOGA +ALON, NOGA +ALON, NOGA +FRANKL, PETER (9) +8 +SPENCER, JOEL H. +SPENCER, JOEL H. +SPENCER, JOEL H. +SPENCER, JOEL H. +FUREDI, ZOLTAN (9) +9 +HAJNAL, ANDRAS +HAJNAL, ANDRAS +HAJNAL, ANDRAS +HAJNAL, ANDRAS +GOULD, RONALD J. (9) +10 +BOLLOBAS, BELA +BOLLOBAS, BELA +BOLLOBAS, BELA +BOLLOBAS, BELA +GRAHAM, RONALD L. (9) +11 +PACH, JANOS +PACH, JANOS +PACH, JANOS +PACH, JANOS +GYARFAS, ANDRAS (9) +12 +STRAUS, ERNST G. +STRAUS, ERNST G. +STRAUS, ERNST G. +KLEITMAN, DANIEL J. +HAJNAL, ANDRAS (9) +13 +KLEITMAN, DANIEL J. +KLEITMAN, DANIEL J. +KLEITMAN, DANIEL J. +STRAUS, ERNST G. +HARARY, FRANK (9) +14 +CHARTRAND, GARY +CHARTRAND, GARY +CHARTRAND, GARY +CHARTRAND, GARY +JACOBSON, MICHAEL S.. (9) +15 +CHUNG, FAN RONG K. +CHUNG, FAN RONG K. +CHUNG, FAN RONG K. +CHUNG, FAN RONG K. +LEHEL, JENO (9) +Centrality evolution of the nodes for various values of the weight. The main observation +is that the proposed measure behavior resembles the one previously observed. This +shows that M-Centrality behavior is not affected by the network scale. +As a main +conclusion, the proposed method is robust as it is not affected by changes in weights +attributes and network size. +Appendix B. Correlation in high ranks between M-Centrality and its +alternatives +The rank-biased overlap (RBO) measure is used to compare the overlap of the two +rankings at incrementally increasing depths [53]. This measure examines the accuracy +of the ranking list paying more attention to high ranks on the list by considering weights +for different ranks and assigning greater weights to high ranks. +The value of RBO +between two ranking lists X and Y is calculated using Eq. B.1. +RBO(X, Y, p) = (1 − p) +n +� +d=1 +pd−1A(X, Y, d), RBO ∈ [0, 1] +(B.1) +where A(X,Y,d) is the value of overlap between two ranking lists X and Y up to +rank d calculated by Eq. B.2, n is the number of distinct ranks on the ranking list, +p is a tunable parameter in (0, 1) range, that models the user’s persistence (at each +depth down the two lists, the user has probability p of continuing to the next rank, +and inversely probability 1 - p of deciding to stop).A high value of RBO reflects a high +correlation in high ranks between the two ranking lists. + +M-Centrality +26 +Table A.3: First 15 nodes sorted by relevance according to M-Centrality, Coreness +and ∆D in Netscience and US airport networks. The numbers between parenthesis +correspond to node centrality values. +Network +Rank +∆D(µ = 0) +M(µ = 0.25) +M(µ = 0.50) +M(µ = 0.75) +Ks(µ = 1) +Netscience +1 +BARABASI, A +BARABASI, A +BARABASI, A +BARABASI, A +GIOT, L (19) +2 +NEWMAN, M +NEWMAN, M +NEWMAN, M +NEWMAN, M +UETZ, P (19) +3 +JEONG, H +JEONG, H +JEONG, H +JEONG, H +CAGNEY, G (19) +4 +YOUNG, M +YOUNG, M +YOUNG, M +YOUNG, M +MANSFIELD, T (19) +5 +BOCCALETTI, S +BOCCALETTI, S +BOCCALETTI, S +UETZ, P +JUDSON, R (19) +6 +OLTVAI, Z +OLTVAI, Z +OLTVAI, Z +CAGNEY, G +KNIGHT, J (19) +7 +SOLE, R +SOLE, R +SOLE, R +MANSFIELD, T +LOCKSHON, D (19) +8 +ALON, U +ALON, U +ALON, U +OLTVAI, Z +NARAYAN, V (19) +9 +DIAZGUILERA, A +DIAZGUILERA, A +KURTHS, J +GIOT, L +SRINIVASAN, M (19) +10 +KURTHS, J +KURTHS, J +DIAZGUILERA, A +JUDSON, R +POCHART, P (19) +11 +LATORA, V +LATORA, V +LATORA, V +KNIGHT, J +QURESHIEMILI, A (19) +12 +HU, G +HU, G +HU, G +LOCKSHON, D +LI, Y (19) +13 +KAHNG, B +KAHNG, B +MUTH, S +NARAYAN, V +GODWIN, B (19) +14 +STAUFFER, D +MUTH, S +VICSEK, T +SRINIVASAN, M +CONOVER, D (19) +15 +MUTH, S +STAUFFER, D +KAHNG, B +POCHART, P +KALBFLEISCH, T (19) +US airport +1 +766 +766 +766 +766 +766 (64) +2 +114 +114 +114 +114 +114 (64) +3 +877 +877 +877 +877 +877 (64) +4 +709 +709 +709 +709 +709 (64) +5 +1200 +1200 +1200 +1200 +1200 (64) +6 +1016 +1016 +1016 +1016 +1016 (64) +7 +500 +500 +500 +500 +500 (64) +8 +389 +389 +389 +389 +389 (64) +9 +711 +711 +711 +711 +711 (64) +10 +1068 +1068 +1068 +1068 +1068 (64) +11 +391 +391 +391 +391 +391 (64) +12 +215 +215 +215 +215 +215 (64) +13 +435 +435 +435 +435 +435 (64) +14 +875 +875 +875 +875 +875 (64) +15 +685 +505 +685 +685 +32 (64) +Table A.4: First 15 nodes sorted by relevance according to M-Centrality, Coreness +and ∆D in E-mail and US Political blogs networks. The numbers between parenthesis +correspond to node centrality values. +Network +Rank +∆D(µ = 0) +M(µ = 0.25) +M(µ = 0.50) +M(µ = 0.75) +Ks(µ = 1) +E-mail network +1 +105 +105 +105 +105 +299 (11) +2 +23 +23 +23 +23 +389 (11) +3 +333 +333 +333 +333 +434 (11) +4 +41 +41 +41 +41 +552 (11) +5 +16 +16 +16 +16 +571 (11) +6 +42 +42 +42 +42 +726 (11) +7 +233 +233 +233 +233 +756 (11) +8 +24 +24 +24 +24 +788 (11) +9 +14 +14 +14 +14 +885 (11) +10 +196 +196 +196 +196 +886 (11) +11 +21 +21 +21 +21 +887 (11) +12 +355 +355 +578 +578 +888 (11) +13 +578 +578 +355 +135 +1 (10) +14 +135 +135 +135 +355 +2 (10) +15 +76 +76 +76 +76 +3 (10) +US Political blogs +1 +blogsforbush.com +blogsforbush.com +blogsforbush.com +blogsforbush.com +aintnobaddude.com (36) +2 +dailykos.com +dailykos.com +dailykos.com +dailykos.com +dailykos.com (36) +3 +drudgereport.com +drudgereport.com +drudgereport.com +drudgereport.com +atrios.blogspot.com (36) +4 +instapundit.com +instapundit.com +instapundit.com +instapundit.com +atrios.blogspot.com/ (36) +5 +talkingpointsmemo.com +talkingpointsmemo.com +talkingpointsmemo.com +talkingpointsmemo.com +talkingpointsmemo.com (36) +6 +atrios.blogspot.com +atrios.blogspot.com +atrios.blogspot.com +atrios.blogspot.com +blog.dccc.org (36) +7 +powerlineblog.com +powerlineblog.com +powerlineblog.com +powerlineblog.com +bodyandsoul.typepad.com (36) +8 +michellemalkin.com +michellemalkin.com +michellemalkin.com +michellemalkin.com +busybusybusy.com (36) +9 +truthlaidbear.com +truthlaidbear.com +truthlaidbear.com +truthlaidbear.com +corrente.blogspot.com (36) +10 +washingtonmonthly.com +washingtonmonthly.com +washingtonmonthly.com +washingtonmonthly.com +washingtonmonthly.com (36) +11 +littlegreenfootballs.com/weblog +littlegreenfootballs.com/weblog +littlegreenfootballs.com/weblog +littlegreenfootballs.com/weblog +anoldsoul.blogspot.com(36) +12 +wizbangblog.com +wizbangblog.com +wizbangblog.com +wizbangblog.com +democrats.org/blog (36) +13 +hughhewitt.com +hughhewitt.com +hughhewitt.com +hughhewitt.com +digbysblog.blogspot.com (36) +14 +juancole.com +juancole.com +juancole.com +juancole.com +juancole.com (36) +15 +lashawnbarber.com +lashawnbarber.com +lashawnbarber.com +lashawnbarber.com +elayneriggs.blogspot.com (36) +A(X, Y, d) = X1:d ∩ X1:d +X1:d ∪ X1:d +, +(B.2) + +M-Centrality +27 +where X1:d (resp. Y1:d) represents the elements present in ranks 1 to d of list X +(resp. Y). +In this work, we investigate the value of RBO between the ranking list produced by +the different centralities and the M-Centrality ranking list. The value of p varies from +0.4 to 0.9, and the value of RBO is calculated in consequence. +The results in Figure B.1 show that for Les Mis´erables, Game Of Thrones, +Paul Erd˝os and E-mail networks, M-Centrality has higher concordance with Gravity +centrality in high ranks. +Note that in the cases of Game Of Thrones and E-mail +networks, this correlation decreases as p increases. +For Dolphins and US airport +networks, M-Centrality achieves the highest concordance with Personalized PageRank, +while for Netscience and Political blogs it is with Collective Influence for all values of +p except for p=0.9 in Political blogs where the correlation in high ranks is registered +with Gravity centrality. +These results show that despite M-Centrality being highly +correlated with Gravity centrality, the correlation in high ranks may vary between the +two depending on the depth of the comparison. +Appendix C. Correlation with the SIR model given transmission rate values +Table C.1 show the numerical results corresponding to Figure 2 for the two most +competitive measures, M-Centrality and Gravity. Although it has been shown previously +that the two methods are highly correlated, one can see that in the majority of cases, +the proposed method performs better than Gravity centrality. +Table C.1: Kendall tau correlation coefficient of M-Centrality and Gravity centrality +with the SIR model. The number 20% refers to β = 0.2×βth. +Network +β +20% +40% +60% +80% +100% +120% +140% +160% +Dolphins +τ(M, SIR) +0.690 +0.689 +0.676 +0.760 +0.746 +0.763 +0.793 +0.877 +τ(Gr, SIR) +0.661 +0.633 +0.667 +0.742 +0.723 +0.749 +0.779 +0.807 +Les Mis´erables +τ(M, SIR) +0.690 +0.737 +0.748 +0.874 +0.852 +0.838 +0.818 +0.873 +τ(Gr, SIR) +0.717 +0.783 +0.801 +0.839 +0.802 +0.824 +0.801 +0.814 +Game Of Thrones +τ(M, SIR) +0.719 +0.736 +0.792 +0.798 +0.816 +0.846 +0.879 +0.861 +τ(Gr, SIR) +0.679 +0.726 +0.728 +0.748 +0.799 +0.831 +0.828 +0.821 +Paul Erd˝os collaborations +τ(M, SIR) +0.746 +0.783 +0.798 +0.765 +0.850 +0.845 +0.850 +0.887 +τ(Gr, SIR) +0.689 +0.749 +0.790 +0.789 +0.801 +0.804 +0.832 +0.840 +Netscience +τ(M, SIR) +0.484 +0.486 +0.485 +0.487 +0.489 +0.488 +0.487 +0.486 +τ(Gr, SIR) +0.409 +0.419 +0.422 +0.428 +0.431 +0.430 +0.432 +0.433 +US airport +τ(M, SIR) +0.393 +0.458 +0.485 +0.516 +0.540 +0.551 +0.544 +0.598 +τ(Gr, SIR) +0.378 +0.441 +0.470 +0.490 +0.506 +0.518 +0.535 +0.550 +E-mail +τ(M, SIR) +0.410 +0.488 +0.523 +0.516 +0.575 +0.617 +0.605 +0.596 +τ(Gr, SIR) +0.393 +0.462 +0.505 +0.512 +0.541 +0.523 +0.519 +0.554 +US Political blogs +τ(M, SIR) +0.443 +0.538 +0.574 +0.643 +0.678 +0.608 +0.687 +0.745 +τ(Gr, SIR) +0.443 +0.518 +0.574 +0.597 +0.621 +0.611 +0.641 +0.647 + +M-Centrality +28 +Figure B.1: +The RBO values between the ranking lists produced by the different +centralities and the obtained ranking list from M-Centrality on the real-world networks +under study. + +(a) Dolphins network +8 +0 +9 +0 +MC-GR +中 +MC-DIL +BO +4 +MC-CR +0 +MC-PPR +MC-COI +2 +0 +X11.4 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +p(b) Les Miserables network +9 +0 +5 +0 +4 +MC-GR +0 +MC-DIL +BO +3 +MC-CR +0 +MC-PPR +MC-COI +2 +0 +0 +0 +0 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +b(c) Game Of Thrones network +8 +0 +9 +0 +MC-GR +中 +MC-DIL +BO +4 +MC-CR +0 +R +MC-PPR +MC-COI +2 +0 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9(e) Netscience network +9 +0 +5 +0 +4 +0 +MC-GR +MC-DIL +BO +3 +MC-CR +0 +R +MC-PPR +MC-COI +2 +0 +0 +0 +0.7 +0.8 +0.4 +0.5 +0.6 +0.9(g) US airport inetwork +5 +0 +4 +0 +MC-GR +3 +MC-DIL +BO +0 +MC-CR +R +MC-PPR +2 +MC-COI +0 +0 +0 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9(h) E-mail network +8 +9 +0 +MC-GR +MC-DIL +RBO +4 +MC-CR +MC-PPR +MC-COI +2 +0 +0.5 +0.6 +0.7 +0.8 +0.9 +0.4(f) Political blogs network +6 +0 +MC-GR +MC-DIL +BO +4 +MC-CR +MC-PPR +MC-COI +2 +0 +0.5 +0.6 +0.7 +0.8 +0.9 +0.4(d) Paul Erdos network +9 +0 +5 +0 +4 +0 +MC-GR +中 +MC-DIL +BO +3 +MC-CR +0 +R +MC-PPR +2 +MC-COI +0 +0 +0 +0.7 +0.4 +0.5 +0.6 +0.8 +0.9M-Centrality +29 +References +[1] Sungmin Lee, Luis EC Rocha, Fredrik Liljeros, and Petter Holme. 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Acm, 2003. +[52] Atul Kumar Srivastava, Rakhi Garg, and PK Mishra. +Discussion on damping factor value in +pagerank computation. International Journal of Intelligent Systems and Applications, 9(9):19, +2017. +[53] William Webber, Alistair Moffat, and Justin Zobel. A similarity measure for indefinite rankings. +ACM Transactions on Information Systems (TOIS), 28(4):20, 2010. + diff --git a/C9AzT4oBgHgl3EQfTvzN/content/tmp_files/load_file.txt b/C9AzT4oBgHgl3EQfTvzN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e286cad5b75429510936a83691d57aed05a69aa0 --- /dev/null +++ b/C9AzT4oBgHgl3EQfTvzN/content/tmp_files/load_file.txt @@ -0,0 +1,2494 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf,len=2493 +page_content='M-Centrality: Identifying key nodes based on global position and local degree variation Ahmed IBNOULOUAFI1,a, Mohamed EL HAZITI2, Hocine CHERIFI3 1 LRIT Laboratory, Associated Unit to CNRST (URAC29) IT Rabat Center - Faculty of Sciences In Rabat, MOHAMMED V UNIVERSITY IN RABAT, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1014 RP, Rabat, Morocco 2 Higher School of technology (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='T) in SALE 3 Laboratoire Electronique, Informatique et Image (Le2i) UMR 6306 CNRS, University of Burgundy, Dijon, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' a Corresponding author: E-mail: ahmedibnoulouafi@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='com Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Identifying influential nodes in a network is a major issue due to the great deal of applications concerned, such as disease spreading and rumor dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' That is why, a plethora of centrality measures has emerged over the years in order to rank nodes according to their topological importance in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Local metrics such as degree centrality make use of a very limited information and are easy to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Global metrics such as betweenness centrality exploit the information of the whole network structure at the cost of a very high computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Recent works have shown that combining multiple metrics is a promising strategy to quantify the node’s influential ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Our work is in this line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In this paper, we introduce a multi-attributes centrality measure called M-Centrality that combines the information on the position of the node in the network with the local information on its nearest neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The position is measured by the K-shell decomposition, and the degree variation in the neighborhood of the node quantifies the influence of the local context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In order to examine the performances of the proposed measure, we conduct experiments on small and large scale real-world networks from the perspectives of transmission dynamics and network connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' According to the empirical results, the M-Centrality outperforms its alternatives in identifying both influential spreaders and nodes essential to maintain the network connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In addition, its low computational complexity makes it easily applied to large scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Keywords: Complex networks, Centrality measures, Influential nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='01256v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SI] 3 Jan 2023 M-Centrality 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Introduction The problem of identifying nodes that are ”central” or ”influential” has attracted wide attention from researchers due to its many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In epidemic spreading, the transmission of the disease depends on the contacts that the infected person has with the susceptible population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Thus, being able to locate and vaccinate the most influential individuals can prevent from a potential outbreak of the disease [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In viral marketing, being able to locate influential individuals can help to optimize the sales of products [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' A straightforward approach towards detecting these central nodes is to quantify their influence using centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The idea of centrality was initially introduced in the context of sociology to look whether there is a relation between the location of an individual in the network and its influence in group processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Since then, various centrality measures have emerged over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' They are employed in a multitude of contexts to rank nodes according to their topological importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' As there is no consensual definition of the centrality of a node within a network, the issue is considered from the multiple concepts reflecting the notion of influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Therefore, we can classify centralities from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' They can be classified according to the underlying approach (geometric, spectral or path-based) [3], the way they quantify influence (locally, globally, position within the network, dynamical processes such as random walks) [4, 5], the computing ideas (iterative refinement) [6], the number of attributes they take into consideration [6] or by taking into account the collective influence of the whole set of nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=', in this case, the problem of finding influential spreaders is targeted as an influence maximization problem [7, 8, 9], where we look for a minimum set of nodes that maximize the spread of information to the whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In geometric measures, influence is related to distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In other words, it depends only on how many nodes exist at every distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Degree [10], closeness [11] and K- shell [12] (and its variations, including MDD [13] and INK [14]) are geometric measures that evaluate influence from respectively a local, global and position point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In [15], authors proposed a local ranking measure, ClusterRank, that quantifies the influence of a node by taking into account not only its direct influence (measured by the number of its followers) and influences of its neighbors, but also its clustering coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' ClusterRank can be applied to directed as well as undirected networks where its superiority in term of locating influential spreaders is significant compared with degree centrality, K-shell decomposition, PageRank and LeaderRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In addition, ClusterRank, only making use of local information, is much more efficient considering computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Spectral measures work on the spectral properties of the graph (eigenvalues and eigenvectors of the adjacency matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Iterative refinement methods identify influential nodes based on the influence of its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This is known as the mutual enhancement effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' PageRank [16], HITS [17], LeaderRank [4] and Personalized PageRank [18] (where the probability of jumping to a node when abandoning the random walk is not uniform, but it is given by a preference vector that favorites nodes over others) are good examples of these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Path-based M-Centrality 3 measures exploit the existence of shortest paths passing through a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Betweenness [19] is the best known centrality of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' It evaluates influence from a global perspective by considering influential nodes those through which transits the largest flow of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' As an alternative to betweenness centrality, a centrality measure called DIL, was proposed in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Instead of working on a global level, DIL centrality ranks nodes based on local information (degree value and the importance of lines) to identify network bridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This measure showed great performances and is adapted to large scale network due to its low complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Note that global and spectral measures are inapplicable to very large graphs due to their computational complexity, while local ones are simple but generally less effective because they only take in consideration the neighborhood of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Another way of approaching the problem of locating influential nodes is to optimize an objective function of influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In this context, Morone and Makse proposed a method called Collective Influence expressed as the product of the reduced degree of a node and the total reduced degree of all nodes at a distance d from the node [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' According to their results, optimal results can be reached at a distance d=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The Collective Influence formula constitutes the main core of their percolation algorithm used to find the minimal set of nodes which are crucial for the global connectivity of the network (their removal cause the destruction of the giant component).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The major drawback of this method is its computational complexity that stands at O(nlog(n)) [8, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Recent works have raised the fact that centrality cannot be apprehended from a single point of view and that combining measures can enhance the performances of ranking methods [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Comprehensive evidence centrality (CEC) [23] is a multi- attributes method that uses the Dempster-Shafer evidence theory to combine degree, betweenness and closeness centralities in order to characterize the influence of a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Another multi-criteria method proposed in [24] uses the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to evaluate the influence of each node from more than one perspective based on four indicators which are the degree, betweenness, closeness and improved K-shell (IKs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Dynamic sensitive centrality (DS) [25] is also a multi-attributes method that takes in consideration topological features (a dynamical process represented by the calculation of the total number of walks of length t from node i to all nodes in the network) and dynamical properties that depend on the spreading rate β over the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' DS centrality can locate influential nodes accurately and performs very well in the early stages of spreading compared to degree, K-shell and eigenvector centralities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In [26], Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' proposed a centrality (labeled Gravity) based on the Isaac Newton classical gravity formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Gravity centrality considers the K-core value of a node as its mass, and the shortest path distance between two nodes in a network is viewed as their distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' To reduce the complexity of their method, only neighbors located in a distance less than or equal to 3 are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This method shows that combining a position based measure (Coreness) and geodesic distance, gives better results in term of locating influential spreaders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The main lesson learned from these studies is that combining multiple attributes is M-Centrality 4 much more accurate than using a single one in order to evaluate the influence capability of a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' However, this raises new questions such as which centralities should be combined and how to combine them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Previous studies show that position [12] and neighborhood [27, 28] are key factors to quantify the node influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Based on this evidence, we propose a new centrality measure combining these complementary aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The measure is called M-Centrality in homage to the M-Theory that unifies all consistent versions of the super-string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' It is a weighted combination of : A ”global” measure that characterizes the location importance of the node in the all network (core or periphery).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We use the K-shell decomposition for its low computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Note, however, that any of its variations can be substituted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' A local measure that characterizes its neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This measure we choose to call ∆D calculates the degree variation in the neighborhood of the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' It is inspired from the preferential attachment process [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Since it cannot be assumed that attributes have equal weights, the Shannon entropy method is used in order to find the appropriate weights for the global and local measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' It is one of the most famous approach for determining the objective attribute weights in multiple-criteria decision problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' To evaluate the proposed centrality measure, we report a series of experiments on real-world networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Extensive comparisons with the most influential alternative measures are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Results clearly show that globally, M-Centrality provides more accurate ranking list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In addition to its effectiveness, one main advantage of this measure as compared to alternative global ranking methods is that it has a low computational complexity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' O(n) (as shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='4), that allows it to be used with large scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The remainder of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In section 2, we review the necessary background on evaluation metrics used to asses different centralities and the preferential attachment concept that inspired our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Section 3 introduces the M-Centrality measure, and it shows how it is related to the preferential attachment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In section 4, the datasets, the experimental setup and experimental results are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The proposed measure is analyzed and compared to the most recent methods, including Gravity (Gr), DIL, Personalized PageRank (PPR), ClusterRank (CR) and Collective influence (COI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Finally, section 5 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Background In this section, we recall the definition of the evaluation measures that are used to compare centralities (Monotonicity, Kendall’s Tau correlation coefficient (τ) and decline rate of network efficiency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Additionally, the model used to simulate an epidemic spreading process in order to evaluate the performances of the different methods in M-Centrality 5 the context of transmission dynamics is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Finally, we present the preferential attachment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Evaluation of nodes ranking methods The main role of centrality measures is to provide a means to rank nodes relative to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Indeed, the numerical values may not be directly interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Nodes are usually ranked in descending order of influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The node with largest influence is ranked first and the one with smallest influence value is ranked last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' There are many ways to evaluate the performances of centrality measures, more or less linked to the underlying applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In this paper, we adopt generic measures (Monotonicity and Kendall’s Tau correlation coefficient) that are commonly employed in the literature to quantify influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Monotonicity measures the ability of a ranking method to assign a different rank to each node, while Kendall tau correlation allows a statistical comparison of the agreement between two rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In addition, we also evaluate the performances of centrality measures from two perspectives, one is based on transmission dynamics where we look for the spreading capabilities of nodes, and the other is based on the network connectivity and the theory that the network damage caused by deleting a node is equivalent to its importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Monotonicity: To quantify the resolution of a ranking method, the monotonicity M [30] of a ranking vector R is defined as follows: M(R) = � 1 − � r∈R nr(nr − 1) n(n − 1) �2 , M ∈ [0, 1] (1) where n is the size of ranking vector R and nr is the number of ties with the same rank r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This metric quantifies the fraction of ties in the ranking list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The monotonicity M(R) is equal to one if the ranking vector R is perfectly monotonic, and it is equal to zero if all nodes in R have the same rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Note that monotonicity allows to quantify the discrimination ability of a centrality measure, however, a ranking with no tie is not necessarily accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Kendall tau correlation: The Kendall’s Tau (τ) correlation coefficient [31] is generally used to compare the performance of different topology-based measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' It measures the ranking consistency of two lists that rank the same set of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Consider X = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=', xn) and Y= (y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=', yn) are two ranked lists that contain n elements, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Any pair of ranks (xi, yi) and (xj, yj) is said concordant if xi > xj and yi > yj , or if xi < xj and yi < yj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' If xi > xj and yi < yj, or if xi < xj and yi > yj, then the pair is said to be discordant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In the case of xi = xj or yi = yj (tied pair), the pair is neither concordant nor discordant, in this case X and Y are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The Kendall’s Tau (τ) correlation M-Centrality 6 coefficient is defined by: τX,Y = nc − nd � (n0 − n1)(n0 − n2) , τ ∈ [−1, 1] (2) where n0 = n(n − 1)/2, n1 = � i ti(ti − 1)/2, n2 = � j tj(tj − 1)/2, nc and nd are respectively the number of concordant pairs and discordant pairs, ti and tj are the number of tied values in the ith and jth group of ties respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We can consider the following ranges in order to qualify the strength of the relation between two rankings: No correlation: τ = 0, Low correlation: τ ∈ ]0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='50[, Moderate correlation: τ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='50, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='70[, High correlation: τ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='70, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='90[, Very high correlation: τ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='90, 1[, Perfect correlation: τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' SIR model: In literature, the susceptible-infected (SI) and susceptible–infected–removed (SIR) models [32, 33] are generally used to simulate an epidemic spreading process in real networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Compared to the SI model, the SIR model is widely used for information dissemination and disease diffusion and in various fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In this paper, we employ the SIR model to estimate the spreading capabilities of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In this model, a node has three possible states: S (susceptible), I (infected) and R (recovered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Initially we start from a single infected node and the other nodes are susceptible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' At each step, the infected node can infect its susceptible neighbors with infection probability β, and then it recovered from the diseases with probability γ (set to 1 in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The spreading process stops when there is no infected node in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' At last, the number of recovered nodes represents the influence of the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In this article, for the SIR epidemic model we consider: For each node we simulate 102 diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The spreading capability of the node will be the average number of infected nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We chose values of infection rate β that vary from 20% to 160% of βth (the epidemic threshold [34]) representing both cases where β < βth and β ≥ βth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Network efficiency: Network efficiency [35] reflects the network connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The better the network efficiency is, the better the network connectivity is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Network efficiency η is defined thus: η = 1 n(n − 1) � i̸=j∈V ηij, (3) M-Centrality 7 where ηij is the efficiency between i and j, ηij = 1 dij , dij is the shortest way between i and j, n is the number of network nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The decline rate of network efficiency ν is defined as: ν = 1 − η η0 , (4) where η is the efficiency of the network after removing nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' η0 is the initial efficiency of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The bigger the ν is, the worse the network connectivity destroyed by removing nodes is and the more important the node removed is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Preferential attachment Growth and preferential attachment are the two mechanisms used in the most prominent approach to reproduce complex networks formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Growth is a network construction process where at each time step, a new node with m links is added to the existing network [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The preferential attachment rule specify that new nodes select old nodes with which they will form links based on their degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' That is, the probability Πn→i that a new node n makes a connection to an existing node i with degree ki is given by : Πn→i(ki) = ki � j∈all kj , (5) where all is the set of nodes to which the new node n could connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Due to this preference, a ”Rich get Richer” process takes place where the nodes with higher degrees will further increase their connexions leading to the emergence of hubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' An important feature of this model introduced by Barab´asi and Albert [36] is that the generated networks display a power-law degree distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The growth mechanism combined with preferential attachment is therefore the most influential explanation for the prevalence of the scale-free networks that are ubiquitous in nature and man-made systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Proposed method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Motivation One of the main application of centrality measures is the assessment of node’s spreading capability in the context of epidemic spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Various approaches based on centralities have been proposed so far, one of them is based on node degrees [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' However, as shown in [30, 38], it is not sufficient since it is just the number of neighboring nodes that is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' An improvement of the approach consists in taking into account the degrees of the neighbors of the node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' a node with neighbors that have high degrees has greater spreading capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' It is worth pointing out, however, that the high degree of a node, or its neighbors, is not sufficient for specification of its spreading capability [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' On M-Centrality 8 the basis of this observation, and driven by the fact that quantifying node influence from multiple points of view, we decided to introduce a new multi-attributes centrality, labeled M-Centrality, that takes in consideration global and local features of a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Of course there is as much combination possible as the number of centralities, but the main reason behind the choice of K-shell and degree in our combination is their great performances in detecting influential spreaders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Indeed the K-shell shows the best results in the case of single origin spreading [12] while the degree operates the best in the case of multiple origins spreading [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' M-Centrality measure The main idea of this approach is that both position and neighborhood attributes play important roles in shaping node influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Therefore, by combining these two attributes, we can enhance the performance of the centrality evaluation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' M-Centrality expresses how influential is a node based on the combination of the local information contained in its neighborhood and a more global information about its position in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' More precisely, the M-Centrality of the node i that we note Mi is the weighted sum of a global measure Ksi and a local measure ∆Di: Mi = µKsi + (1 − µ)∆Di, 0 ≤ µ ≤ 1 (6) Ksi is the coreness index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' It is computed using K-core decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This global measure characterizes the position (core or periphery) of the node i in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Note that any of its variation can be substituted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' ∆Di is a new measure that we introduce to quantify the degree variation at a local level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' It takes inspiration from the preferential attachment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We first calculate the degree variation (di,j) between node i and each of its neighbors as follows: di,j = |Ni|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' ��� kj − ki � j∈Ni kj ���, j ∈ Ni (7) where ki and kj represent the degree of nodes i and j respectively, Ni is the neighborhood of the node i and |Ni| the number of i neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We use the absolute value to eliminate negative values in case of kj > ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The degree variation in the neighborhood of node i is given by: ∆Di = � j di,j, (8) µ is a tailored weighting factor that is estimated from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Unlike most traditional multi-attribute ranking methods that consider all attributes as equally important (equal weights), we propose to compute the weight µ by targeting the problem in a multi attribute decision-making framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Among the various solutions, we choose an entropy technique that is known for its great performances M-Centrality 9 in attributes weights determination [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The weight computation process proceeds as follows: First, we normalize the global and local measure attributes of the M-Centrality: r1j = Ksj �n j=1 Ksj and r2j = ∆Dj �n j=1 ∆Dj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' where n is the size of the network Second, we build the matrix R2,n defined by: R2,n = � r11 r12 r13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' r1n r21 r22 r23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' r2n � (9) Third, we compute the entropy Ei of the ith attribute: Ei = − 1 ln(n) n � j=1 rij ln(rij), i = 1, 2 and 1 ≤ j ≤ n (10) Finally, the weight of the two attributes is computed: wi = 1 − Ei 2 − �2 i=1 Ei , i = 1, 2 (11) According to the properties of entropy, 0 ≤ wi ≤ 1 and �2 i=1 wi = 1, thus µ = w1 and 1 - µ = w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Relation with the preferential attachment Since the centrality we propose takes inspiration from the preferential attachment phe- nomenon, we will now start by developing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (5) to establish that link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' But first, let us agree on the following notations: Ni: The set of node i neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' N c i : The complement of the set Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' V={v1, v2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=', vn}: The set of network nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' all = V - {vi}: The set of all network nodes except the node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' all= Ni ∪ N c i and Ni ∩ N c i = ∅ Ni = all - N c i By developing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (5): ( � j∈all kj)Πn→i(ki) = ki ( � j∈all kj � j∈all kj − � j∈Nc i kj )Πn→i(ki) = ki � j∈all kj − � j∈Nc i kj ( � j∈all kj � j∈Ni kj )Πn→i(ki) = ki � j∈Ni kj M-Centrality 10 We note ki � j∈Ni kj by Πlocal n→i(ki) to obtain: Πlocal n→i(ki) = ( � j∈all kj � j∈Ni kj )Πn→i(ki).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (12) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (12) expresses the preferential attachment at a local level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This result allows us to construct the local measure of M-Centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' By developing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (7) we obtain: di,j = |Ni|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' ��� kj � j∈Ni kj − ki � j∈Ni kj ��� di,j = |Ni|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' ��� kj � j∈Ni kj − ( � j∈all kj � j∈Ni kj )Πn→i(ki) ��� Replacing ( � j∈all kj � j∈Ni kj )Πn→i(ki) by Πlocal n→i(ki) and kj � j∈Ni kj by ¯kNi j gives: di,j = |Ni|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='|¯kNi j − Πlocal n→i(ki)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (13) Finally, replacing ∆Di by its value in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (6) we obtain: Mi = µKsi + (1 − µ) � j |Ni|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='|¯kNi j − Πlocal n→i(ki)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (14) This expression shows that the M-Centrality is linked to the preferential attachment phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Computational complexity The calculation of Ks index for all nodes has a complexity of O(n), where n is the size of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The calculation of ∆D for all nodes has a complexity of O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The calculation of the entropy matrix and the weight µ has a complexity of O(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The calculation of M-Centrality for all nodes has a complexity of O(n) + O(n) + O(2n)= O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Experimental results To evaluate the efficiency of the proposed method, eight real-world networks‡ (small [43, 44, 45, 46] and large scale [47, 48, 49, 50]) are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' All datasets are considered undirected and unweighted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' also, the largest connected component was used in the spreading process using the SIR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The statistical properties relative to these ‡ The networks used can be found via this link: https://icon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='edu/ and http://vlado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='fmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='uni- lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='si/pub/networks/doc/erdos/ for Paul Erd˝os collaborations network M-Centrality 11 networks are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Concerning the evaluation process, it comprises the ability to detect key nodes§, monotonicity, rank correlation, impact of nodes removal on network efficiency and finally the spreading capabilities of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The results of the proposed centrality are compared to Gravity (Gr), DIL, Personalized PageRank (PPR)∥, ClusterRank (CR) and Collective influence (COI) centralities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Table 1: The statistical properties of the networks under study, where < k > is the average degree of the network, kmax the highest degree, Ksmax the highest coreness, σ the size of the giant component and βth the epidemic threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Network Type Number of nodes Number of edges < k > kmax Ksmax βth σ Dolphins Social 62 159 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='12 12 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='139 62 Les Mis´erables Co-appearance 77 254 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='59 36 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='083 77 Game Of Thrones 107 352 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='57 36 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='075 107 Paul Erd˝os collaborations Collaboration 492 1417 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='76 42 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='058 446 Netscience 1589 2742 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='45 34 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='052 379 US Political blogs Web-graph 1490 16715 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='43 351 36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='013 1222 E-mail Communication 1133 5451 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='62 71 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='048 1133 US airport 2010 Traffic 1574 17215 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='87 314 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='008 1572 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Evaluation of nodes ranking 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Small scale networks Since there is no consensus on the concept of centrality, real networks with known information about the node’s importance are commonly used as benchmarks to assess the effectiveness of centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We present the results of the experimental evaluation on three well documented networks (Dolphins, Les Mis´erables and Game Of Thrones) and a network with no information about central nodes (Paul Erd˝os collaborations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We choose to concentrate on these small scale networks in order to better understand the behavior of the centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' First, let’s study the influence of the weight variation µ on the M-Centrality measure for the four networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='2 illustrate the M-Centrality evolution of the nodes ranking for various values of the weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We can clearly distinguish three major behaviors corresponding to the following cases µ = 0, 0 < µ <1, and µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' At the extremes, M-Centrality reduces to ∆D and Coreness while in between these two it tends to adapt to both local and global topological properties of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The major shortcoming of this is the fact that µ can vary in a wide range without impacting the nodes rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This is a very interesting result because it shows that a node needs to be strategically positioned in the network along with a high degree variation in its neighborhood in order to be considered influential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For more details about ranking, we suggest to refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' § For all the Tables that follow, the nodes marked in color (blue, red, yellow, green, violet, orange and olive) are the most central according to literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For networks with unknown information about central nodes, we mark in gray are the nodes that appear frequently in the ranking lists ∥ The parameters used are: (1) a damping factor α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='15 as experiments suggest that small changes in α have little effect in practice[51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (2) for the preference vector, we choose to favor the Hubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' M-Centrality 12 The next step is to determine the optimal weight value of the M-Centrality measure using the entropy weighted technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The values obtained for the four networks are reported in Further analysis section, Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For Dolphins (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Les Mis´erables and Game of Thrones) the weight µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='44 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='33 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='36) puts more emphasis on the local measure ∆D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The main reason for this is that Coreness assigns the same rank to many nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Therefore, its entropy is smaller than the one associated to the local measure ∆D, and so is its weight in the M-centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This result makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Indeed, ∆D is more efficient at identifying key characters of the novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' However, combining Coreness with degree variation allows to distinguish the characters that are grouped in the same category by the Coreness centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For Paul Erd˝os collaborations, both global and local information are given the same importance by the entropy weighted technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' After we study the behavior of M-Centrality based on µ values, we focus now on its capability of detecting key nodes and then compare the results with the alternative measures presented previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For Dolphins, the information provided in [43] reports that the female denoted SN100 plays an important role holding the community together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Table 2 shows that DIL is the only centrality measure that puts SN100 at the top of the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' It is understandable given the fact that its disappearance split the network into two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' ClusterRank, Personalized PageRank and Collective Influence are the only ones that fail in identifying SN100 in the top 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In fact the first one ranks it at the 46th position while the second and third ones rank it at the 19th position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For M-Centrality, it is in the top 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' However, this result suggests that the measures do not exploit adequately the information about bridges that is well encoded in the DIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Further improvement for M-Centrality needs to be made in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For Les Mis´erables, The ex-convict, Jean Valjean, is a central character of the novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' He spends a great deal of time running away from Inspector Javert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' He is also closely tied to his adopted daughter, Cosette, and her future husband, Marius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Fantine, Cosette’s mother, Mr and Mme Thenardier and their son Gavroche are also important characters of the novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We report on Table 2 the top 15 nodes sorted by relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' M-Centrality is the only measure that succeeds in ranking Valjean as the most influential character of the novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Additionally, it identifies all the key characters of the novel in the top 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For Gravity, DIL and Personalized PageRank, they all rank Gavroche, Valjean and Marius in the top 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Thenardier and Javert are always in the top 15 but with very various rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We may also notice that Collective Influence and ClusterRank give the worst ranking as they clearly fail to detect main characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The major shortcoming of these rankings is that (except for M-Centrality) Cosette does not appear in the top 15 major nodes and that the central hub (Valjean) is not necessarily the main bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Moving on to the third network, Game Of Thrones, no one can deny the important role played by Tyrion Lannister in the story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Indeed, all the centralities (except ClusterRank) identify this character as the most important one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We also can see that M-Centrality performs the best as it identifies, along with Tyrion, a majority of other key characters, including his brother and sister Jamie and Cersei Lannister, Jon Snow, M-Centrality 13 Table 2: First 15 nodes sorted by relevance according to the centrality measures in the networks Dolphins and Les Mis´erables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' M (µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='44) and M (µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='33) refer to the M-Centrality ranking corresponding to the weight obtained by the entropy weighted technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Networks Les Mis´erables Dolphins Rank M (µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='44) Gr DIL CR PPR COI M (µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='33) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='DIL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='CR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='PPR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='COI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Valjean ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Collective influence manages to identify some major characters, with Daenerys ranked at the top of the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' However, we may notice the absence of Tyrion in the top 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Again, ClusterRank gives the poorest ranking results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Overall, two important observations need to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' First, the fact that only the proposed method succeeds in ranking Daenerys in the top 15 (3rd position).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Second, for Jon Snow it is only M-Centrality that managed to give him importance by ranking him 2nd most influential character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Paul Erd˝os collaborations network is a collaboration graph of mathematicians where two mathematicians are joined by an edge whenever they co-authored a paper together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' As the concept of ”central” nodes is relative, one may consider influence in term of the number of co-authors (degree), others may consider nodes located in the core, or those without the network will split in two or more sub-graphs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This can intuitively seem a good approach, but in reality it tends to be biased since nodes that are considered central in term of connections (Hubs) in the topological context, can be less influential when talking about spreading capabilities where nodes located in the core are more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Being said, at this stage of the paper and especially for networks with unknown information about central nodes, we make the assumption that all the rankings provided by the centralities under study are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Later, these rankings will be exploited in studying the impact of deleting keys nodes on network structure and efficiency and thus we can conclude which measure(s) provide(s) the most suitable ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' From Table 3, the only observation we can make is that despite all centralities have different conceptions of influence, some nodes (marked in gray) appear frequently in all ranking results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' There are also some concordances between the different centralities on ranking certain nodes in the top 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For example M-Centrality and Gravity (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' DIL, ClusterRank, Personnalized PageRank and Collective Influence) agree on 11 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 7, 3, 8 and 1) out of 15 rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' M-Centrality 14 Table 3: First 15 nodes sorted by relevance according to the centrality measures in the networks Game Of Thrones and Paul Erd˝os collaborations network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' M (µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='36) and M (µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='50) refer to the M-Centrality ranking corresponding to the weight obtained by the entropy weighted technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Networks Game Of thrones Paul Erd˝os collaborations network Rank M (µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='36) Gr DIL CR PPR COI M (µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='50) Gr DIL CR PPR COI 1 Tyrion Tyrion Tyrion Eddard Tyrion Daenerys HARARY, FRANK GRAHAM, RONALD L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' RODL, VOJTECH SZEMEREDI, ENDRE GRAHAM, RONALD L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' POMERANCE, CARL 2 Jon Sansa Sansa Meryn Sansa Mance GRAHAM, RONALD L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' RODL, VOJTECH GRAHAM, RONALD L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' TROTTER, WILLIAM T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=', JR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' RODL, VOJTECH HAJNAL, ANDRAS 3 Daenerys Robb Jaime Ilyn Jaime Samwell POMERANCE, CARL ALON, NOGA ALON, NOGA LEHEL, JENO ALON, NOGA CHARTRAND, GARY 4 Sansa Jaime Robb Joffrey Robb Cersei TUZA, ZSOLT TUZA, ZSOLT TUZA, ZSOLT FRANKL, PETER SPENCER, JOEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' HARARY, FRANK 5 Robb Tywin Cersei Balon Cersei Tyrion RODL, VOJTECH SPENCER, JOEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' SPENCER, JOEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' JACOBSON, MICHAEL S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' TUZA, ZSOLT STRAUS, ERNST G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 6 Tywin Arya Arya Petyr Tywin Jon SOS, VERA T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' HARARY, FRANK FUREDI, ZOLTAN RODL, VOJTECH FUREDI, ZOLTAN TUZA, ZSOLT 7 Jaime Cersei Joffrey Gregor Joffrey Jorah ALON, NOGA FUREDI, ZOLTAN CHUNG, FAN RONG K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' GOULD, RONALD J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' CHUNG, FAN RONG K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' GYARFAS, ANDRAS 8 Samwell Robert Tywin Cersei Arya Sandor SPENCER, JOEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' CHUNG, FAN RONG K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' GYARFAS, ANDRAS FUREDI, ZOLTAN SZEMEREDI, ENDRE RODL, VOJTECH 9 Catelyn Joffrey Robert Aerys Robert Rhaegar HAJNAL, ANDRAS BOLLOBAS, BELA SZEMEREDI, ENDRE SAKS, MICHAEL E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' LOVASZ, LASZLO SOS, VERA T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 10 Cersei Catelyn Catelyn Sandor Catelyn Joffrey BOLLOBAS, BELA FAUDREE, RALPH J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' FAUDREE, RALPH J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' ALON, NOGA PACH, JANOS BABAI, LASZLO 11 Mance Stannis Sandor Arya Sandor Beric PACH, JANOS SOS, VERA T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' PACH, JANOS GRAHAM, RONALD L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' HARARY, FRANK TURAN, PAL 12 Arya Eddard Eddard Jaime Stannis Davos STRAUS, ERNST G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' PACH, JANOS LOVASZ, LASZLO KUBICKA, EWA MARIE BABAI, LASZLO SPENCER, JOEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 13 Robert Jon Jon Pycelle Eddard Jaime KLEITMAN, DANIEL J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' SZEMEREDI, ENDRE BABAI, LASZLO KUBICKI, GRZEGORZ GYARFAS, ANDRAS SARKOZY, ANDRAS 14 Joffrey Sandor Gregor Sansa Gregor Tywin CHARTRAND, GARY LOVASZ, LASZLO FRANKL, PETER GYARFAS, ANDRAS FAUDREE, RALPH J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' LOVASZ, LASZLO 15 Bran Gregor Stannis Stannis Jon Edmure CHUNG, FAN RONG K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' HAJNAL, ANDRAS JACOBSON, MICHAEL S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' SCHELP, RICHARD H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' FRANKL, PETER RENYI, ALFRED A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Now we turn to the comparative evaluation of the centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Table 4 shows the monotonicity of the ranking methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Results show that M-Centrality, Gravity and Personalized PageRank clearly outperform the other measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In other words, few nodes are assigned the same rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' DIL is more competitive than ClusterRank centrality except in the case of Game Of Thrones network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Collective Influence also gives good performances in distinguishing between nodes importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Table 5 reports the rank correlation of the various centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The main result is that M-Centrality is highly and positively correlated with Gravity, with τ values ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='71 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This is reasonable since they are both a variant of Coreness centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Globally, the proposed method is moderately correlated with ClusterRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The lowest correlation value is registered between M-Centrality and Collective Influence (τM,COI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='46) in the case of Les Mis´erables network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In other words, Collective Influence centrality behaves very differently than the proposed method for this network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Note that it is evident if we refer to the top fifteen nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Further investigations about the nature of the relation between the ranking produced by the M-Centrality and the one of its alternatives are reported in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Table 4: Monotonicity M of centrality measures for the small scale real-world networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The weight of the M-Centrality is computed by the entropy weighted technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Network M(M) M(Gr) M(DIL) M(CR) M(PPR) M(COI) Dolphins 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='873 0.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='922 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='879 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Large scale networks After we study the behavior of M-Centrality on small scale networks, we move on to present the results of the experimental evaluation on four large scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' It includes E-mail, Netscience, US airport and US Political blogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' As the performances of the proposed measure have been established previously, it will be interesting to test its effectiveness on much larger graphs that are not necessarily well M-Centrality 15 Table 5: Kendall’s tau (τ) rank correlation coefficient for the small scale real-world networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Network τ(M, Gr) τ(M, DIL) τ(M, CR) τ(M, PPR) τ(M, COI) Dolphins 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='716 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='699 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='589 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='517 Les Mis´erables 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='612 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='748 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='566 Paul Erd˝os collaborations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='623 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='857 documented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Details concerning the impact of the weight variation on M-Centrality are reported in Appendix A, Tables A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='3 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' First, we determine the key nodes identified by the different centralities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Tables 6 and 7 present our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The main observation is that each centrality identifies various key nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' However, some nodes appear more frequently in the top 15 of multiple centralities, this can possibly suggest their potential importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The first network, E- mail, represents the exchange of emails among members of the Rovira i Virgili University in Spain, in 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The nodes marked in gray appear in 4 out of 5 ranking results except ClusterRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' However, an important thing to notice about these nodes is the fact that none of them is strategically located in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In other words, they are not in the core of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Netscience is a network of co-authorships between scientists whose research centers on the properties of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Edges join every pair of individuals whose names appear together as authors of a paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The nodes marked in gray are the ones who appear in all ranking results except M-Centrality and Collective Influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In addition, they all belong to the core of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This result suggests that in this network, the proposed measure and Collective Influence quantify node influence differently from its alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Note that these two methods agree both on ranking BARABASI, A, NEWMAN, M and JEONG,H in the top 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Table 6: First 15 nodes sorted by relevance according to the centrality measures in the networks E-mail and Netscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Networks E-mail Netscience Rank M (µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='40) Gr DIL CR PPR COI M (µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='50) Gr DIL CR PPR COI 1 105 105 105 886 105 16 BARABASI,' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='blogspot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='com techievampire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='net/wppol Next we examine the monotonicity of the different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The results are reported in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Again M-Centrality and Gravity clearly offer the best performances in all networks with a very high monotonicity score outperforming the one of the other centralities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We also notice the poor results of Personalized PageRank and Collective Influence in Netscience network, which can be interpreted as a sign of incapability to distinguish between nodes influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Table 9 shows the correlation between M-Centrality and other centralities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' One can see that in the four networks, M-Centrality is very correlated with Gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We also notice that the proposed measure exhibits poor relation with Personalized PageRank, especially in Netscience network (τ(M, PPR) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For the results concerning the concordance in high ranks between the ranking list produced by M-Centrality and the one of its alternatives, we suggest the reader to refer to Appendix B Table 8: Monotonicity M of the centrality measures for the large scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Network M(M) M(Gr) M(DIL) M(CR) M(PPR) M(COI) Netscience 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='910 0.' metadata={'source': 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measures is evaluated in the context of network vulnerability and transmission dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In this work we explore the two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' First, we evaluate the efficiency of the proposed method by examining the impact of removing the top most important nodes on network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Table 10 shows the rest graph obtained after deleting the key nodes identified by each centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The main observation is that M-Centrality outperforms the other centralities, with the exception of the two collaborations networks where Collective Influence seems to be more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Indeed the removal of the most important nodes identified by the proposed measure has high impact on network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' These results suggest that the ranking list of the proposed measure is more consistent and accurate than the one of its alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Table 10: Number of connected components C obtained after removing the most important nodes according to ranking lists produced by the various centrality measures.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Game Of Thrones ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Paul Erd˝os collaborations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='48 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='61 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Netscience ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='462 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='411 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='406 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='395 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='418 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='490 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='US airport 2010 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='286 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='203 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='203 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='203 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='94 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='E-mail ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='US Political blogs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='362 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='327 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='292 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='274 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='296 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='314 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='After we study the structural damage caused by the removal of important nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' we move on to study the impact of deleting important nodes on network efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Figure 1 shows the relationship between the decline rate of network efficiency and the number of nodes removed from the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Two main observations can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' First, we can see that the decline rate of network efficiency is rising with the increase of the number of nodes removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Second, the proposed measure clearly performs the best compared to its alternatives, with ClusterRank giving the worst results in all the networks under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For Dolphins network, the removal of node SN100 (identified by DIL) seems to cause more damage to the network than nodes Grin and Hook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This comforts the fact that SN100 is a key member in holding the group together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Gravity and DIL give good performances in all stages of the removal process, while Personalized PageRank fails in the last stages (after the removal of the 8th node).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In Les Mis´erables, removing Valjean (identified by M-Centrality) causes more damage than Gavroche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In the stages 2, 3 and 4 of the removal process, we notice that Marius and Enjolras are more important M-Centrality 18 than Myriel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The fifth stage of the removal process rises the fact that Javert is clearly more central than Bossuet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This suggests that the most suitable ranking would be Valjean, Gavroche, Marius, Enjolras and Javert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For Game Of Thrones, the top 10 nodes identified by M-Centrality are clearly the most important ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Gravity, DIL and Personalized PageRank give competitive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Globally all centralities are very competitive with a slight advantage of the proposed method due to its consistent ranking as previously shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For the remaining networks, the performance gap between M-Centrality and its alternatives is consequent and more visible especially in the cases of Political blogs, US airport and E-mail networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In Netscience and Paul Erd˝os, Collective Influence gives better results in the first network and is as competitive as M-Centrality in the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This is concordant with the results presented in Table 10 and can be explained by the fact that in this type of networks, the importance of an author is closely related to the importance of authors with whom he collaborates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This topological property is captured perfectly by the definition of the Collective Influence centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Being said, given the computational complexity, monotonicity, structural damage caused by the removal of key nodes and network efficiency scores, the proposed centrality gives better results than its alternatives, which comforts the previous ranking results presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This suggests a more consistent and accurate ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We also notice that the highest vulnerability to the removal of important nodes identified by M-Centrality is registered for Game Of Thrones (90%) and Les Mis´erable (87%) while the lower one is for E-mail network (18%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This reflects that some networks are more resilient to targeted attacks than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Evaluation of nodes spreading capability with SIR model In addition to network vulnerability, we test the effectiveness of the methods by evaluating their Kendall tau correlation with the SIR model for different values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' A greater correlation indicates the greater accuracy of the centrality in quantifying the spreading capability of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Figure 2 shows the rank correlation results of different ranking methods on the previously studied networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Note that M-Centrality is compared with the most recent and effective methods in detecting influential spreaders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For the real-world networks investigated, the first observation to make is that the correlation of different methods with the SIR epidemic model increases as the transmission rate increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Second, for small values of β, the correlation is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This is due to the fact that the epidemic is limited to the neighborhood of nodes and thus couldn’t spread all over the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Third, when β>βth, we can observe that the variations in the transmission rate have in general little effect on the correlation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This can be explained by the fact that when the transmission rate is larger than the epidemic threshold, the epidemic starting from any node can reach all the network nodes and the role of topology in the spreading will be overridden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Thus, distinguishing between nodes spreading capabilities become much harder.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='top nodes removedM-Centrality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Figure 2: The accuracy of five centrality measures in evaluating the spreading influence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='of nodes according to the SIR model in the eight real-world networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' quantified by the Kendall’s Tau coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The dot line correspond to the epidemic threshold βth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (b) Les Misérables network 0 5 0 MC 中 Gr O 公 DIL 0 0 CR PPR COI ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='10 0.' metadata={'source': 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+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='20 B(a) Dolphins network 0 5 0 MC Gr DIL 0 0 CR PPR COI ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='20 BM-Centrality 21 case for all centralities except M-Centrality that seems to have the ability to capture these small variations in the transmission rate due to its local component ∆D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This is an interesting feature of the proposed method that needs to be perfected in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We also can see that despite all centralities being very competitive, better results are achieved by M-Centrality and Gravity with a slight advantage for the first one, see Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1 for the numerical results corresponding to Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This is reasonable since they are both the improved methods of k-shell method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Although the M-Centrality evaluates the node influence from multiple attributes, the proposed method weight the position and neighborhood attributes by using the entropy method, leading to a better rank correlation and thus better performances than its alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Overall, in the cases of the three small networks and Political blogs, the less competitive and inconsistent centralities are Collective Influence and ClusterRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For Netscience, E-mail and Paul Erd˝os collaborations, it is DIL centrality that fails to deliver suitable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' These results confirm that the nodes located in the core of the network are more important when considering epidemic spreading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The main problem of centralities is that they characterize the importance of nodes related to the topological features, which is perfect when we are considering the structural importance of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' But they may not perform well in processes involving transmission dynamics which consider more about the dynamics of nodes [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This can explain why in Figure 2 the correlation between the different centralities and the SIR model is not perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Further Analysis We conclude the analysis by highlighting the common features of the M-Centrality revealed by the investigations on real-world networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In order to measure a node influence, the M-Centrality combines a global information about its position in the network with the local information of the degree variation in its neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' According to the weight value, more or less importance is given to these complementary aspects of a node influence, see Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In any case, varying the value of the mixing proportion µ allows to tune the influence of the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For µ = 0, its influence depends only on the topology of its neighborhood, while for µ = 1, it is related to its global position in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Between these two extremes, M-Centrality tends to adapt to both local and global topological properties of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' So far, the empirical results demonstrate the effectiveness of the entropy weighted technique to estimate the optimum mixing proportion of the M-Centrality components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Moreover, small variations around the optimal weight value does not significantly alter the ranking of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Table 11: Entropy associated with the global (EKs) and local measure (E∆D) of M- Centrality for the real-world networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Network Dolphins Les Mis´erables Game Of Thrones Paul Erd˝os collaborations Netscience US airport E-mail Political blogs EKs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='5 E∆D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='5 M-Centrality 22 These results show that M-Centrality is able to extract the most relevant information from the network topology at both the global and local level in order to measure the influence of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The local influence as measured by the degree variation in the neighborhood, allows to distinguish between the multiple nodes that share the same position in the network according to the Coreness centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For the real-world networks under study, it appears that M-Centrality shows a high and positive correlation, especially with Gravity centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This is due to the fact that they are both a variant of Coreness centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' However, the results show that the proposed measure performs better in term of quantifying node influence topologically and in transmission dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The proposed measure, M-Centrality, is more subtle in ranking nodes than Coreness centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This subtlety can be summarized in four major points: (i) Its hybrid nature (combine global and local information) that leads to a more accurate ranking of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' To illustrate this behavior, let’s consider nodes having the same degree and Coreness in these networks, and let’s see if M-Centrality can discern differences between them in term of influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Table 12 reports centrality informations about different nodes that share the same degree centrality and Coreness values in the three well documented networks under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Results shows that these nodes can be always distinguished by the M-centrality because the information about the degree variation in their neighborhood is always different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For example, node Topless and node SN4 of the Dolphins network are ranked similarly by the Coreness and degree centralities, while they have different influence according to M-Centrality (Topless is considered more central than SN4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This is due to the fact that the degree variation in the neighborhood of Topless (∆DTopless = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='28) is higher than the one of SN4 (∆DSN4 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' So the distinction between nodes having the same Coreness and degree is made subtly thanks to ∆D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' We also notice that M-Centrality is higher when the nodes are strategically positioned in the core of network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (ii) Invariance to change of attribute weights as shown by the wide range of µ values for which ranking accuracy is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (iii) The very high monotonicity scores achieved on all the networks studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In fact the hybrid nature of the method and the entropy weighting strategy play a major role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (iv) Presents more correlation with the SIR model in the case of epidemic spreading, which results in a better detection of influential spreaders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' conclusion In this paper, a new centrality measure combining global information related to the position of the node in the network and local information linked to its neighborhood is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The node position in the network is measured by its Coreness and the M-Centrality 23 Table 12: M-Centrality, degree variation in the neighborhood (∆D), degree and Coreness centralities of some nodes sharing the same degree and Coreness values in the Dolphins, Les Mis´erables and Game Of Thrones networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Network Nodes ID M-Centrality ∆D Degree Coreness Dolphins Topless, SN4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='25, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='93 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='28, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='70 11 4 Mus, Notch 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='12, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='41, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='31 3 3 TR120, TR88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='51, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='11, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='85 2 2 Whitetip, Zig 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='93, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='87, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='66 1 1 Les Mis´erables Brujon, Dahlia 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='40, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='10, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='18 7 7 Mme Magloire, Anzelma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='63, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='44, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='28 3 3 Perpetue, Magnon 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='71, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='57, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='55 2 2 Gribier, Jondrette 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='83, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='75,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='50 1 1 Game Of Thrones Bronn, Theon 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='44, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='12, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='09 4 4 Viserys, Eddison 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='69, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='61 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='51, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='40 3 3 Jeyne, Chataya 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='88, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='87 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='81, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='80 2 2 Doran, Orell 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='98, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='97,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='96 1 1 local information is quantified by the degree variation in its neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Rather than assigning the same importance to both components of the M-Centrality, a weighting strategy based on their respective entropy is applied in order to tune their relative importance to the network topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' An extensive comparative evaluation has been performed on eight real-world networks of different scales, and with existing as well as unknown knowledge about influential nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The experimental analysis provides strong evidence of the effectiveness of the M-Centrality as compared to recent and robust centrality measures such as Gravity, DIL, Personalized PageRank, ClusterRank and Collective Influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Despite the fact that it shows strong correlation with Gravity centrality, M-Centrality gives better results in term of capturing topological and dynamical influence of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' It is therefore more subtle in measuring node influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Moreover the proposed method exhibits a low complexity (O(n)) which makes it suitable for large scale networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This work can be extended in various directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' First of all, other variants of K-shell, including MDD and INK can be substituted to the position component of the measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Further progress can be obtained by a finer evaluation of the degree variation in the neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Indeed, all the neighbors are given the same weight independently of their context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Finally, extensions to weighted and directed networks must be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Impact of the weight µ on M-Centrality Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1 reports for Dolphins and Les Mis´erables networks the top 15 nodes sorted according to their M-Centrality values computed using various weights ranging from µ = 0 to µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In Dolphins, for µ = 0, the M-Centrality reduces M = ∆D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The proposed measure identify nodes Grin, Trigger, Topless, Jet and Web as the 5 most important (See Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' It also succeed in identifying SN100 in the top 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' When the weight µ increases, the rank of SN100 remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For µ = 1, there is four groups of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The biggest group with the highest coreness value (Ks = 4) contains 36 individuals including node SN100, Grin, Jet and Kringel that are considered among the M-Centrality 24 most influential by ∆D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The second one with Ks = 3 contains 9 individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The third one with Ks = 2 contains 8 individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The last one (Ks = 1) is made of 9 individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' These results confirm the inability of Coreness to distinguish accurately the nodes in a networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For Les Mis´erables network, when µ = 0, the M-Centrality measures the degree variation in the neighborhood of the nodes (M = ∆D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' It ranks Valjean as the most central, followed by Gavroche, Marius, Javert, Fantine, Mr and Mme Thenardier and Cosette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Those are main characters of the novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' All the other nodes are not so central that the measure may suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' At the other extreme, (µ = 1) the M-Centrality reduces to the Coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Among the top fifteen nodes, 12 share a Coreness value of 9 and the 3 remaining have also a common coreness of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Note that there is some characters in this list such as Gavroche and Marius (Ks = 9) that are ranked more important than characters such as Valjean, Javert and Thenardier (Ks = 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' When the weight µ ranges between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='25 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='75 the top 15 characters are always the same with the exception of Cosette that disappear from the list when µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Note that a high variation of µ is needed in order to observe some evolutions of the ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1: First 15 nodes sorted by relevance according to M-Centrality, Coreness and ∆D in Dolphins and Les Mis´erables networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The numbers between parenthesis correspond to node centrality values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Network Rank ∆D(µ = 0) M(µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='25) M(µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='50) M(µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='75) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Ks(µ = 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Dolphins ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Grin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Grin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Grin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Grin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Beak (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Trigger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Trigger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Trigger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Trigger ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Beescratch (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Topless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Topless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Topless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Topless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='DN21 (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Jet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Jet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Jet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Jet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='DN63 (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Web ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Web ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Web ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Web ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Double (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Feather (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Scabs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Scabs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Scabs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Scabs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Fish (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Patchback ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Patchback ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Patchback ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Patchback ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gallatin (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Kringel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Kringel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Kringel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Kringel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Grin (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN63 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Haecksel (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Beescratch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Beescratch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Beescratch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Beescratch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Hook (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Stripes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Stripes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Stripes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Stripes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Jet (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN100 (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gallatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gallatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gallatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gallatin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Knit (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Shmuddel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='SN9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Kringel (4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Les Mis´erables ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Valjean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Valjean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Valjean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Valjean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gavroche (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gavroche ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gavroche ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gavroche ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gavroche ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Marius (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Myriel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Myriel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Myriel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Marius ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Mabeuf (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Marius ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Marius ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Marius ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Javert ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Enjolras (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Javert ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='J avert ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Javert ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Thenardier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Combeferre (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Fantine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Fantine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Fantine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Fantine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Prouvaire (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Thenardier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Thenardier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Thenardier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Enjolras ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Feuilly (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Cosette ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Enjolras ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Enjolras ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Bossuet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Courfeyrac (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Enjolras ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Cosette ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Bossuet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Courfeyrac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Bahorel (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='MlleGillenormand ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Bossuet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Cosette ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Mabeuf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Bossuet (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gillenormand ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='MmeThenardier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Courfeyrac ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Myriel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Joly (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='MmeThenardier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Montparnasse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Mabeuf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Prouvaire ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Grantaire (9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Bamatabois ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='MlleGillenormand ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Montparnasse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Bahorel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Valjean (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Bossuet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gueulemer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Prouvaire ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Joly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Thenardier (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Montparnasse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Babet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gueulemer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Grantaire ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Javert (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='For the two remaining small scale networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Game Of Thrones and Paul Erd˝os collaborations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' the same previous observations can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' TableA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='2 shows that µ can vary in a wide range without changing the top nodes positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This is interesting because the estimate of µ can be very approximative without affecting too much the M-Centrality behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For the large scale networks under study, Tables A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='3 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='4 illustrate the M- M-Centrality 25 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='2: First 15 nodes sorted by relevance according to M-Centrality, Coreness and ∆D in Game Of Thrones and Paul Erd˝os collaborations networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The numbers between parenthesis correspond to node centrality values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Network Rank ∆D(µ = 0) M(µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='25) M(µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='50) M(µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='75) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Ks(µ = 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Game Of Thrones ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Tyrion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Tyrion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Tyrion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Tyrion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Jaime (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Jon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Jon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Jon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Jon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Robert (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Daenerys ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Daenerys ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Sansa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Sansa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Tyrion (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Sansa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Sansa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Robb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Robb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Tywin (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Robb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Robb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Daenerys ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Tywin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Arya (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Tywin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Tywin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Tywin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Jaime ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Cersei (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Jaime ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Jaime ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Jaime ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Daenerys ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Gregor (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Samwell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Samwell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Samwell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Catelyn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Joffrey (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Mance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Catelyn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Catelyn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Cersei ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Sandor (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Catelyn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Cersei ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Cersei ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Arya ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Catelyn (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Cersei ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Mance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Arya ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Robert ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Robb (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Arya ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Arya ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Mance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Joffrey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Sansa (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Robert ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Robert ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Robert ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Samwell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Stannis (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Joffrey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Joffrey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Joffrey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Stannis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Eddard (7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Bran ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Bran ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Bran ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Bran ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Bran (6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Paul Erd˝os collaborations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='HARARY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' FRANK HARARY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' FRANK HARARY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' FRANK HARARY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' FRANK ALON,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' NOGA (9) 2 POMERANCE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' CARL GRAHAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' RONALD L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' GRAHAM, RONALD L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' GRAHAM, RONALD L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' BABAI, LASZLO (9) 3 GRAHAM, RONALD L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' POMERANCE, CARL POMERANCE, CARL TUZA, ZSOLT BOLLOBAS, BELA (9) 4 TUZA, ZSOLT TUZA, ZSOLT TUZA, ZSOLT RODL, VOJTECH BURR, STEFAN ANDRUS (9) 5 RODL, VOJTECH RODL, VOJTECH RODL, VOJTECH POMERANCE, CARL CHUNG, FAN RONG K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (9) 6 SOS, VERA T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' SOS, VERA T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' SOS, VERA T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' SOS, VERA T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' FAUDREE, RALPH J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (9) 7 ALON, NOGA ALON, NOGA ALON, NOGA ALON, NOGA FRANKL, PETER (9) 8 SPENCER, JOEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' SPENCER, JOEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' SPENCER, JOEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' SPENCER, JOEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' FUREDI, ZOLTAN (9) 9 HAJNAL, ANDRAS HAJNAL, ANDRAS HAJNAL, ANDRAS HAJNAL, ANDRAS GOULD, RONALD J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (9) 10 BOLLOBAS, BELA BOLLOBAS, BELA BOLLOBAS, BELA BOLLOBAS, BELA GRAHAM, RONALD L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' (9) 11 PACH, JANOS PACH, JANOS PACH, JANOS PACH, JANOS GYARFAS, ANDRAS (9) 12 STRAUS, ERNST G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' STRAUS, ERNST G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' STRAUS, ERNST G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' KLEITMAN, DANIEL J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' HAJNAL, ANDRAS (9) 13 KLEITMAN, DANIEL J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' KLEITMAN, DANIEL J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' KLEITMAN, DANIEL J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' STRAUS, ERNST G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' HARARY, FRANK (9) 14 CHARTRAND, GARY CHARTRAND, GARY CHARTRAND, GARY CHARTRAND, GARY JACOBSON, MICHAEL S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='. (9) 15 CHUNG, FAN RONG K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' CHUNG, FAN RONG K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' CHUNG, FAN RONG K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' CHUNG, FAN RONG K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' LEHEL, JENO (9) Centrality evolution of the nodes for various values of the weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The main observation is that the proposed measure behavior resembles the one previously observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This shows that M-Centrality behavior is not affected by the network scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' As a main conclusion, the proposed method is robust as it is not affected by changes in weights attributes and network size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Correlation in high ranks between M-Centrality and its alternatives The rank-biased overlap (RBO) measure is used to compare the overlap of the two rankings at incrementally increasing depths [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' This measure examines the accuracy of the ranking list paying more attention to high ranks on the list by considering weights for different ranks and assigning greater weights to high ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The value of RBO between two ranking lists X and Y is calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' RBO(X, Y, p) = (1 − p) n � d=1 pd−1A(X, Y, d), RBO ∈ [0, 1] (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1) where A(X,Y,d) is the value of overlap between two ranking lists X and Y up to rank d calculated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='2, n is the number of distinct ranks on the ranking list, p is a tunable parameter in (0, 1) range, that models the user’s persistence (at each depth down the two lists, the user has probability p of continuing to the next rank, and inversely probability 1 - p of deciding to stop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='A high value of RBO reflects a high correlation in high ranks between the two ranking lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' M-Centrality 26 Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='3: First 15 nodes sorted by relevance according to M-Centrality, Coreness and ∆D in Netscience and US airport networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The numbers between parenthesis correspond to node centrality values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Network Rank ∆D(µ = 0) M(µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='25) M(µ = 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='505 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='685 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='685 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='32 (64) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='4: First 15 nodes sorted by relevance according to M-Centrality, Coreness and ∆D in E-mail and US Political blogs networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The numbers between parenthesis correspond to node centrality values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Network Rank ∆D(µ = 0) M(µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='25) M(µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='50) M(µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='75) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='Ks(µ = 1) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='blogspot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='com (36) A(X, Y, d) = X1:d ∩ X1:d X1:d ∪ X1:d , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='2) M-Centrality 27 where X1:d (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Y1:d) represents the elements present in ranks 1 to d of list X (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' In this work, we investigate the value of RBO between the ranking list produced by the different centralities and the M-Centrality ranking list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The value of p varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='4 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='9, and the value of RBO is calculated in consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The results in Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1 show that for Les Mis´erables, Game Of Thrones, Paul Erd˝os and E-mail networks, M-Centrality has higher concordance with Gravity centrality in high ranks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Note that in the cases of Game Of Thrones and E-mail networks, this correlation decreases as p increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' For Dolphins and US airport networks, M-Centrality achieves the highest concordance with Personalized PageRank, while for Netscience and Political blogs it is with Collective Influence for all values of p except for p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='9 in Political blogs where the correlation in high ranks is registered with Gravity centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' These results show that despite M-Centrality being highly correlated with Gravity centrality, the correlation in high ranks may vary between the two depending on the depth of the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Correlation with the SIR model given transmission rate values Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1 show the numerical results corresponding to Figure 2 for the two most competitive measures, M-Centrality and Gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Although it has been shown previously that the two methods are highly correlated, one can see that in the majority of cases, the proposed method performs better than Gravity centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='1: Kendall tau correlation coefficient of M-Centrality and Gravity centrality with the SIR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' The number 20% refers to β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='2×βth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' Network β 20% 40% 60% 80% 100% 120% 140% 160% Dolphins τ(M, SIR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='690 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='676 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='760 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='793 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='877 τ(Gr, SIR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='633 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='742 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='749 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='779 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='807 Les Mis´erables τ(M, SIR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='690 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='748 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='874 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='873 τ(Gr, SIR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='717 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='824 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='814 Game Of Thrones τ(M, SIR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='719 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='736 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='816 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='879 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='861 τ(Gr, SIR) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content='679 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' A similarity measure for indefinite rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} +page_content=' ACM Transactions on Information Systems (TOIS), 28(4):20, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfTvzN/content/2301.01256v1.pdf'} diff --git a/D9AyT4oBgHgl3EQf4vqB/content/tmp_files/2301.00792v1.pdf.txt b/D9AyT4oBgHgl3EQf4vqB/content/tmp_files/2301.00792v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6366ef1ec6a12cfc2d39b9c1025957c641f0e970 --- /dev/null +++ b/D9AyT4oBgHgl3EQf4vqB/content/tmp_files/2301.00792v1.pdf.txt @@ -0,0 +1,714 @@ +The Undesirable Dependence on Frequency of Gender Bias Metrics Based +on Word Embeddings +Francisco Valentini +ICC (UBA-CONICET) +Maestría en Data Mining (UBA) +Buenos Aires, Argentina +ft.valentini@gmail.com +Germán Rosati +Escuela IDAES (UNSAM) +Buenos Aires, Argentina +grosati@unsam.edu.ar +Diego Fernandez Slezak +ICC (UBA-CONICET) +Buenos Aires, Argentina +dfslezak@dc.uba.ar +Edgar Altszyler +ICC (UBA-CONICET) +Maestría en Data Mining (UBA) +Buenos Aires, Argentina +ealtszyler@dc.uba.ar +Abstract +Numerous works use word embedding-based +metrics to quantify societal biases and stereo- +types in texts. +Recent studies have found +that word embeddings can capture semantic +similarity but may be affected by word fre- +quency. In this work we study the effect of +frequency when measuring female vs. male +gender bias with word embedding-based bias +quantification methods. +We find that Skip- +gram with negative sampling and GloVe tend +to detect male bias in high frequency words, +while GloVe tends to return female bias in low +frequency words. +We show these behaviors +still exist when words are randomly shuffled. +This proves that the frequency-based effect ob- +served in unshuffled corpora stems from prop- +erties of the metric rather than from word as- +sociations. +The effect is spurious and prob- +lematic since bias metrics should depend ex- +clusively on word co-occurrences and not indi- +vidual word frequencies. Finally, we compare +these results with the ones obtained with an al- +ternative metric based on Pointwise Mutual In- +formation. We find that this metric does not +show a clear dependence on frequency, even +though it is slightly skewed towards male bias +across all frequencies. +1 +Introduction +Word embeddings are one of the most commonly +used techniques to measure semantic closeness be- +tween words in a corpus. In recent years, they +have been widely used in Computational Social +Science applications to measure societal biases and +stereotypes (Caliskan et al., 2017; Garg et al., 2018; +Kozlowski et al., 2019; Lewis and Lupyan, 2020; +Charlesworth et al., 2021). +For practical purposes, we consider bias to be +the degree to which the language used to describe +groups or things is different. Bias is typically mea- +sured by computing the difference between the +mean similarity of words of two context groups +A and B with respect to a target word x: +BiasWE = mean +a∈A cos(vx, va) − mean +b∈B cos(vx, vb), +(1) +where vi is the word embedding of word i and +cos(vi, vj) is the cosine similarity between vectors. +Gender bias has long been one of the most stud- +ied biases with this method. In this context, A and +B are usually defined as gendered nouns and pro- +nouns (Caliskan et al., 2017; Lewis and Lupyan, +2020). A representative example is Garg et al. +(2018), where the female vs. male bias of pro- +fessions in historical corpora is found to correlate +with the percentage of women employed in each +profession over time. +Not as widely used as word embeddings, Point- +wise Mutual Information (PMI) is a metric of word +similarity that can also be used to study biases +(Gálvez et al., 2019; Aka et al., 2021; Valentini +et al., 2021). Valentini et al. (2021) define the PMI- +based bias metric as +BiasPMI = PMI(x, A) − PMI(x, B), +where +PMI(x, Y ) = log +P(x, Y ) +P(x)P(Y ). +P(x, Y ) is the probability of co-occurrence be- +tween the word x with any one in Y in a window +of a predefined number of words, and P(x) and +arXiv:2301.00792v1 [cs.CL] 2 Jan 2023 + +P(Y ) are the probability of occurrence of the word +x and any word in Y , respectively. Valentini et al. +(2021) show that BiasPMI can be expressed as +BiasPMI = log P(x|A) +P(x|B). +(2) +That is, BiasPMI can be interpreted as how much +more likely it is to find words in x in the context of +words in A than in the context of words in B, in a +log scale. Thus, it captures exclusively first-order +associations and can be computed via maximum +likelihood using co-occurrence counts from the text +(Valentini et al., 2021). +Some recent works have studied the relationship +between word frequencies and word embeddings. +In particular, embeddings seem to encode word fre- +quency even after normalization (Schnabel et al., +2015), vector norm depends on word frequency +(Wilson and Schakel, 2015), top principal compo- +nent directions encode frequency in different ways +(Mu and Viswanath, 2018) and vectors of high- +frequency and low-frequency words lie in different +regions of the embedding space (Gong et al., 2018). +These studies are nevertheless inconclusive in +the sense that they do not determine clearly to what +extent the association observed is caused by actual +attributes of corpora or by undesirable properties of +embedding training. Hence, an answer to the origin +of the relation between embeddings and frequency +is yet to be found. What is more, the repercussions +of this effect in applications relevant to Computa- +tional Social Sciences such as bias quantification +have not yet been explored. +We make three main contributions. First, we +show that frequency has an association with gen- +der bias when measured with word embedding- +based metrics: both Skip-gram with negative sam- +pling (SGNS) and GloVe-based bias metrics tend +to detect male bias in high frequency words, while +GloVe also yields female bias on average in low- +frequency words. Second, we show that the de- +pendence of the embedding-based gender bias on +frequency holds when tokens in the corpus are +randomly shuffled. This proves that the depen- +dence on frequency is an artifact of the metric it- +self i.e. that embedding-based bias metrics can +encode frequency spuriously. Third, we find that +the PMI-based gender bias metric does not present +this frequency-based effect but is slightly skewed +towards male bias across all frequency ranges.1 +1Code for the paper is available at https://github.com/ +ftvalentini/EmbeddingsBiasFrequency +Our analyses are restricted to the English lan- +guage and are based on a binary understanding of +gender (see Limitations). +2 +The effect of frequency on gender bias +Our objective is to study the association between +gender bias and frequency in the widely used +embedding-based metrics and in the alternative +PMI-based metric. Therefore, in a first experiment, +we analyze this in two pretrained word embeddings, +GloVe (Pennington et al., 2014) and Word2Vec +with SGNS (Mikolov et al., 2013). +We do this by studying the distribution of bias +in different bins of frequency of words in the vo- +cabulary. Bias is computed with equation 1 with +the female and male context words lists used in +Caliskan et al. (2017), and we refer to this as fe- +male bias or gender bias. For each frequency bin, +we also compute the ratio between the mean and +the sample standard deviation (SD). These are Co- +hen’s d effect sizes of the mean of each group under +the null hypothesis of absence of bias on average +(Cohen, 1988). Here we use it as a normalized mag- +nitude of the deviation of the distribution from zero. +We use this methodology to assess the association +between gender bias and frequency hereinafter. See +Appendix B for further details. +There is a clear association between gender +BiasWE with the pretrained embeddings and tar- +get word frequency (Figure 1). +GloVe embed- +dings present a monotonic relationship between +frequency and gender bias, such that the top 103.5 +words tend to have male bias with large effect sizes, +whereas less frequent words have mean female bias +with medium to large effect sizes. In the SGNS em- +beddings the effect is small and positive in less +frequent words, but in the top 100 words there is a +large shift towards male bias. +Even if there is literature which has studied the +relationship between frequency and word vectors +(see section 1), this result is still startling: a priori, +we wouldn’t expect the gender bias of words to +correlate so strongly with frequency, because word +similarity should be more closely related to seman- +tics and co-occurrences in the training corpus than +with the individual frequencies of words. +To validate this behavior, we train GloVe and +SGNS embeddings from scratch with the English +Wikipedia and study the association between gen- +der bias and word frequency. We compare this with +the results obtained with BiasPMI (equation 2). + +Figure 1: Female bias distribution vs. +words’ fre- +quency rank in pretrained GloVe (top panel) and +Word2Vec with SGNS (bottom panel). +Words are +grouped into bins according to their rank in a log-scale, +so that the most frequent words are in the leftmost bin +and the less frequent, in the rightmost. +We use fre- +quency ranks as raw frequencies are not available for +pretrained embeddings. Blue dots represent the means +and blue values are the effect sizes (mean to SD ratio). +The plots are not comparable in either axis because the +corpus, the vocabulary and the training methodology of +each set of embeddings are different. +2.1 +Comparing BiasWE with BiasPMI +Methods and data We measure the gender bias +of words in the vocabulary of the 2021 English +Wikipedia with BiasPMI and BiasWE and assess the +association with word frequency. We train SGNS +and GloVe vectors to compute BiasWE, whereas +the frequency counts from the corpus are used to +compute BiasPMI. Refer to appendices A and B for +details on the corpus and the methods, respectively. +Results The relation between BiasWE and fre- +quency in pretrained embeddings (Figure 1) holds +qualitatively when training embeddings from +scratch (top and middle panels in Figure 2). GloVe +embeddings yield a negative relationship between +female bias and frequency, while in SGNS we find +an average male bias with medium to large effect +sizes in high frequency words. +When using BiasPMI no frequency bin is +strongly biased on average (bottom panel in Figure +2). There is however a slight skew towards male +bias, such that all frequency ranges present small +negative effect sizes. Furthermore, the variability +of bias tends to increase as the frequency of target +words decreases: this behavior is attributable to +the fact that PMI is usually high and noisy in low +frequency words (Jurafsky and Martin, 2009). +This analysis is not enough to determine that +the effect of frequency on embedding-based bias +metrics is a spurious artifact generated by the em- +beddings. It still might be the case that higher +frequency words are actually more male-biased +than lower frequency words due to second-order or +higher associations, thus yielding plots like those +on the top and middle panels of Figure 2. We con- +duct the following study to assess this. +2.2 +The undesirable dependency on +frequency +Methods and data We create five randomly shuf- +fled, independent versions of the Wikipedia corpus +where tokens are randomly located across the text. +In these corpora words keep their frequency but +lose their context because co-occurrences are com- +pletely random. We estimate bias with BiasWE and +BiasPMI in each of the corpora and consider the +average of the five values as the gender bias of each +word. We analyze the relationship between the gen- +der bias metrics and frequency in this setting. By +shuffling the words in the corpus, contexts become +meaningless, thus any association found between +bias and frequency in this setting is explained only +by the frequencies of the words. We highlight that +it is problematic and undesirable for a metric to +detect biases in a setting where they do not exist. +Results In this controlled experimental setup, +BiasWE presents a strong association with target +word frequency (Figure 3): average male bias +grows as frequency increases for both SGNS and +GloVe, with large effect sizes from around frequen- +cies 104 onwards. Low frequency words present +female bias on average when measured with GloVe, +while they tend to have a slight male bias with small +effect sizes when using SGNS. +Conversely, measuring bias with PMI in the shuf- +fled corpora does not produce a clear dependence +on frequency. The average bias is roughly constant +for all frequencies, with small negative effect sizes; + +4.06 -3.12 -2.04 -159 -1.07 -0.51 0.03 0.53 0.99 +1.36 +0.1 +0.0 +-0.1 +103 +104.5 +5 +4.5 +3.5 +(10 +10° +(104 +(104 +0.10 +0.05 +0.00 +-0.05 +-0.10 +Lg01 +3.5 +4.51 +6.481 +10° +10° +10° +10° +(10 +(104 +(10)Figure 2: Female bias vs. +frequency in Wikipedia. +Bias is measured with BiasWE using GloVe (top panel), +BiasWE using SGNS (middle panel), and BiasPMI (bot- +tom panel). Words in the vocabulary are grouped in +bins according to their frequencies in log-scale. Blue +dots represent the means and blue values are the effect +sizes (mean to SD ratio). +that is, there is a slight skew towards male bias +across all frequencies. +3 +Discussion and Conclusion +In this work we revealed the existence of a spurious +frequency-based distortion in gender bias metrics +based on the cosine similarity between word em- +beddings. Both SGNS and GloVe-based gender +bias metrics tend to detect male bias in high fre- +quency words, while GloVe also yields female bias +on average in low frequency words. +To determine whether this effect is indeed an +undesirable artifact of the embedding-based metric +we assessed the relation between gender bias and +frequency in shuffled corpora, where words lose +Figure 3: +Female bias vs. +frequency in shuffled +Wikipedia. The bias of each word is computed as the +average of five estimates, one for each of five shuffles +performed. Words are grouped in bins according to +their frequencies. Blue dots represent the means and +blue values are the effect sizes (mean to SD ratio). +their context but keep their frequency. Results re- +veal that the dependence on frequency is caused by +the metric and does not originate from actual prop- +erties of the texts. This shows that popular gender +bias measurements can detect bias even when there +is none. Additionally, we found that an alternative +PMI-based bias metric does not show a clear depen- +dence on frequency, even though it shows a slight +tendency towards male bias. +According to these results, we consider the +PMI-based bias metric has an advantage over the +embedding-based metrics, which adds to the ad- +vantages of interpretability and hypothesis testing +(Valentini et al., 2021). However, as PMI captures +exclusively first-order associations and is unable to +capture synonyms, it may be required to include +several terms associated to the context words in + +0.1 +0.0 +0.1 +103 +10% +soL) +[102 +0.10 +08 -0.06 -0. +03 0.05 0.01 -0.16 -0 +66 +0.05 +0.00 +-0.05 +104 +106 +900) +103 +(103 +soL) +[102 +(104 +0.34 +0.23 +-021 +0.29 +0119 +2 +4 +105.5, +104 +4.51 +.121 +104 +4.5 +5.5 +g0l) +3.5 +(10 +(10 +(10) +(100.050 +116 0.88 0.07 -1/12 -3/45 -527 -549 -5136 -282 +0.025 +0.000 +0.025 +-0.050 +4.51 +(103 +10 +[102 +0.31 -021 -0.23 -0.75 -2/33 -4/71 -5/11 -698 -17.84 +0.00 +0.03 +0.06 +104 +103 +sol) +[102 +(103 +(104 +0.50 +0.25 +0.00 +-0.25 +-0.50 +121 +3.51 +4.5 +10 +10° +2.5 +3.5 +(10 +(10°order to measure some biases. +Male nouns and pronouns are usually more fre- +quent than female ones in large corpora (Twenge +et al., 2012; Gálvez et al., 2019). For example, +in the Wikipedia corpus, he appears 11.8 million +times, while the frequency of she is 3.5 million (re- +fer to Appendix B for the frequencies of the other +gendered context words). +The disparity in frequencies of male and female +contexts is a type of bias in itself and can be mea- +sured by counting word occurrences. In contrast, +the bias we study refers to the stereotyped contexts +in which male and female entities are portrayed, +and should be independent of individual word fre- +quencies. +When words are shuffled, the biases associated +with the contexts of female and male context words +are eliminated, but the disparities in frequencies are +maintained. We propose that bias metrics capture +this disparity in frequencies of female and male +context words. In the case of the embedding-based +metric, this hypothesis is supported by the existing +evidence that embeddings encode word frequency +in addition to semantics. +We believe the random-shuffling experiment is +general enough to show that the frequency effect +would still exist with other word lists, types of bi- +ases and domains, as long as the frequencies of +the context words differ. This result is important +because the context words’ frequencies are disre- +garded when measuring biases with embeddings. +Our findings have important implications for +bias measurement applications, as they cast doubt +on the reliability of widely used bias metrics when +the frequencies of the words involved are very dif- +ferent. We believe that more effort should be put +into designing new bias detection methods that do +not suffer from this weakness. +Limitations +We use sets of context words typically used in +the gender bias literature. These words imply a +binary understanding of gender, excluding other +gender representations from the bias measurement. +Moreover, we focus exclusively on the English +Wikipedia corpus and do not apply methods on cor- +pora of other domains, which might yield different +distributions of gender bias. +We report results using default hyperparame- +ters. This intends to mimic the typical experimental +setup found in the Computational Social Science +literature. Hyperparameters are left at their default +values because there is no ground truth for biases, +i.e. there are no annotations indicating the level of +bias of words. +The studies conducted in this work can be +adapted to other languages, other biases and other +corpora. We hope further research can assess the +frequency-based distortion in these settings as well +as the influence of hyperparameter choices. +References +Osman Aka, Ken Burke, Alex Bauerle, Christina Greer, +and Margaret Mitchell. 2021. Measuring model bi- +ases in the absence of ground truth. In Proceedings +of the 2021 AAAI/ACM Conference on AI, Ethics, +and Society. ACM. +Aylin +Caliskan, +Joanna +J. +Bryson, +and +Arvind +Narayanan. 2017. Semantics derived automatically +from language corpora contain human-like biases. +Science, 356(6334):183–186. +Tessa ES Charlesworth, Victor Yang, Thomas C Mann, +Benedek Kurdi, and Mahzarin R Banaji. 2021. Gen- +der stereotypes in natural language: Word embed- +dings show robust consistency across child and adult +language corpora of more than 65 million words. +Psychological Science, 32(2):218–240. +Jacob Cohen. 1988. Statistical power analysis for the +behavioral sciences. Routledge. +Ramiro H. Gálvez, Valeria Tiffenberg, and Edgar Alt- +szyler. 2019. Half a century of stereotyping associa- +tions between gender and intellectual ability in films. +Sex Roles, 81(9):643–654. +Nikhil Garg, Londa Schiebinger, Dan Jurafsky, and +James Zou. 2018. +Word embeddings quantify +100 years of gender and ethnic stereotypes. +Pro- +ceedings of the National Academy of Sciences, +115(16):E3635–E3644. +Chengyue Gong, Di He, Xu Tan, Tao Qin, Liwei Wang, +and Tie-Yan Liu. 2018. Frage: Frequency-agnostic +word representation. In Advances in Neural Infor- +mation Processing Systems, volume 31. Curran As- +sociates, Inc. +Daniel Jurafsky and James H. Martin. 2009. Speech +and Language Processing (2nd Edition). Prentice- +Hall, Inc., USA. +Austin C. Kozlowski, Matt Taddy, and James A. Evans. +2019. The geometry of culture: Analyzing the mean- +ings of class through word embeddings. American +Sociological Review, 84(5):905–949. +Omer Levy, Yoav Goldberg, and Ido Dagan. 2015. Im- +proving distributional similarity with lessons learned +from word embeddings. Transactions of the Associ- +ation for Computational Linguistics, 3:211–225. + +Molly Lewis and Gary Lupyan. 2020. +Gender +stereotypes are reflected in the distributional struc- +ture of 25 languages. +Nature Human Behaviour, +4(10):1021–1028. +Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Cor- +rado, and Jeff Dean. 2013. Distributed representa- +tions of words and phrases and their compositional- +ity. In Advances in Neural Information Processing +Systems, volume 26. Curran Associates, Inc. +Jiaqi Mu and Pramod Viswanath. 2018. +All-but-the- +top: Simple and effective postprocessing for word +representations. +In International Conference on +Learning Representations. +Jeffrey Pennington, Richard Socher, and Christopher D. +Manning. 2014. Glove: Global vectors for word rep- +resentation. In Empirical Methods in Natural Lan- +guage Processing (EMNLP), pages 1532–1543. +Radim ˇReh˚uˇrek and Petr Sojka. 2010. +Software +Framework for Topic Modelling with Large Cor- +pora. In Proceedings of the LREC 2010 Workshop +on New Challenges for NLP Frameworks, pages 45– +50, Valletta, Malta. ELRA. http://is.muni.cz/ +publication/884893/en. +Tobias Schnabel, Igor Labutov, David Mimno, and +Thorsten Joachims. 2015. Evaluation methods for +unsupervised word embeddings. +In Proceedings +of the 2015 Conference on Empirical Methods in +Natural Language Processing, pages 298–307, Lis- +bon, Portugal. Association for Computational Lin- +guistics. +Jean M Twenge, W Keith Campbell, and Brittany Gen- +tile. 2012. +Male and female pronoun use in us +books reflects women’s status, 1900–2008. +Sex +roles, 67(9):488–493. +Francisco Valentini, Germán Rosati, Damián Blasi, +Diego Fernandez Slezak, and Edgar Altszyler. 2021. +On the interpretation and significance of bias met- +rics in texts: a pmi-based approach. arXiv preprint +arXiv:2104.06474. +Benjamin J Wilson and Adriaan MJ Schakel. 2015. +Controlled experiments for word embeddings. arXiv +preprint arXiv:1510.02675. +A +Corpus +We use the April 2021 Wikipedia dump2 and re- +move articles with less than 50 tokens. We remove +non-alpha-numeric symbols and apply sentence +splitting. The corpus contains around 1.7 billion to- +kens and 78.1 million documents (sentences) after +pre-processing. +2https://archive.org/download/enwiki-20210401 +B +Methods +We measure female vs. male gender using gendered +nouns and pronouns (Caliskan et al., 2017; Lewis +and Lupyan, 2020), namely, A={female, woman, +girl, sister, she, her, hers, daughter} and B={male, +man, boy, brother, he, him, his, son}. +Tables 1 and 2 display the frequency of each of +these words in the pre-processed Wikipedia corpus. +Word +Frequency +her +3,720,408 +she +3,517,570 +daughter +294,043 +female +282,159 +woman +236,954 +sister +179,511 +girl +141,616 +hers +5,706 +Table 1: Frequencies of female context words in the +Wikipedia corpus +Word +Frequency +he +11,815,189 +his +9,603,118 +him +1,811,552 +son +541,828 +man +443,881 +brother +287,544 +male +181,471 +boy +124,326 +Table 2: Frequencies of male context words in the +Wikipedia corpus +We exclude words with fewer than 100 oc- +currences, which yields a vocabulary of 222,144 +words. Table 3 displays the distribution of these +words according to their frequencies, excluding the +female and male context words. + +Frequency +# words +[102, 102.5] +116,340 +(102.5, 103] +54,187 +(103, 103.5] +26,617 +(103.5, 104] +13,144 +(104, 104.5] +6,579 +(104.5, 105] +3,255 +(105, 105.5] +1,448 +(105.5, 106] +441 +(106, 108.12] +117 +Table 3: Number of words in each frequency range in +the Wikipedia corpus +In section 2 we use pretrained GloVe embed- +dings trained on Wikipedia 2014 and Gigaword 5 +(Pennington et al., 2014), and Word2Vec SGNS em- +beddings trained on Google News (Mikolov et al., +2013), both with 300 dimensions. +All methods employed in sections 2.1 and 2.2 +(GloVe, SGNS and PMI) use a window size of 10 +and remove out-of-vocabulary tokens before the +corpus is processed into word-context pairs (Levy +et al., 2015). +For SGNS we use the Word2Vec implementa- +tion available in the Gensim library ( ˇReh˚uˇrek and +Sojka, 2010) with default hyperparameters. GloVe +is trained with Pennington et al. (2014)’s imple- +mentation with 100 iterations. +For PMI, we count co-occurrences with the +GloVe module (Pennington et al., 2014) and set +the smoothing parameter ϵ to 0.01, so that it can +be computed whenever there are no co-occurrences +between the target word and any of the context +words. +All computations were performed on a desktop +machine with 4 cores Intel Core i5-4460 CPU @ +3.20GHz and 32 GB RAM. Training took around +30 minutes per iteration with GloVe and 2 hours +per epoch with SGNS. + diff --git a/D9AyT4oBgHgl3EQf4vqB/content/tmp_files/load_file.txt b/D9AyT4oBgHgl3EQf4vqB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..67d2e47f1f36732cb802b9ec3d555f11ad0057e7 --- /dev/null +++ b/D9AyT4oBgHgl3EQf4vqB/content/tmp_files/load_file.txt @@ -0,0 +1,386 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf,len=385 +page_content='The Undesirable Dependence on Frequency of Gender Bias Metrics Based on Word Embeddings Francisco Valentini ICC (UBA-CONICET) Maestría en Data Mining (UBA) Buenos Aires, Argentina ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='valentini@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='com Germán Rosati Escuela IDAES (UNSAM) Buenos Aires, Argentina grosati@unsam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='ar Diego Fernandez Slezak ICC (UBA-CONICET) Buenos Aires, Argentina dfslezak@dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='uba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='ar Edgar Altszyler ICC (UBA-CONICET) Maestría en Data Mining (UBA) Buenos Aires, Argentina ealtszyler@dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='uba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='ar Abstract Numerous works use word embedding-based metrics to quantify societal biases and stereo- types in texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Recent studies have found that word embeddings can capture semantic similarity but may be affected by word fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' In this work we study the effect of frequency when measuring female vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' male gender bias with word embedding-based bias quantification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We find that Skip- gram with negative sampling and GloVe tend to detect male bias in high frequency words, while GloVe tends to return female bias in low frequency words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We show these behaviors still exist when words are randomly shuffled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' This proves that the frequency-based effect ob- served in unshuffled corpora stems from prop- erties of the metric rather than from word as- sociations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' The effect is spurious and prob- lematic since bias metrics should depend ex- clusively on word co-occurrences and not indi- vidual word frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Finally, we compare these results with the ones obtained with an al- ternative metric based on Pointwise Mutual In- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We find that this metric does not show a clear dependence on frequency, even though it is slightly skewed towards male bias across all frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' 1 Introduction Word embeddings are one of the most commonly used techniques to measure semantic closeness be- tween words in a corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' In recent years, they have been widely used in Computational Social Science applications to measure societal biases and stereotypes (Caliskan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Kozlowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Lewis and Lupyan, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Charlesworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' For practical purposes, we consider bias to be the degree to which the language used to describe groups or things is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Bias is typically mea- sured by computing the difference between the mean similarity of words of two context groups A and B with respect to a target word x: BiasWE = mean a∈A cos(vx, va) − mean b∈B cos(vx, vb), (1) where vi is the word embedding of word i and cos(vi, vj) is the cosine similarity between vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Gender bias has long been one of the most stud- ied biases with this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' In this context, A and B are usually defined as gendered nouns and pro- nouns (Caliskan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Lewis and Lupyan, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' A representative example is Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' (2018), where the female vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' male bias of pro- fessions in historical corpora is found to correlate with the percentage of women employed in each profession over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Not as widely used as word embeddings, Point- wise Mutual Information (PMI) is a metric of word similarity that can also be used to study biases (Gálvez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Aka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Valentini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Valentini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' (2021) define the PMI- based bias metric as BiasPMI = PMI(x, A) − PMI(x, B), where PMI(x, Y ) = log P(x, Y ) P(x)P(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' P(x, Y ) is the probability of co-occurrence be- tween the word x with any one in Y in a window of a predefined number of words, and P(x) and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='00792v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='CL] 2 Jan 2023 P(Y ) are the probability of occurrence of the word x and any word in Y , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Valentini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' (2021) show that BiasPMI can be expressed as BiasPMI = log P(x|A) P(x|B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' (2) That is, BiasPMI can be interpreted as how much more likely it is to find words in x in the context of words in A than in the context of words in B, in a log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Thus, it captures exclusively first-order associations and can be computed via maximum likelihood using co-occurrence counts from the text (Valentini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Some recent works have studied the relationship between word frequencies and word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' In particular, embeddings seem to encode word fre- quency even after normalization (Schnabel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2015), vector norm depends on word frequency (Wilson and Schakel, 2015), top principal compo- nent directions encode frequency in different ways (Mu and Viswanath, 2018) and vectors of high- frequency and low-frequency words lie in different regions of the embedding space (Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' These studies are nevertheless inconclusive in the sense that they do not determine clearly to what extent the association observed is caused by actual attributes of corpora or by undesirable properties of embedding training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Hence, an answer to the origin of the relation between embeddings and frequency is yet to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' What is more, the repercussions of this effect in applications relevant to Computa- tional Social Sciences such as bias quantification have not yet been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We make three main contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' First, we show that frequency has an association with gen- der bias when measured with word embedding- based metrics: both Skip-gram with negative sam- pling (SGNS) and GloVe-based bias metrics tend to detect male bias in high frequency words, while GloVe also yields female bias on average in low- frequency words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Second, we show that the de- pendence of the embedding-based gender bias on frequency holds when tokens in the corpus are randomly shuffled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' This proves that the depen- dence on frequency is an artifact of the metric it- self i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' that embedding-based bias metrics can encode frequency spuriously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Third, we find that the PMI-based gender bias metric does not present this frequency-based effect but is slightly skewed towards male bias across all frequency ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='1 1Code for the paper is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='com/ ftvalentini/EmbeddingsBiasFrequency Our analyses are restricted to the English lan- guage and are based on a binary understanding of gender (see Limitations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' 2 The effect of frequency on gender bias Our objective is to study the association between gender bias and frequency in the widely used embedding-based metrics and in the alternative PMI-based metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Therefore, in a first experiment, we analyze this in two pretrained word embeddings, GloVe (Pennington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2014) and Word2Vec with SGNS (Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We do this by studying the distribution of bias in different bins of frequency of words in the vo- cabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Bias is computed with equation 1 with the female and male context words lists used in Caliskan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' (2017), and we refer to this as fe- male bias or gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' For each frequency bin, we also compute the ratio between the mean and the sample standard deviation (SD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' These are Co- hen’s d effect sizes of the mean of each group under the null hypothesis of absence of bias on average (Cohen, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Here we use it as a normalized mag- nitude of the deviation of the distribution from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We use this methodology to assess the association between gender bias and frequency hereinafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' See Appendix B for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' There is a clear association between gender BiasWE with the pretrained embeddings and tar- get word frequency (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' GloVe embed- dings present a monotonic relationship between frequency and gender bias, such that the top 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5 words tend to have male bias with large effect sizes, whereas less frequent words have mean female bias with medium to large effect sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' In the SGNS em- beddings the effect is small and positive in less frequent words, but in the top 100 words there is a large shift towards male bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Even if there is literature which has studied the relationship between frequency and word vectors (see section 1), this result is still startling: a priori, we wouldn’t expect the gender bias of words to correlate so strongly with frequency, because word similarity should be more closely related to seman- tics and co-occurrences in the training corpus than with the individual frequencies of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' To validate this behavior, we train GloVe and SGNS embeddings from scratch with the English Wikipedia and study the association between gen- der bias and word frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We compare this with the results obtained with BiasPMI (equation 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Figure 1: Female bias distribution vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' words’ fre- quency rank in pretrained GloVe (top panel) and Word2Vec with SGNS (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Words are grouped into bins according to their rank in a log-scale, so that the most frequent words are in the leftmost bin and the less frequent, in the rightmost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We use fre- quency ranks as raw frequencies are not available for pretrained embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Blue dots represent the means and blue values are the effect sizes (mean to SD ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' The plots are not comparable in either axis because the corpus, the vocabulary and the training methodology of each set of embeddings are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='1 Comparing BiasWE with BiasPMI Methods and data We measure the gender bias of words in the vocabulary of the 2021 English Wikipedia with BiasPMI and BiasWE and assess the association with word frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We train SGNS and GloVe vectors to compute BiasWE, whereas the frequency counts from the corpus are used to compute BiasPMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Refer to appendices A and B for details on the corpus and the methods, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Results The relation between BiasWE and fre- quency in pretrained embeddings (Figure 1) holds qualitatively when training embeddings from scratch (top and middle panels in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' GloVe embeddings yield a negative relationship between female bias and frequency, while in SGNS we find an average male bias with medium to large effect sizes in high frequency words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' When using BiasPMI no frequency bin is strongly biased on average (bottom panel in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' There is however a slight skew towards male bias, such that all frequency ranges present small negative effect sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Furthermore, the variability of bias tends to increase as the frequency of target words decreases: this behavior is attributable to the fact that PMI is usually high and noisy in low frequency words (Jurafsky and Martin, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' This analysis is not enough to determine that the effect of frequency on embedding-based bias metrics is a spurious artifact generated by the em- beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' It still might be the case that higher frequency words are actually more male-biased than lower frequency words due to second-order or higher associations, thus yielding plots like those on the top and middle panels of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We con- duct the following study to assess this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='2 The undesirable dependency on frequency Methods and data We create five randomly shuf- fled, independent versions of the Wikipedia corpus where tokens are randomly located across the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' In these corpora words keep their frequency but lose their context because co-occurrences are com- pletely random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We estimate bias with BiasWE and BiasPMI in each of the corpora and consider the average of the five values as the gender bias of each word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We analyze the relationship between the gen- der bias metrics and frequency in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' By shuffling the words in the corpus, contexts become meaningless, thus any association found between bias and frequency in this setting is explained only by the frequencies of the words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We highlight that it is problematic and undesirable for a metric to detect biases in a setting where they do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Results In this controlled experimental setup, BiasWE presents a strong association with target word frequency (Figure 3): average male bias grows as frequency increases for both SGNS and GloVe, with large effect sizes from around frequen- cies 104 onwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Low frequency words present female bias on average when measured with GloVe, while they tend to have a slight male bias with small effect sizes when using SGNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Conversely, measuring bias with PMI in the shuf- fled corpora does not produce a clear dependence on frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' The average bias is roughly constant for all frequencies, with small negative effect sizes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='06 -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='12 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='04 -159 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='07 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='1 103 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5 (10 10° (104 (104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='10 Lg01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='51 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='481 10° 10° 10° 10° (10 (104 (10)Figure 2: Female bias vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' frequency in Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Bias is measured with BiasWE using GloVe (top panel), BiasWE using SGNS (middle panel), and BiasPMI (bot- tom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Words in the vocabulary are grouped in bins according to their frequencies in log-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Blue dots represent the means and blue values are the effect sizes (mean to SD ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' that is, there is a slight skew towards male bias across all frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' 3 Discussion and Conclusion In this work we revealed the existence of a spurious frequency-based distortion in gender bias metrics based on the cosine similarity between word em- beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Both SGNS and GloVe-based gender bias metrics tend to detect male bias in high fre- quency words, while GloVe also yields female bias on average in low frequency words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' To determine whether this effect is indeed an undesirable artifact of the embedding-based metric we assessed the relation between gender bias and frequency in shuffled corpora, where words lose Figure 3: Female bias vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' frequency in shuffled Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' The bias of each word is computed as the average of five estimates, one for each of five shuffles performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Words are grouped in bins according to their frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Blue dots represent the means and blue values are the effect sizes (mean to SD ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' their context but keep their frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Results re- veal that the dependence on frequency is caused by the metric and does not originate from actual prop- erties of the texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' This shows that popular gender bias measurements can detect bias even when there is none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Additionally, we found that an alternative PMI-based bias metric does not show a clear depen- dence on frequency, even though it shows a slight tendency towards male bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' According to these results, we consider the PMI-based bias metric has an advantage over the embedding-based metrics, which adds to the ad- vantages of interpretability and hypothesis testing (Valentini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' However, as PMI captures exclusively first-order associations and is unable to capture synonyms, it may be required to include several terms associated to the context words in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='1 0.' metadata={'source': 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4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5 10 10° 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5 (10 (10°order to measure some biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Male nouns and pronouns are usually more fre- quent than female ones in large corpora (Twenge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Gálvez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' For example, in the Wikipedia corpus, he appears 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='8 million times, while the frequency of she is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5 million (re- fer to Appendix B for the frequencies of the other gendered context words).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' The disparity in frequencies of male and female contexts is a type of bias in itself and can be mea- sured by counting word occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' In contrast, the bias we study refers to the stereotyped contexts in which male and female entities are portrayed, and should be independent of individual word fre- quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' When words are shuffled, the biases associated with the contexts of female and male context words are eliminated, but the disparities in frequencies are maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We propose that bias metrics capture this disparity in frequencies of female and male context words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' In the case of the embedding-based metric, this hypothesis is supported by the existing evidence that embeddings encode word frequency in addition to semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We believe the random-shuffling experiment is general enough to show that the frequency effect would still exist with other word lists, types of bi- ases and domains, as long as the frequencies of the context words differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' This result is important because the context words’ frequencies are disre- garded when measuring biases with embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Our findings have important implications for bias measurement applications, as they cast doubt on the reliability of widely used bias metrics when the frequencies of the words involved are very dif- ferent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We believe that more effort should be put into designing new bias detection methods that do not suffer from this weakness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Limitations We use sets of context words typically used in the gender bias literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' These words imply a binary understanding of gender, excluding other gender representations from the bias measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Moreover, we focus exclusively on the English Wikipedia corpus and do not apply methods on cor- pora of other domains, which might yield different distributions of gender bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We report results using default hyperparame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' This intends to mimic the typical experimental setup found in the Computational Social Science literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Hyperparameters are left at their default values because there is no ground truth for biases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' there are no annotations indicating the level of bias of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' The studies conducted in this work can be adapted to other languages, other biases and other corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We hope further research can assess the frequency-based distortion in these settings as well as the influence of hyperparameter choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' References Osman Aka, Ken Burke, Alex Bauerle, Christina Greer, and Margaret Mitchell.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' On the interpretation and significance of bias met- rics in texts: a pmi-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='06474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Benjamin J Wilson and Adriaan MJ Schakel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Controlled experiments for word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' arXiv preprint arXiv:1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='02675.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' A Corpus We use the April 2021 Wikipedia dump2 and re- move articles with less than 50 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' We remove non-alpha-numeric symbols and apply sentence splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' The corpus contains around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='7 billion to- kens and 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='1 million documents (sentences) after pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' 2https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='org/download/enwiki-20210401 B Methods We measure female vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' male gender using gendered nouns and pronouns (Caliskan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Lewis and Lupyan, 2020), namely, A={female, woman, girl, sister, she, her, hers, daughter} and B={male, man, boy, brother, he, him, his, son}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Tables 1 and 2 display the frequency of each of these words in the pre-processed Wikipedia corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Word Frequency her 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='408 she 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='517,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='570 daughter 294,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='043 female 282,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='159 woman 236,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='954 sister 179,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='511 girl 141,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='616 hers 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='706 Table 1: Frequencies of female context words in the Wikipedia corpus Word Frequency he 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='815,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='189 his 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='603,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='118 him 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='811,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='552 son 541,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='828 man 443,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='881 brother 287,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='544 male 181,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='471 boy 124,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='326 Table 2: Frequencies of male context words in the Wikipedia corpus We exclude words with fewer than 100 oc- currences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' which yields a vocabulary of 222,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='144 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Table 3 displays the distribution of these words according to their frequencies, excluding the female and male context words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Frequency # words [102, 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5] 116,340 (102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5, 103] 54,187 (103, 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5] 26,617 (103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5, 104] 13,144 (104, 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5] 6,579 (104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5, 105] 3,255 (105, 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5] 1,448 (105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='5, 106] 441 (106, 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='12] 117 Table 3: Number of words in each frequency range in the Wikipedia corpus In section 2 we use pretrained GloVe embed- dings trained on Wikipedia 2014 and Gigaword 5 (Pennington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2014), and Word2Vec SGNS em- beddings trained on Google News (Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2013), both with 300 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' All methods employed in sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='2 (GloVe, SGNS and PMI) use a window size of 10 and remove out-of-vocabulary tokens before the corpus is processed into word-context pairs (Levy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' For SGNS we use the Word2Vec implementa- tion available in the Gensim library ( ˇReh˚uˇrek and Sojka, 2010) with default hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' GloVe is trained with Pennington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' (2014)’s imple- mentation with 100 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' For PMI, we count co-occurrences with the GloVe module (Pennington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=', 2014) and set the smoothing parameter ϵ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='01, so that it can be computed whenever there are no co-occurrences between the target word and any of the context words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' All computations were performed on a desktop machine with 4 cores Intel Core i5-4460 CPU @ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content='20GHz and 32 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} +page_content=' Training took around 30 minutes per iteration with GloVe and 2 hours per epoch with SGNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQf4vqB/content/2301.00792v1.pdf'} diff --git a/DdE2T4oBgHgl3EQf9gki/content/tmp_files/2301.04228v1.pdf.txt b/DdE2T4oBgHgl3EQf9gki/content/tmp_files/2301.04228v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d76ac9b0a58132847415073661286c5433877f9 --- /dev/null +++ b/DdE2T4oBgHgl3EQf9gki/content/tmp_files/2301.04228v1.pdf.txt @@ -0,0 +1,1813 @@ +Harvesting L2 Caches in Server Processors +Majid Jalili, and Mattan Erez +The University of Texas at Austin +{majid,mattan.erez}@utexas.edu +Abstract—We make three observations in modern processors: +(1) LLC capacity is getting larger (up to 1GB); (2) core counts +are increasing (up to 128 cores), accumulating a more significant +amount of private L2 cache capacity on the chip; and (3) overall +processor utilization in the cloud remains very low despite many +efforts, leaving many large private caches unused. To enable +better use of these beefy processors, we propose to open up a +logical path for LLC evictions to unused private caches. In other +words, instead of writing LLC evictions back to slow and busy +main memory, we send some of them that are still alive up to idle +L2 caches to avoid unnecessary long and costly main memory. +Our scheme takes the importance of applications (user-facing +vs. background), and system load into account to provide each +application with a fair share of idle resources. Our results show +that we can improve system performance by up to 2× (geomean +of 10%) for single-application runs. Also, for mixes of user-facing +and background jobs, our scheme improves the P99 latency of +user-facing tasks by up to 32% (geomean of 15%), and the IPC +of background jobs by up to 50% (geomean of 10%). +I. INTRODUCTION +CPU manufacturers are increasing the L2/LLC sizes and +number of cores to respond to the ever-increasing demand +for computation. For instance, comparing two high-end Intel +processors (Xeon Platinum 8180 and Xeon Platinum 8380), we +observe that total cache capacity has increased from 66.5MB to +110MB. AMD processors also follow a similar trends: EPYC- +7773X has 800MB on-chip memory with 64 cores. At the +same time, these CPUs are operating at very low utilization, +40% at Azure [10], and 20-50% at Alibaba [14]. This shift +toward deeper and larger caches combined with low utilization +opens opportunities for novel cache management mechanisms. +We propose L2 Harvester (L2H), a completely software- +transparent scheme built on top of the current rigid memory +hierarchy that harvests idle cache resources, reducing the +average load latency by up to 30%. L2H moves a fraction of +LLC capacity/conflict evictions to unused private L2 caches +instead of writing them back to main memory. Thus, later +L2 misses can find data in other L2 caches, reducing off- +chip transactions. L2H has the benefits of the classic memory +hierarchy such as software transparency, simplicity of design, +and isolation, while increasing cache utilization. +Cache underutilization is prevalent because the main con- +sumers of server processors are public clouds, where core +utilization still remains low in spite of numerous efforts [6], +[10], [12], [22], [38], [41]. The low core utilization leaves +many large private L2 caches unused for the majority of the +time. For instance, for a typical 64-core CPU with 1MB/core +L2 caches, if the utilization is around 50%, then 32MB +cache capacity is wasted just in one CPU. Given that there +are millions of servers running in public clouds including +Azure, AWS, Alibaba, and Google, this huge underutilization +translates to gigabytes of scarce on-chip cache wastage that +could have otherwise been used to reduce memory access +latency. +The prior solution to increasing cache utilization is to break +the classic cache hierarchy and fully or partially flatten it by +allocating cache space based on application demands. Jenga +[36] proposes a reconfigurable cache where flat, distributed, +and heterogeneous cache banks are controlled and managed +by hardware and run-time. Flattening the cache allows Jenga +to use all cache space, but this increases the complexity of +the memory sub-system. Similarly, IBM Z16 [3] offers a 4- +level cache hierarchy where L2 and L3 can be reconfigured +to be part of an L2 cache. Z16 is shipped with Processor +Resource/Systems Manager that decides how to use these large +caches. In both cases, the memory hierarchy is no longer +software transparent and adds huge complexity to current +designs. +Instead in L2H, on LLC evictions, we predict if the block +is not dead and will be accessed soon. If so, the block is sent +to a load balancer to decide if there is any idle core. Upon +finding an available L2 cache, the block is written to the lender +L2 cache, and the snoop filter metadata is updated as normal. +Later, if a request to this block is received, the lender cache +responds and satisfies the request. Note that L2H relies on +the current coherence mechanism to locate the cache block, +and does not need any special support from the hardware or +runtime system, thus it is superior to designs such as Jenga +[36] and IBM Z16 [3]. +L2H utilizes two lightweight predictors to find the dead +blocks: (1) a bloom filter-based predictor that tracks recently +evicted addresses, identifying those misses that could have +been avoided with a larger cache; and (2) MPPP [18]: a +perceptron-based dead block predictor that combines different +features such as address and program counter to predict if a +block has exhausted its useful lifetime. The two predictors +complement each other: MPPP covers the bloom filter when +it is not warmed up, and the bloom filter makes up for the +MPPP sensitivity to thresholds when the system load is high. +We consult both predictors and make our decision based +on a simple algorithm: if the load in the system is high, both +predictors should agree if a block is not dead in order for the +block to be sent up to a private L2 cache. If the load is low, +the block is sent up if either predicts the block is not dead. +We take into account the importance and criticality of appli- +cations being run to give them a fair share of private L2 caches. +1 +arXiv:2301.04228v1 [cs.AR] 10 Jan 2023 + +TABLE I +INTEL AND AMD CPU GENERATIONS. +Intel (2016-2020) +AMD (2017-2022) +SKX +CSX +ICX +Rome +Milan(X) +Genoa +L1 +32KB +32KB +48KB +32KB +32KB +32KB +L2/core +1MB +1MB +1.25MB +512KB +512KB +1MB +L3/core +1.37MB +1.37MB +1.5MB +4-8MB +4-12MB +4-16MB +Cores +4-28 +2-56 +8-40 +8-64 +8-64 +8-96 +Total +66.5MB +133MB +110MB +288MB +800MB +1100MB +SKU +Xeon-P8180 +Xeon-P9282 +Xeon-P8380 +EPYC-7H12 +EPYC-7773X +N/A +User-facing applications maximally use the extra cache space +as they have the highest priority in the system. Background +jobs can also get extra space if the load balancer detects that +the user-facing applications are not cache-sensitive, and can +yield the extra space. +We evaluate L2H under different utilization scenarios. First, +when the CPU load is very low (<25%) and running one +application. This allows the application to take up all private +L2 caches in the system, representing the upper-bound benefit +of L2H. Then, we move to more complex scenarios where a +mix of critical and background jobs are run. L2H must make +decisions regarding what blocks are dead and how to split the +private L2 caches. +We implement L2H in gem5 [23] and run applications from +different domains (datacenter, scientific, and graph analytics). +Our experimental result shows that for a single application +with multiple lenders, we improve P99 latency by 2×. Also, +for mixes of user-facing and background jobs, L2H improves +P99 and throughput by up to 32% and 50%. +To summarize our main contributions: +• We demonstrate that a substantial amount of cache ca- +pacity is wasted in modern processors due to a rigid +hierarchal design, and conservative resource allocation in +the cloud. +• We architect and evaluate an effective, yet low-cost L2 +harvesting mechanism that enables a logical path from +LLC evictions to private L2 caches. This allows the idle +cores to lend their unused L2 caches, thus keeping more +data blocks on the chip. +• We incorporate two dead block prediction schemes in +the L2 harvester to identify those capacity/conflict-caused +evictions that are worth keeping on chip. We also devise +a simple load balancer that distributes data blocks over +unused resources by taking system load and criticality of +applications into the account. +• We evaluate our proposed method and compare it to a +conventional hierarchy with a larger LLC. Our evaluation +results show that a quad-core system with 2MB/core +LLC and 1.25 MB/core L2 cache benefiting from L2H +improves system performance by up to 2× over the +baseline. Also, we show that L2H provides competitive +system performance compared to a baseline with a 50% +larger LLC (12MB). +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Fraction of Servers +Fraction of Cache Capacity Wasted +Utilization +CSX +ICX +Milan-X +Genoa +CDF Servers +Cortez, Eli et al. +Fig. 1. +Total cache wasted by different server processors under various +utilization. The CDF (red line) is taken from [10]. +TABLE II +TOTAL L2 CACHE CAPACITY OF 3 SUPERCOMPUTERS IN THE TACC +DATACENTER. +Systems +Nodes +Processor +Core/node +Total L2 (GB) +Frontera [2] +8008 +Xeon 8280 +56 +438 +Lonestar6 [4] +560 +EPYC 7763 +128 +35 +Chamealon [1] +10000 +Haswell +96 +469 +II. MOTIVATION +According to Microsoft Azure and Alibaba, datacenter core +utilization is very low. Servers run at 40% or lower utilization +at 90% of the time at Azure [10], and between 20%-50% most +of the time at Alibaba [14]. This over-allocation stems from +the fact that VMs should have enough cores and resources if +the load surges rapidly. +In addition, CPU manufacturers are increasing the L2/LLC +sizes and the number of cores. Table I exhibits three gener- +ations of Intel and AMD server CPUs. We can observe that +both manufacturers’ L2/LLC and core counts have steadily +increased over generations. L2 and LLC sizes are reaching +1.25MB/core, 2MB/core for Intel processors, and 1MB/core +and 4MB/core for AMD processors. Combined with the fact +that core counts are also increasing, we can see that the third +generation of Intel processors are accumulating 110MB total +cache capacity, while AMD is reaching over giga bytes of +on-chip cache storage. +To better understand the current situation in datacenters, +Figure 1 shows the total cache capacity wasted by different +server processors under various utilization levels. On the x- +axis, we show the utilization. We assume that all processors +have 32 cores. Thus, the minimum utilization is when there is +one application running taking one core and the whole LLC +( 1 +32 = 0.03), and maximum utilization is when all 32 cores +are active ( 32 +32 = 1). To calculate the total cache wasted (first +y-axis), we subtract the used cache capacity under each load +from the total cache capacity available on the chip (32 ×(L1+ +L2)+LLC). For example, if there are two cores running, and +L1=48KB, L2=1MB/core, and LLC=8MB, then wasted cache +is 32×(48KB+1MB)+8MB−2×(48KB+1MB)−8MB. +On the second y-axis, we show the CDF of core utilization on +Azure [10]. +As can be seen from Figure 1, 50% of Azure Icelake +machines waste around 40% of the total cache capacity. Given +2 + +0 +10 +20 +30 +40 +50 +50 +150 +250 +350 +450 +P95 (ms) +QPS +moses +2MB +4MB +8MB +16MB +22MB +0 +1 +2 +3 +4 +2 +6 +10 +14 +18 +22 +Time/iteration (sec) +LLC Size (MB) +PageRank +u20 +u21 +g20 +g21 +0 +100 +200 +300 +400 +500 +600 +700 +2MB +4MB +8MB +16MB +20MB +22MB +Time (sec) +LLC Size (MB) +520.omnetpp +Fig. 2. Impact of LLC size on applications performance. +that Icelake machines have an L2 capacity of 1.25MB/core +(see Table I), for 32 cores, around 35MB of total on-chip cache +capacity is wasted that could otherwise be used to keep data +blocks on the chip and boost up system performance. AMD +processors also suffer from similar issues but at smaller scale. +For instance, Rome wastes around 10% of cache capacity +under the load of 40%. The main reason is that AMD has +smaller L2/core, and very large L3/core capacity. However, in +Genoa, we observe that AMD is enlarging the L2/core from +512KB/core in Milan to 1MB/core. +To put L2 cache waste into perspective, Table II shows 3 +supercomputers in the TACC datacenter (Frontera, Lonestart6, +and Chameleon). The table shows the main processor types as +well as the number of nodes and total L2 capacity (GB). As +can be seen, the total cache capacity in a small-scale datacenter +like TACC can be somewhere between 35GB (Lonestart6) to +469GB (Chameleon). Hence, if the utilization is around 50% +on average, a substantial amount of a very scarce resource +like L2 cache is being wasted (234.5GB in Chameleon and +17.5GB in Lonestart6). Note that public clouds such as AWS, +Azure, Google, and Alibaba are operating significantly larger +datacenters, so we are projecting the L2 waste reaches to +terabytes. +A larger cache capacity can help reduce the long memory +access latency. We conduct a cache study on a real machine +to measure how much cache capacity impacts system perfor- +mance. The machine is an Intel(R) Xeon(R) Gold 6242 CPU +with 22MB 11-way LLC cache. We run one application and +change the cache size using Intel Cache Allocation Technology +(CAT) from one way (2MB) to 11 ways (22MB). We set the +core frequency to 3.9GHz. +Figure 2 shows the performnace for three applications: +(1) moses from TailBench [19], where we sweep the system +load in terms of query per second (qps) and cache size; (2) +PageRank from gapbs [7] with 4 different synthetic inputs +(u: uniform graph, and g: Kronecker graph), and two different +sizes (20, and 21); and (3) 520.omnetpp from SPEC CPU 2017 +[5]. +For moses we make two observations: (1) with larger caches, +the saturation point (point that P95 increases sharply) is pushed +to higher qps (further to the right). For example, we can see +that the knee point for 22MB occurs at 450, while for the 2M, +the server is saturated at qps=300; 1.5× improvement in the +maximum load; (2) at similar loads before the saturation point +the larger caches provides better P95 latency. For instance, +when qps=250, we see that 2MB LLC provides P95 of 12ms, +while the 22MB cache shows P95 of 8ms. +For PageRank we observe that a larger LLC reduces the +execution time significantly. For example, for the largest graph +(g21) the execution time is halved when increasing the LLC +size from 2MB to 12MB. We also see that for LLC sizes +of greater than 12MB, the execution times remain fixed. +Finally, for 520.omnetpp we observe similar sensitivity to +cache size. the execution time constantly reduces from 610 +seconds for 2MB LLC, to 420 seconds for 22MB cache. Our +conclusion is that larger cache help applications from different +domains, thus wasting a huge amount of on-chip cache is not +reasonable, and we need to devise schemes to allow the unused +L2 caches to be utilized when possible. +III. L2 HARVESTER µARCHITECTURE +We propose L2H, a simple yet effective mechanism for har- +vesting L2 caches, that provides performance improvement for +memory-bound applications. In this section, we first overview +the design of L2H. Then, we discuss the algorithm behind +detecting the dead blocks, and how we distribute the blocks +over idle cores. +A. L2H: Overview and Organization +Figure 3 shows the overview of L2H. Without loss of +generality, we assume there are 4 cores connected to LLC +banks with a shared bus. LLC has MPPP dead block predictor +[18]. L2H sits between LLC and the memory controller and +tracks the writebacks. If a block is detected by the predictor +to be not dead, is sent to the load balancer. Then, the load +balancer decides where this block can be written to. If there +is any idle core that can lend its L2 cache, the load balancer +pushes the block up to the lender. Otherwise, if the block is +dead, or if there is no free L2, the block is written back to +main memory. Thus, in the next reference to this block, there +might be a private L2 cache that responds to the request and +thereby saves one off-chip transfer. +L2H needs four pieces of information to perform prediction +and load balancing: (1) L2 MPKIs; (2) Critical Task Map +(CTM): a bit mask that determines if the application being +run on a core is critical, “1” determines the application being +run at core n is critical. This bit mask is provided by the +user or system administrator and is updated as soon as a new +application is assigned to cores; (3) Idle Core Map (ICM): +a bit mask that determines if a core is idle and can lend its +L2 cache. This is updated by the cores if core has nothing +3 + +Harvester +Load +Balancer +Predictor +Eviction +MPKIs +Writeup +L2 +Memory Controller +Writeback +Dead/Alive? +Core 0 +Core 1 +Core 2 +Core 3 +Critical Task Map +LLC Bank 0 +LLC Bank 1 +LLC Bank 2 +LLC Bank 3 +LLC +C0 C1 C2 C3 +C0 C1 C2 C3 +Idle Core Map +MPPP +Dead/Alive? +Fig. 3. L2 harvester architecture. MPPP [18] is the state-of-the-art dead block +predictor. +to execute; and (4) The output of the MPPP [18] dead block +predictor. +B. L2H Structures +Predictor The purpose of the predictor is to determine if a +block is dead, and thus it is not worth keeping on chip. This +is particularly important for streaming applications because +redirecting all cache blocks to upper levels will waste power, +increase traffic, and elevate congestion on coherence. +Figure 4 shows the structure of our predictor. We combine +two predictors to find dead blocks: (1) a bloom filter-based +predictor; and (2) the multi-perspective perceptron predictor +(MPPP) [18]. The functionality of the bloom filter-based +predictor is simple. We insert the missed addresses into the +bloom filter. To make a prediction, we just need to look up the +address, if the address was not found in the filter, we conclude +the block is dead. Because we have not seen a reference to this +block recently. We reset the bloom filter periodically to make +sure the false positive rate stays low. Unfortunately, after each +reset, the bloom filter starts declaring all blocks dead as they +have not been seen, thus we need to address this shortcoming. +Morpheus [11] uses a bloom filter for hit/miss prediction, +and addresses this problem by using two separate bloom filters +with different reset intervals. So, when one of them is being +warmed up, the other one services the requests, and vice versa. +However, we found that we get better accuracy if we combine +our bloom filter with another type of dead block predictor (e.g., +perceptron-based dead block predictor). The two predictors +complement each other: MPPP covers the bloom filter when +it is not warmed up, and the bloom filter makes up for the +MPPP sensitivity to thresholds when the system load is high. +We use MPPP to solve the reset problem of the bloom filter. +MPPP [18] is a perceptron-based technique that predicts the +future reuse of cache blocks. MPPP combines several features +including program counter and address to form weight tables. +Then taking summations of entries from each table, it predicts +if a block is: (a) not dead, (b) dead on arrival, and can bypass +the cache, and (c) dead, and can be evicted from the cache. +MPPP uses three thresholds to make the prediction based on +the aggregated values taken from the weight tables. +Predictor +Bloom Filter +Block Address +From MPPP +Seen? +Dead? +Average L2 MPKI +if RC < Warmup_TH +Prediction = MPPP_Dead // 1- MPPP +else +if Avg L2 MPKI > MPKI_TH +Prediction = Seen & !MPPP_Dead // 2 +else +Prediction = Seen | !MPPP_Dead // 3 +Rest Counter +(RC) +Fig. 4. L2 harvester predictor. +Our experiments show that MPPP works well when MPKI +in the system is not very high, but it becomes very sensitive to +the thresholds when MPKI is very high. The issue is that when +there are many misses, the MPPP tables are updated more +frequently; we increase the value for one entry and decrement +for the rest (usually cache associativity -1). This lead to a +situation where MPPP observes smaller aggregated values. +Hence, differentiating dead blocks becomes more challenging. +However, this is a situation where the bloom filter works well, +because it warms up faster, and can help to detect the addresses +that have been evicted recently. +Hence, while the bloom filter is being warmed up, we use +MPPP to find the dead blocks, and we rely on the bloom +filter when the load is high and MPPP becomes sensitive to +the thresholds. The second advantage is that for challenging +applications, we can refer to both predictors to decide if a +block is dead to increase the accuracy. +As can be seen from Figure 4, when a block arrives, and if +the bloom filter is not warmed up (RC < Warmup TH), +then we have no options other than relying on MPPP +for prediction (Prediction = MPPP Dead). Otherwise +(if the bloom is warmed up RC +> +Warmup TH), +then we have both predictors available to make a predic- +tion. In such a case, if the load is high (L2 MPKI > +MPKIT H) both predictors should agree on the outcome +(Prediction = Seen & !MPPP Dead). Otherwise, the +block is not dead, if either predictor predicts so (Prediction = +Seen | !MPPP Dead). +Load Balancer The purpose of the load balancer is two-fold: +(1) find a lender and make decision if a block must be sent +up; (2) redirect dead blocks, and non-critical live blocks to the +main memory if the system load is high. Figure 6 shows the +structure and algorithm of the load balancer. +The load balancer takes as the input five pieces of in- +formation: (1) output of the predictor as a boolean signal +called Dead; (2) average L2 MPKI of caches running user- +facing applications; (3) first idle L2 cache obtained from Idle +Core Map (ICM) using a round-robin scheme; (4) a boolean +signal named Critical if this block belongs to the core +running critical applications; and (5) total number of critical +4 + +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +20 +40 +60 +80 +100 +120 +Sendup Likelihood +Critical L2 MPKI +Fig. 5. Probability of sending a non-critical block to a private L2 cache as a +function of critical applications MPKI. +Load Balancer +if Num Idle == 0 or Dead +Write to DRAM // no option other than DRAM +else +if Critical +Send to First Idle // All critical blocks are sent up +else +Chance = 0.95𝑀𝑃𝐾𝐼 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙𝑠 +if Chance < Rand(0, 1) // Chance based on MPKI +Send non-critical to First Idle +else +Send non-critical to DRAM +Dead? +Criticals L2 MPKI +Idle Core Map +Critical Task Map +First Idle +Num Idles +Critical? +Fig. 6. The overview of the load balancer. +applications running at the moment in the system obtained +from Critical Task Map (CTM). +The intuition behind the load balancer algorithm is to give +critical applications with maximum L2 capacity and provide +the non-critical applications with as much as the capacity +that will not negatively impact the critical applications. The +algorithm works as follows: if there is no idle core, or if the +block is dead, we must write the block back to main memory. +If there is an idle core, and if the block belongs to critical +applications, it will be pushed to the first idle resources. +On the other hand, if the block is not critical, we probabilis- +tically send the block to a private L2 cache with a probability +that decays as critical L2 MPKI grows. The intuition is that +requests should not be sent up when L2 MPKI is high. +We arbitrarily choose an exponentially decaying probability +density function (Chance = 0.95MP KI) as shown in Figure 5. +Hence, if the MPKI is low for critical applications, we give a +fraction of the capacity to the non-critical applications. As the +MPKI for critical applications increases, the chance for non- +critical applications decreases. For example, if the MPKI=20, +the chance of sending a non-critical application reduces to +30%, while for MPKIs > 40, non-critical blocks will be +barely sent to the private caches. +C. L2 Harvester Operation +Figure 7 shows how the harvester works in practice. As +Figure 7 (a) shows, in Step +1 a cache block is evicted from +L2-0 +(Borrower) +L2-1 +(Lender) +LLC +Bus +1 +2 +5 +4 +Snoop +Filter +6 +7 +3 +L2-0 +(Borrower) +L2-1 +(Lender) +LLC +Bus +2 +1 +Snoop +Filter +3 +4 +8 +5 +6 +(a) +(b) +Fig. 7. (a) L2H operations; and (b) Circular problem. +its private L2 cache, sends over the bus and checks the snoop +filter in Step +2 to find its destination port. The snoop filter +directs the block to the LLC. This block stays in the LLC until +it is evicted in Step 4 . The L2 harvester decides to send it to +L2-1. The block lookups the snoop filter in Step +5 , updates +its location to be L2-1, and is filled in the lender in Step +6 . +Later, when a request to this block arrives, the snoop filter +redirects the request to the lender (L2-1), and the response is +sent back by the lender to the borrower in Step 8 . Note that +we do not change the functionality of the snoop filter; this +operation is treated as a normal transfer to L2-1. +Possible Circular Harvesting L2H may create a circular +situation where a block stays on the chip and never gets evicted +despite not being useful. Figure 7 (b) shows such a scenario. +Similar to the previous example, assume that in Step +1 a +block is redirected to a lender. Thus, it updates the snoop +filter (Step +2 ) and fills in the cache (Step +3 ). Eventually, +this block gets evicted and is sent to the LLC in Step +5 . +Upon eviction from the LLC, it may again be redirected to a +private L2 cache based on a prediction. This loop can happen +infinitely, and this cache block will never depart the chip, even +though it is not touched. To address this problem, we add one +extra bit to the L2 cache tag store indicating if a block has +been redirected to the upper-level cache. Then, when we are +evicting this cache from the private cache, instead of writing +it back to the LLC, we bypass the LLC in Step 6 and write +it directly to the main memory. We find this approach to help +because this block has been given a second chance already +and can be evicted from the cache to avoid creating circular +harvesting. +IV. EVALUATION METHODOLOGY +A. Simulator Configuration +We use the gem5 full-system cycle-level simulator to con- +duct the experiments [23]. We model a 3-level cache hierarchy +where L1 and L2 are inclusive and private and L3 (the LLC) +is non-inclusive and shared. L1, L2, and L3 are parallel caches +where tag and data stores are accessed in parallel. L1 is 16- +way 48KB/core with 1-cycle access latency, L2 is 16-way +1.25MB/core with 12-cycle access latency, and the LLC is +16-way 2MB/core at 25-cycle access latency. We use one +prefetcher per level: L1 uses AMPM [16], L2 runs DCPT +[13], and LLC uses STeMS [34]. L1, L2, and the LLC have +16, 32, 64 MSHR entries. +5 + +TABLE III +FEATURES USED TO FORM MPPP TABLES [18]: FEATURE(LRU STACK +POSITION, START BIT, END BIT, [nthaccess], [XOR WITH PC]). +bias(6,0) +addr(9,9,14,5,1) +addr(9,12,29,0) +addr(13,21,29,0) +addr(14,17,25,0) +lastmiss(6,0) +lastmiss(18,0) +offset(13,0,4,0) +offset(14,0,6,0) +offset(16,0,1,0) +pc(6,13,31,4,0) +pc(9,11,7,16,0) +pc(13,16,24,17,0) +pc(16,2,10,2,0) +pc(16,4,46,9,0) +pc(17,0,13,5,0) +We also find that always enabling these prefetchers signif- +icantly degrades system performance for some applications +(e.g., 505.mcf) because the prefetchers contend too strongly +with demand requests. We, therefore, implement two prefetch +throttling mechanisms. In the first scheme, we reserve 25% +of MSHR entries for demand accesses, which decreases the +prefetch rate and maintains some minimum demand request +service. The second throttling mechanism is that we monitor +the performance of the prefetcher periodically and disable a +prefetcher when its accuracy drops below 40%. Specifically, +in each epoch of 10 million accesses, the prefetchers operate +for the first 1 million accesses, then the prefetcher accuracy +determines if the prefetcher remains enabled for the following +9 million accesses. +We use the MPPP [18] dead block predictor for the LLC. +Similarly to the original design, we use all 16 features sug- +gested by the authors for multicores as listed in Table IV-A. +We use 256 randomly selected sets to train the model. When +a block is accessed in the cache, all features are extracted +from the address, and program counters and used to index +the weight tables. Then, we sum up all weights and if it +exceeds a threshold, the block is declared dead. To train the +model, when any of the sampled sets are accessed (fills or +hits in the sampler as suggested in the paper), we extract the +features from the access. Then, we use the features to look +up the tables and increase the counters. Also, we decrement +the counters associated for those blocks that are impacted by +sampled access’s promotion. +MPPP [18] does not explicitly provide the thresholds in the +paper. Hence, in order to find the threshold to declare a block +dead, we ran 10 experiments, each running 4 randomly chosen +applications and swept the thresholds comparing the MPPP +suggestions with those of the bloom filter. We found that if +the summation of features is greater than 320, MPPP exhibits +the best performance. We refer the reader to MPPP [18] for +more detail. +For the bloom filter, we use the structure proposed by +Sanchez et al. [29]. The bloom filter has 4096 entries and +4 hash functions. This bloom filter uses a high-quality hash +functions (H3 [9]). Given that redirecting evictions is not on +a critical path, we do not use parallel bloom filter lookup, and +instead use a single-port structure to save power and area. +The main memory is DDR4-3200. There is one command +and address bus, with timings based on a DDR4-3200 8Gbit +device (Micron MT40A1G8) in an 8 × 8 configuration. The +total channel capacity is 16GB. This maintains a reasonable +core-to-memory ratio for the simulations. +The core has 320, 128, and 128 ROB, LQ, and SQ entries, +respectively. The core frequency is set to 3.66GHz. Fetch-, +commit-, and writeback-widths are all set to 8. The branch +predictor is TAGE SC L [32]. The TLB has 128 entries, and +there are 8 page-table walkers. +B. Benchmarks +We evaluate the applications of: (1) Tailbench [19] rep- +resenting user-facing jobs in datacenters; (2) SPEC CPU +2017 [5] representing background jobs; and (3) gapbs graph +analytics benchmarks. We mainly choose applications that are +memory-bound and benefit from larger cache capacity, but also +include some compute-bound applications to show how the +proposed solution behaves in such scenarios. +We choose 2 memory-bound applications from Tailbnech +(moses and img-dnn) and one compute-bound application +(massstree). moses is a statistical machine translation (SMT) +system. The input is randomly-chosen dialogue snippets from +the opensubtitles.org English-Spanish corpus. moses has high +L2 and LLC MPKIs of 26, and 22, respectively. img-dnn is +a handwriting recognition that uses OpenCV under the hood. +The input to this application is chosen randomly from MNIST +dataset. img-dnn shows L2 and LLC MPKIs of 20, and 18, +respectively. We also evaluate masstree fast key-value store +applications written in C++. This application has MPKIs of 6 +and 5, respectively. masstree is driven with the Yahoo Cloud +Serving Benchmark. +We choose 5 memory-bound applications from SPEC CPU +2017: 502.gcc, 505.mcf, 519.lbm, 520.omnetpp, and 549.fo- +tonik3d. We also run 3 compute-bound applications: 500.perl- +bench, 531.deepsjeng, and 521.wrf. From gapbs, we choose +3 applications: the page rank algorithm to find the web page +ranking (pr), the betweenness centrality score for approximate +calculations all vertices in a graph by only computing the +shortest paths from a subset of the vertices (bc); and single- +source shortest paths that computes the distances of the short- +est paths from a given source vertex to every other reachable +vertex (sssp). +We drive pr, bc, and sssp with synthetic graphs: (1) u: +a synthetically generated graph by the Erddos–Reyni model +(Uniform Random); and (2) g: a synthetically generated graph +by the Kronecker synthetic graph generator. We set the input +size to be 220 and 221. Note that all applications of gapbs are +memory-bound, and thus we do not have any compute-bound +representative application from this suite. +C. Single-Application Runs +We run moses, masstree, and img-dnn for 250 requests on +gem5: We launch Tailbnech in integrated mode, where both +client and server are running within one process. Then, we +warm up the internal data structures by running 1000 requests +in fast-simulation mode via KVM CPUs. After the warm-up +is finished, we switch the simulator CPU model to the most +accurate version (detailed OOO), and continue the simulation +until 250 requests are serviced. Due to the fact that clients +and the server are run in one process, architectural statistics +are not accurate. Hence, we record request timestamps while +6 + +TABLE IV +EVALUATED SYSTEM CONFIGURATION. +Processor +Single and Quad-core, 3.66 GHz, Ubuntu 20.04 OS. +ROB:320, LQ:128, SQ:128, Fetch-width=8 +L1 Cache +48kB 8-way; LRU; 1 cycles. Prefetcher: AMPM [16] +L2 Cache +1MB 8-way; LRU; 12 cycles, Prefetcher: DCPT [13] +L3 Cache +2MB/core; 16-way; LRU; 12 cycles. Prefetcher: STeMS +[34] +Main Memory +16 GB: DDR4-3200 x64, 8x8 Micron MT40A1G8 +TABLE V +MULTI-PROGRAM APPLICATIONS. +User-facing +img-dnn +qps=200, 300, 400 +masstree +qps=200, 300, 500 +Background +bc u20, pr u20, sssp u20, sjeng, omnet, lbm, mcf, perl +the applications are running on top of the simulator, and copy +them back to the host, and calculate the P99 of simulated 250 +requests. +For SPEC CPU, we use the SimPoint methodology [15] +to find representative regions of each application. We use 2 +SimPoints of 250 million instructions each and 250 million +instructions for warmup. For gapbs, we run each application +10 times after the graph was generated. +D. Multi-Applications Runs +We use Tailbench to represent the user-facing latency- +critical applications, and SPEC CPU 17 and gapbs applications +as background tasks. Due to gem5 limitations, simulating +more than 4 cores is very slow and difficult. Hence, we limit +our study to 4 cores. For user-facing applications we choose +one application from img-dnn, masstree, and moses, and one +application from SPEC CPU 2017, or gapbs. We leave two +cores idle each can provide 1.25MB L2 cache. Similar to the +single-application scenario, we run the user-facing applications +for 250 requests and make sure the background job continues +to run until the simulation is finished. We create 50 random +mixes out of the applications listed in Table V. +E. Systems +We compare three 4-core systems: (1) baseline with an 8MB +LLC; (2) the baseline configuration but with a 12MB LLC; +and (3) L2H with an 8MB LLC. Depending on the number of +applications running, L2H can borrow 3, 2, or 1 L2 caches. +Hence, the total L2 and L3 capacity for L2H is 8MB LLC ++ 3×1.25MB=11.75MB at most when it borrows three L2 +caches, and 8MB LLC+ 1×1.25MB=9.25MB, when it borrows +one L2 cache. +V. EVALUATION RESULTS +A. Single Application with Three Lenders +Performance In this section, we analyze a scenario where +one application is running, and there are three idle cores (25% +utilization) lending their private L2 caches. Figure 8 shows the +impact of LLC configuration on the latency-throughput curves +in terms of P99 latency. We compare three LLC configurations +(8MB, 12MB, or 8MB+L2H with 3.75MB of borrowed L2 +capacity) on three user-facing applications (img-dnn, moses, +and masstree). +img-dnn benefits from the larger cache the most. L2H +closely follows the 12MB LLC, while the gap between these +two and the 8MB LLC stays fairly constant (2X better +P99). The reason for such a large performance improvement +can stems from the large reduction in MPKI. As shown +in Figure 10, the img-dnn MPKI decreases from 26 to 2 +when the LLC size reaches 12MB. This implies that img- +dnn working set size fits in the larger LLC, and thus a huge +P99 improvement is realized. L2H could provide the needed +capacity for such applications almost for free with a 33% +smaller LLC size (8MB vs. 12MB). +moses performance is shown in Figure 8 (middle). At the +lowest qps (100), moses shows 7% and 5% lower P99 for +a 12MB LLC and L2H compared to the baseline with 8MB +LLC, respectively. L2H closely tracks the 12MB LLC. +Figure 8 (right) shows the performance of masstree, whose +MPKI is very low (1.1). This application is not memory +bound, so we do not expect to see improvement in P99 +when the LLC grows. We aldo expect L2H to not negatively +impact the P99 latency. As expected, all three systems show +very similar P99 latency, meaning L2H does not interfere +with compute-bound applications. We observe similar behavior +(not shown) across other compue-bound applications as well +(shore, xapian, specJBB, and silo). +In addition to lowering the P99 latency, extra cache space +can increase the maximum supported load: the qps after which +the P99 latency increases sharply. For img-dnn the saturation +point is pushed to higher qps by the 12 MB LCC and L2H: +baseline with 8MB LLC has a rapid increase in P99 for +qps>200, but the saturation point occurs at qps=500 for both +L2H and the 12MB LLC. +Figure 9 shows system performance on the gapbs and SPEC +CPU 2017 benchmark suites. The harmonic mean speedups for +L2H are 15% and 1.7% for gapbs, and SPECU CPU 2017, +respectively. Among gapbs application, page rank with the +u:21 input exhibits the largest speedup (2.77× for the 12MB +LLC and 1.26 × for L2H). As with img-dnn case, the MPKI +of pr u21 decreases from 36 to 15. +SPEC CPU applications also benefit from larger caches, but +to a lesser extent. We found that only 3 applications somewhat +benefit from larger caches in this benchmark suite: omnet +6.2%, 505.mcf 4.5%, and lbm 3.9%. However, the majority of +applications do not significantly benefit from the larger caches. +We found two reasons for this behavior: (1) some applications +are cache-friendly, but an 8MB LLC is sufficient for them; and +(2) other applications such as perl and wrf are not memory- +bound, and their MPKIs are less than 2. +MPKI Figure 10 shows the MPKI for the three LLC configu- +rations. The normalized geo-mean performance of the 12MB +LLC and L2H are 15% and 12% better than the baseline with +an 8MB LLC, respectively. Note that L2H achieves this 12% +better MPKI with 33% less LLC size (8MB vs. 12MB). This +brings a substantial saving in terms of area, power and cost. +7 + +0 +50 +100 +150 +200 +0 +200 +400 +600 +800 +1000 +P99 (ms) +qps +img-dnn +LLC=8MB +LLC=12MB +L2H +100 +120 +140 +160 +180 +200 +220 +240 +0 +50 +100 +150 +200 +250 +300 +350 +P99 (ms) +qps +moses +LLC=8MB +LLC=12MB +L2H +0 +10 +20 +30 +40 +50 +60 +70 +80 +0 +200 +400 +600 +800 +P99 (ms) +qps +masstree +LLC=8MB +LLC=12MB +L2H +Fig. 8. Impact of LLC size, and QPS on applications performance for (1) baseline with 8MB LLC; (2) baseline with 12MB LLC; and (3) L2H with 8MB +LLC, and 3 lenders each 1.25MB. +1 +1.1 +1.2 +1.3 +1.4 +1.5 +bc_u20 +bc_u21 +bfs_u20 +bfs_u21 +pr_u20 +pr_u21 +sssp_u20 +sssp_u21 +tc_u20 +tc_u21 +H-mean +Speedup Norm. to LLC=8MB +LLC=12MB +L2H +2.77 +1 +1.02 +1.04 +1.06 +1.08 +lbm +mcf +omnet +gcc +fotonik +perl +sjeng +wrf +H-mean +Speedup Norm. to LLC=8MB +LLC=12MB +L2H +Fig. 9. Impact of LLC size on applications performance for (1) baseline with 8MB LLC; (2) baseline with 12MB LLC; and (3) L2H with 8MB LLC, and 3 +lenders each 1.25MB. +0 +10 +20 +30 +40 +50 +60 +MPKI +LLC=8MB +LLC=12MB +L2H +Fig. 10. MPKI for 3 systems: (1) an 8MB LLC; (2) a 12 MB LLC; and (3) L2H. +0 +0.2 +0.4 +0.6 +0.8 +1 +500.perl +502.gcc +505.mcf +519.lbm +520.omnet +521.wrf +531.sjeng +549.fotonik +bc_u20 +bc_u21 +bfs_u20 +bfs_u21 +pr_u20 +pr_u21 +sssp_u20 +sssp_u21 +tc_u20 +tc_u21 +img_200 +img_300 +img_425 +img_575 +img_750 +masstree_200 +masstree_400 +masstree_600 +masstree_750 +moses_125 +moses_200 +moses_250 +Accuracy +Fig. 11. Prediction accuracy. Fraction of blocks +that the load balancer sends to private L2 caches +that satisfy a request. +We make two observations: (1) there are applications such as +pr and img-dnn whose MPKIs are reduced significantly due +to fitting the whole working set in the cache; and (2) there are +applications with various MPKI ranging from 1 to 55 in our +evaluation, stressing the load balancer properly. +Prediction Accuracy Figure 11 shows the prediction accuracy +of L2H. We calculate the accuracy by counting how many +blocks are sent up and what fraction of those are requested by +the borrower. The average prediction accuracy for memory- +bound applications is 89%. The averages are 96%, 75%, +and 70% for gapbs (applications with the highest MPKIs), +Tailbench, and SPEC CPU 2017, respectively. There are some +applications with low prediction accuracy such as perl, gcc, +and wrf, but given that their MPKIs are very low (< 2), the +mispredictions have insignificant impact. +Traffic Analysis L2H sends data blocks to upper-level caches +based on a heuristic. Although the prediction accuracy is high, +we need to carefully study any increased traffic on the sahred +interconnect. Figure 12 shows the traffic for the 12MB cache +and L2H normalized to the baseline traffic of the 8MB LLC: +the First bar is the 12MB LLC and the second bar is L2H. +We also separate the actual packets from the snoop packets, +as they usually have different sizes and purposes: the dark +blue represents actual packets and the light blue represents the +snoop packet seen on the interconnect connecting L2 caches +to LLC. +As can be seen from Figure 12, the 12MB LLC has +consistently lower or equal traffic. This is expected because the +larger cache keeps more data blocks on the chip than the 8MB +LLC, so it does not generate more traffic. In terms of packet +count, we can see that the majority of packets are data packets +and not snoop, as there is only one application running. Given +that we are running in full-system mode, the OS processes are +running on the cores and may share data blocks, but this is +negligible. Hence, overall, the 12MB LLC has less traffic. +On the other hand, the geo-mean for L2H is 24% increase +in traffic. This increase in traffic is expected as the blocks are +sent up and distributed over private L2 caches. However, the +behavior of L2H is very dynamic: some applications, such as +mcf generate more traffic (42% more), while others, like img- +dnn generate less traffic (-20%). Compute-bound applications +(those applications for which L2H has no impact) exhibit no +change in traffic. To understand this behavior better, we show +the breakdown of packets for two applications in Figure 13 +and Figure 14. +The increase in traffic comes from two sources: (1) sending +blocks up to a private cache, indicated as WriteUp requests +in Figure 13 and Figure 14; (2) evicting a block that has +been sent to a private cache (without first reusing it). The +load balancer and the predictor accuracy determine how many +8 + +0 +0.25 +0.5 +0.75 +1 +1.25 +1.5 +Normliazed to LLC=8MB +Packet +Snoop +First Bar: LLC=12MB +Second Bar: L2H +Fig. 12. Traffic increase on the bus between L2s and LLC normalized to a baseline with 8MB LLC. The first bar is a baseline with 12MB, and the second +bar is L2H. The dark color is the data packets, and the light blue is the snoop packets. +0 +1 +2 +3 +4 +5 +6 +7 +Millions of Transactions +505.mcf +8MB +12MB +L2H +Fig. 13. Example of high prediction accuracy (96%), and high traffic (42%). +Breakdown of packets seen on the bus between L2s and LLC for 505.mcf. +L2H is the only one that has WriteUp packets. +WriteUpRequest are generated. Given that prediction accuracy +is high in L2H, we believe that extra traffic generated by +WriteUps will actually help performance. +L2H increases snoop traffic because it first checks the +snoop filter before sending up a block. This ensures that +data is not needlessly replicated. Depending on the data +block status (clean or writeback clean), this snoop request is +either CleanEvict or WritebackClean. This snoop check is the +main reason why we see an increase in WritebackClean and +CleanEvict in Figure 13 and Figure 14. +We observe that for mcf (prediction accuracy=96%, traffic +increase=42%), WritbackClean, and CleanEvict are substan- +tially higher than the baselines, leading to a situation where +the total traffic increases by 42%. On the other hand, for img- +dnn because the larger cache can fit the working set size, the +CleanEvict for L2H stays very close to that of the 12MB LLC, +helping to reduce the total traffic by 20%. +B. Two Applications with Two Lenders +Performance We now focus on a more complex scenario, +where there are two applications running: core 0 runs a user- +facing application and core 1 runs a background job. Thus, +there are two idle cores (50% utilization). Figure 15 shows +the reduction in P99 for the user-facing application (top) and +speedup for the background job (bottom). We normalize both +to the baseline with an 8MB LLC. We sort the workloads +in ascending order to yield S-curves. For the P99, the lower +is the better, while for the speedup the higher is the better. +We observe that P99 decreases to almost 60%, while the +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +Millions of Transactions +img-dnn-qps=200 +8MB +12MB +L2H +Fig. 14. Example of mediocre prediction accuracy (75%), and low bus traffic +(-20%). Breakdown of packets seen on the bus between L2s and LLC for +img-dnn. L2H is the only one that has WriteUp packets. +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0 +10 +20 +30 +40 +50 +P99 Reduction over LLC=8MB +Mixes +Core 0 (User-facing) +LLC=12MB +L2H +0 +0.5 +1 +1.5 +2 +0 +10 +20 +30 +40 +50 +Speedup Over LLC=8MB +Mixes +Core 1 (Background Job) +LLC=12MB +L2H +Fig. 15. +System performance s-curve, normalized to the baseline with an +8MB LLC. (Top) Normalized P99 latency of user-facing jobs; the lower, the +better; (b) Background job speedup. +background job is sped up by up to 50%. We also show +the 12MB LLC configuration. As can be seen, L2H closely +follows the behavior of the larger 12MB LLC. +To better understand the results, we take a deeper look at +two mixes shown in Figure 16. The first mix has img-dnn as +the user-facing job and omnet as the background job. From the +single-application experiments (Figure 10), we expect these +9 + +0 +50 +100 +150 +200 +250 +150 +200 +250 +300 +350 +400 +P99 (ms) +qps +img-dnn, omnet +LLC=8MB +LLC=12MB +L2H +0 +20 +40 +60 +80 +100 +200 +300 +400 +500 +P99 (ms) +qps +masstree, lbm +LLC=8MB +LLC=12MB +L2H +Fig. 16. P99 latency of two pairs of appilications. +0 +0.2 +0.4 +0.6 +0.8 +1 +1 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Prediction Accuracy +Mixes +Fig. 17. Multi-application prediction accuray. +two applications to be very sensitive to cache size. In this +experiment, we vary the request rate from 200 to 400 qps. +We make two observations: (1) as expected the absolute P99 +latency increases compared to a single-application run (from +50ms to 126ms). However, the server is not saturated and (2) +L2H helps P99 stay very close to that with the 12MB LLC. +For example, at qps=400, the P99 latency for the baseline with +an 8MB LLC is around 200ms while the L2H keeps it very +close to that of the 12MB LLC at 150ms. This is significantly +given that our result shows that omnet IPC also improves by +4% at the same time. It is evident that the load balancer has +helped both applications to share the extra space provided by +the idle cores. +Prediction Accuracy Figure 17 shows the prediction accuracy +for all 50 mixes. The average prediction accuracy is 86% +and ranges from 62% to 99%. Overall, the high prediction +accuracy carries from the single-application experiments. We +also measure how often the bloom filter is not warmed up and +we need to refer or MPPP (15%), the load is high and we must +get the same output from both predictors (40%), and finally +how often we need one predictor to send a block up (45%). +We observe that all three situations are serviced well, given +the high prediction accuracy. +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0 +25 +50 +75 +100 +125 +150 +175 +IPC +Time +Core 0 - bc (Foreground Job) +LLC=8MB +LLC=12MB +L2H +MPKI=5 +MPKI=15 +MPKI=41 +t0 t1 t2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +0 +25 +50 +75 +100 +125 +150 +175 +IPC +Time +Core 1 - mcf (Background Job) +LLC=8MB +LLC=12MB +L2H +MPKI=5 +MPKI=15 +MPKI=41 +Fig. 18. +Load balancer analysis. When MPKI is low, L2H approaches the +baseline with 12MB LLC as the foreground job yields extra space. When +MPKI is high, the background job approaches the baseline with 8MB LLC +as the load balancer gives extra space to the background job. +C. Load Balancer Analysis +One major benefit of L2H is software transparency. The +load balancer plays an important role to achieve this goal. To +better understand how the load balancer works in practice, +we designed a simple experiment where we varied the critical +application MPKI to reveal how the load balancer works. +Figure 18 shows the absolute IPC for two applications and +3 systems (baseline with an 8MB LLC, a 12MB LLC, and +L2H) over time: foreground job bc, and background job mcf. +We pick bc to be the foreground job because the input to this +workload can be changed such that the MPKI changes. We call +this workload foreground, and not user-facing because this is +not a usual user-facing application. We could not find any +Tailbnech applications whose MPKI changes easily. We use +bc with input g19 to have the foreground job show MPKI=5, +input u20 to reach the MPKI to 15, and input u21 to increase +the MPKI to 41. We annotate the figure to show these three +MPKI regions over time. +Based on the load balancer algorithm and Figure 5, we ex- +pect that in this first region (MPKI=5), the background job gets +the majority of the extra space as the critical application has +very low MPKI and is driven with a small graph (0.955 = 0.77 +of alive mcf blocks are sent up). We observe that in this region, +all three system show very close IPC for bc, and all provide +enough cache for this application. Interestingly, for MPKI=5 +and mcf, we notice that L2H is very close to the baseline with +12MB LLC, and 13% better than the baseline with 8MB LLC. +Hence, the load balancer has redirected data blocks properly +and fairly to private L2 caches in this region. +In the second region (MPKI=15), we expect that all bc alive +blocks and 0.9515 = 0.46 of mcf alive blocks get the chance +to stay on the chip because now the foreground MPKI has +10 + +increases. We make two observations in this region: (1) bc +gets more space allowing it to follow the baseline with 21MB +cache. Also, this extra space allows L2H and the 12 MB LLC +to execute faster; the peaks are shifting to the right for the +8MB LLC; and (2) now the mcf sits between the 8MB and +12MB LLCs because it now must yield the extra space. +Finally, in the third region (MPKI=41), baseline the 12MB +LLC and L2H continue to execute faster than baseline +LLC=8MB for bc. The difference between the peaks is now +more visible; The peak at t0 for the 12MB LLC arrives earlier +than L2H (t1), and the baseline 8MB (t2). In this region, +the background job approaches the baseline with 8MB LLC, +mainly because the load balancer does not allow it to send the +blocks up (0.9541 = 0.12). +D. Storage Overhead Analysis +L2H uses two predictors. The bloom filter can store 4096 +entries. It has 4 tables, each 4K, summing up to a total of +16KB storage overhead per processor. MPPP uses 256 sampled +sets, adding up to 68.63KB. Other components in L2H are +fairly small. Idle Core Map (ICM) and Critical Task Map each +requires n bits, where n is the number of cores (e.g., 128 bits = +16B). We store the sendup likelihood in a lookup table to avoid +computation. This needs 100×2B=200B storage. Overall, L2H +needs 84.85KB storge for a 128-core processor. +VI. RELATED WORK +The insight behind Morpheus [11] is similar to that of L2H, +but for GPUs. The authors observe that increasing the number +of SMs is not always useful and system performance stays +constant after a certain number of SMs. They propose to not +activate several SMs, and instead borrow some resources such +as cache or register files from idle SMs. Apart from applying +this idea to a different context (GPU vs. CPU in L2H), the +differences are two-fold. First, idleness in L2H comes from +natural underutilization in the cloud, while Morpheus needs to +deactivate SMs to be able to borrow resources. This requires +Morpheus to run profiling to find the optimal number of SMs +for each application. Second, GPUs lack coherent caches, +substantially increasing complexity and requiring extensive +changes to the GPU microarchitecture. In contrast, L2H relies +on existing mechanisms and adds off-the-critical path predic- +tors at the LLC. Overall, both techniques address important +underutilization scenarios, but very different ones. +Jenga [36], and Eva [8] address underutilization in caches +by redesigning a new reconfigurable virtual cache hierarchy. +Jenga defines a pool of caches where a run-time decides +how each of them should be used. They propose an adaptive +hierarchy allocation which finds the exact number of cache +banks as well as the right cache level. They also propose a +placement strategy called Bandwidth-aware data placement, +where they try to put data blocks in the hierarchy where it +makes more sense in terms of bandwidth. Jenga breaks the +rigid hierarchy in the interest of reconfigurability, where L2H +keeps the classic memory hierarchy but opens the path to +use all levels automatically. Jenga requires OS and run-time +support, while L2H is completely transparent to software. +D2D [31] split data hierarchy from metadata hierarchy +allowing the data blocks to be found in the memory hierarchy +with a single lookup. Separating metadata from data allows the +authors to propose optimizations for data placement. However, +D2D cannot utilize the unused cache, instead helps to find the +block faster. +IBM Z16 [3], the latest generation of IBM mainframe pro- +cessors, has 4 levels of caches L1=128KB, L2=32 MB, L3=up +to 256 MB, and L4=2048 MB. L3 and L4 are called virtual +caches similar to Jenga’s definition. They can be allocated on +any of the share part of any L2 cache. Hence, with proper +run-time management, the L2 waste can be reduced. For IBM +z16 to work, the IBM Processor Resource/Systems Manager +(PR/SM) scheduler and the z/OS WLM and dispatcher must +work together to enable and use the large caches. IBM also +optimizes the lithography to reduce the cache access latency. +Z16 also needs a translation layer to be able to find the data +block in banked caches scattered throughout the chip. We +believe that the classic hierarchy offers a simpler design, and +can be fixed to make better use of the caches with L2H. +CATCH [24] proposes a criticality-aware tiered cache hier- +archy, where the authors argue that having a large L2 cache +is not an efficient design choice as L2 is not large enough +to capture the working set completely, nor as fast as the L1 +cache. Instead, CATCH proposes to remove the L2 cache and +compensate for its loss with new inter-level prefetchers to +move data in a timely manner between a larger LLC and the +L1 caches. We argue that the L2 is still very valuable. First, +it is very effective for some applications [17]. Second, L2 is +very effective in reducing the number of coherence requests +as it is usually inclusive of L1 cache. Thus, keeping L2 is a +good design choice, and its low hit ratio can be compensated +for by borrowing/lending space from/to neighboring cores. +Dead block prediction is another way to increase LLC +utilization. A cache block is dead if it has exhausted its useful +lifetime in the cache, and can be evicted to make space for +other blocks [18], [20], [21], [35]. Using perceptron-based +prediction proposed in some prior work [18], [35]. Using +sampling to detect dead blocks suggested by authors of [21]. +Cache partitioning is a strategy to provide quality of service +for over-provisioning datacenters cores [22], [25]–[28], [30], +[33], [37], [39], [40], [42], [43]. They use Intel Cache Allo- +cation Technology to partition LLC on a real machine or a +cluster of machines. L2H is orthogonal to cache partitioning, +although try to provide fairness for datacenter applications. +VII. CONCLUSION +We propose L2 Harvester (L2H), a simple approach to +harvest unused L2 caches in low-utilization beefy server +processors. We make this observation that number of cores +and cache sizes (both L2 and LLC) are constantly increasing +while the core utilization struggles to catch up in public clouds +(mostly <40% in Azure, and around 20-50% in Alibaba). To +address this shortcoming, we devise a mechanism to detect +11 + +LLC evictions that are not dead, and redirect them to upper L2 +caches, if the system load permits. L2H is implemented with +minimal changes to the current architecture. 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[Online]. +Available: https://doi.org/10.1145/2872362.2872394 +13 + diff --git a/DdE2T4oBgHgl3EQf9gki/content/tmp_files/load_file.txt b/DdE2T4oBgHgl3EQf9gki/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..58ec8c9466c49bd0df9304cecc8a19c9fafbd532 --- /dev/null +++ b/DdE2T4oBgHgl3EQf9gki/content/tmp_files/load_file.txt @@ -0,0 +1,1138 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf,len=1137 +page_content='Harvesting L2 Caches in Server Processors Majid Jalili, and Mattan Erez The University of Texas at Austin {majid,mattan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='erez}@utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='edu Abstract—We make three observations in modern processors: (1) LLC capacity is getting larger (up to 1GB);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (2) core counts are increasing (up to 128 cores), accumulating a more significant amount of private L2 cache capacity on the chip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (3) overall processor utilization in the cloud remains very low despite many efforts, leaving many large private caches unused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' To enable better use of these beefy processors, we propose to open up a logical path for LLC evictions to unused private caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' In other words, instead of writing LLC evictions back to slow and busy main memory, we send some of them that are still alive up to idle L2 caches to avoid unnecessary long and costly main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Our scheme takes the importance of applications (user-facing vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' background), and system load into account to provide each application with a fair share of idle resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Our results show that we can improve system performance by up to 2× (geomean of 10%) for single-application runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Also, for mixes of user-facing and background jobs, our scheme improves the P99 latency of user-facing tasks by up to 32% (geomean of 15%), and the IPC of background jobs by up to 50% (geomean of 10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' INTRODUCTION CPU manufacturers are increasing the L2/LLC sizes and number of cores to respond to the ever-increasing demand for computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For instance, comparing two high-end Intel processors (Xeon Platinum 8180 and Xeon Platinum 8380), we observe that total cache capacity has increased from 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='5MB to 110MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' AMD processors also follow a similar trends: EPYC- 7773X has 800MB on-chip memory with 64 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' At the same time, these CPUs are operating at very low utilization, 40% at Azure [10], and 20-50% at Alibaba [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This shift toward deeper and larger caches combined with low utilization opens opportunities for novel cache management mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We propose L2 Harvester (L2H), a completely software- transparent scheme built on top of the current rigid memory hierarchy that harvests idle cache resources, reducing the average load latency by up to 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H moves a fraction of LLC capacity/conflict evictions to unused private L2 caches instead of writing them back to main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Thus, later L2 misses can find data in other L2 caches, reducing off- chip transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H has the benefits of the classic memory hierarchy such as software transparency, simplicity of design, and isolation, while increasing cache utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Cache underutilization is prevalent because the main con- sumers of server processors are public clouds, where core utilization still remains low in spite of numerous efforts [6], [10], [12], [22], [38], [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The low core utilization leaves many large private L2 caches unused for the majority of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For instance, for a typical 64-core CPU with 1MB/core L2 caches, if the utilization is around 50%, then 32MB cache capacity is wasted just in one CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Given that there are millions of servers running in public clouds including Azure, AWS, Alibaba, and Google, this huge underutilization translates to gigabytes of scarce on-chip cache wastage that could have otherwise been used to reduce memory access latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The prior solution to increasing cache utilization is to break the classic cache hierarchy and fully or partially flatten it by allocating cache space based on application demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Jenga [36] proposes a reconfigurable cache where flat, distributed, and heterogeneous cache banks are controlled and managed by hardware and run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Flattening the cache allows Jenga to use all cache space, but this increases the complexity of the memory sub-system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Similarly, IBM Z16 [3] offers a 4- level cache hierarchy where L2 and L3 can be reconfigured to be part of an L2 cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Z16 is shipped with Processor Resource/Systems Manager that decides how to use these large caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' In both cases, the memory hierarchy is no longer software transparent and adds huge complexity to current designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Instead in L2H, on LLC evictions, we predict if the block is not dead and will be accessed soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' If so, the block is sent to a load balancer to decide if there is any idle core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Upon finding an available L2 cache, the block is written to the lender L2 cache, and the snoop filter metadata is updated as normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Later, if a request to this block is received, the lender cache responds and satisfies the request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Note that L2H relies on the current coherence mechanism to locate the cache block, and does not need any special support from the hardware or runtime system, thus it is superior to designs such as Jenga [36] and IBM Z16 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H utilizes two lightweight predictors to find the dead blocks: (1) a bloom filter-based predictor that tracks recently evicted addresses, identifying those misses that could have been avoided with a larger cache;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (2) MPPP [18]: a perceptron-based dead block predictor that combines different features such as address and program counter to predict if a block has exhausted its useful lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The two predictors complement each other: MPPP covers the bloom filter when it is not warmed up, and the bloom filter makes up for the MPPP sensitivity to thresholds when the system load is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We consult both predictors and make our decision based on a simple algorithm: if the load in the system is high, both predictors should agree if a block is not dead in order for the block to be sent up to a private L2 cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' If the load is low, the block is sent up if either predicts the block is not dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We take into account the importance and criticality of appli- cations being run to give them a fair share of private L2 caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='04228v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='AR] 10 Jan 2023 TABLE I INTEL AND AMD CPU GENERATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Intel (2016-2020) AMD (2017-2022) SKX CSX ICX Rome Milan(X) Genoa L1 32KB 32KB 48KB 32KB 32KB 32KB L2/core 1MB 1MB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='25MB 512KB 512KB 1MB L3/core 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='37MB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='37MB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='5MB 4-8MB 4-12MB 4-16MB Cores 4-28 2-56 8-40 8-64 8-64 8-96 Total 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='5MB 133MB 110MB 288MB 800MB 1100MB SKU Xeon-P8180 Xeon-P9282 Xeon-P8380 EPYC-7H12 EPYC-7773X N/A User-facing applications maximally use the extra cache space as they have the highest priority in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Background jobs can also get extra space if the load balancer detects that the user-facing applications are not cache-sensitive, and can yield the extra space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We evaluate L2H under different utilization scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' First, when the CPU load is very low (<25%) and running one application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This allows the application to take up all private L2 caches in the system, representing the upper-bound benefit of L2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Then, we move to more complex scenarios where a mix of critical and background jobs are run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H must make decisions regarding what blocks are dead and how to split the private L2 caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We implement L2H in gem5 [23] and run applications from different domains (datacenter, scientific, and graph analytics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Our experimental result shows that for a single application with multiple lenders, we improve P99 latency by 2×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Also, for mixes of user-facing and background jobs, L2H improves P99 and throughput by up to 32% and 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' To summarize our main contributions: We demonstrate that a substantial amount of cache ca- pacity is wasted in modern processors due to a rigid hierarchal design, and conservative resource allocation in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We architect and evaluate an effective, yet low-cost L2 harvesting mechanism that enables a logical path from LLC evictions to private L2 caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This allows the idle cores to lend their unused L2 caches, thus keeping more data blocks on the chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We incorporate two dead block prediction schemes in the L2 harvester to identify those capacity/conflict-caused evictions that are worth keeping on chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We also devise a simple load balancer that distributes data blocks over unused resources by taking system load and criticality of applications into the account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We evaluate our proposed method and compare it to a conventional hierarchy with a larger LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Our evaluation results show that a quad-core system with 2MB/core LLC and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='25 MB/core L2 cache benefiting from L2H improves system performance by up to 2× over the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Also, we show that L2H provides competitive system performance compared to a baseline with a 50% larger LLC (12MB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='9 1 Fraction of Servers Fraction of Cache Capacity Wasted Utilization CSX ICX Milan-X Genoa CDF Servers Cortez, Eli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Total cache wasted by different server processors under various utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The CDF (red line) is taken from [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' TABLE II TOTAL L2 CACHE CAPACITY OF 3 SUPERCOMPUTERS IN THE TACC DATACENTER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Systems Nodes Processor Core/node Total L2 (GB) Frontera [2] 8008 Xeon 8280 56 438 Lonestar6 [4] 560 EPYC 7763 128 35 Chamealon [1] 10000 Haswell 96 469 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' MOTIVATION According to Microsoft Azure and Alibaba, datacenter core utilization is very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Servers run at 40% or lower utilization at 90% of the time at Azure [10], and between 20%-50% most of the time at Alibaba [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This over-allocation stems from the fact that VMs should have enough cores and resources if the load surges rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' In addition, CPU manufacturers are increasing the L2/LLC sizes and the number of cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Table I exhibits three gener- ations of Intel and AMD server CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We can observe that both manufacturers’ L2/LLC and core counts have steadily increased over generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2 and LLC sizes are reaching 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='25MB/core, 2MB/core for Intel processors, and 1MB/core and 4MB/core for AMD processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Combined with the fact that core counts are also increasing, we can see that the third generation of Intel processors are accumulating 110MB total cache capacity, while AMD is reaching over giga bytes of on-chip cache storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' To better understand the current situation in datacenters, Figure 1 shows the total cache capacity wasted by different server processors under various utilization levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' On the x- axis, we show the utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We assume that all processors have 32 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Thus, the minimum utilization is when there is one application running taking one core and the whole LLC ( 1 32 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='03), and maximum utilization is when all 32 cores are active ( 32 32 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' To calculate the total cache wasted (first y-axis), we subtract the used cache capacity under each load from the total cache capacity available on the chip (32 ×(L1+ L2)+LLC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For example, if there are two cores running, and L1=48KB, L2=1MB/core, and LLC=8MB, then wasted cache is 32×(48KB+1MB)+8MB−2×(48KB+1MB)−8MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' On the second y-axis, we show the CDF of core utilization on Azure [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' As can be seen from Figure 1, 50% of Azure Icelake machines waste around 40% of the total cache capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Given 2 0 10 20 30 40 50 50 150 250 350 450 P95 (ms) QPS moses 2MB 4MB 8MB 16MB 22MB 0 1 2 3 4 2 6 10 14 18 22 Time/iteration (sec) LLC Size (MB) PageRank u20 u21 g20 g21 0 100 200 300 400 500 600 700 2MB 4MB 8MB 16MB 20MB 22MB Time (sec) LLC Size (MB) 520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='omnetpp Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Impact of LLC size on applications performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' that Icelake machines have an L2 capacity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='25MB/core (see Table I), for 32 cores, around 35MB of total on-chip cache capacity is wasted that could otherwise be used to keep data blocks on the chip and boost up system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' AMD processors also suffer from similar issues but at smaller scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For instance, Rome wastes around 10% of cache capacity under the load of 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The main reason is that AMD has smaller L2/core, and very large L3/core capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' However, in Genoa, we observe that AMD is enlarging the L2/core from 512KB/core in Milan to 1MB/core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' To put L2 cache waste into perspective, Table II shows 3 supercomputers in the TACC datacenter (Frontera, Lonestart6, and Chameleon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The table shows the main processor types as well as the number of nodes and total L2 capacity (GB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' As can be seen, the total cache capacity in a small-scale datacenter like TACC can be somewhere between 35GB (Lonestart6) to 469GB (Chameleon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Hence, if the utilization is around 50% on average, a substantial amount of a very scarce resource like L2 cache is being wasted (234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='5GB in Chameleon and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='5GB in Lonestart6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Note that public clouds such as AWS, Azure, Google, and Alibaba are operating significantly larger datacenters, so we are projecting the L2 waste reaches to terabytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' A larger cache capacity can help reduce the long memory access latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We conduct a cache study on a real machine to measure how much cache capacity impacts system perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The machine is an Intel(R) Xeon(R) Gold 6242 CPU with 22MB 11-way LLC cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We run one application and change the cache size using Intel Cache Allocation Technology (CAT) from one way (2MB) to 11 ways (22MB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We set the core frequency to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='9GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Figure 2 shows the performnace for three applications: (1) moses from TailBench [19], where we sweep the system load in terms of query per second (qps) and cache size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (2) PageRank from gapbs [7] with 4 different synthetic inputs (u: uniform graph, and g: Kronecker graph), and two different sizes (20, and 21);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (3) 520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='omnetpp from SPEC CPU 2017 [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For moses we make two observations: (1) with larger caches, the saturation point (point that P95 increases sharply) is pushed to higher qps (further to the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For example, we can see that the knee point for 22MB occurs at 450, while for the 2M, the server is saturated at qps=300;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='5× improvement in the maximum load;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (2) at similar loads before the saturation point the larger caches provides better P95 latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For instance, when qps=250, we see that 2MB LLC provides P95 of 12ms, while the 22MB cache shows P95 of 8ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For PageRank we observe that a larger LLC reduces the execution time significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For example, for the largest graph (g21) the execution time is halved when increasing the LLC size from 2MB to 12MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We also see that for LLC sizes of greater than 12MB, the execution times remain fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Finally, for 520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='omnetpp we observe similar sensitivity to cache size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' the execution time constantly reduces from 610 seconds for 2MB LLC, to 420 seconds for 22MB cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Our conclusion is that larger cache help applications from different domains, thus wasting a huge amount of on-chip cache is not reasonable, and we need to devise schemes to allow the unused L2 caches to be utilized when possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2 HARVESTER µARCHITECTURE We propose L2H, a simple yet effective mechanism for har- vesting L2 caches, that provides performance improvement for memory-bound applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' In this section, we first overview the design of L2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Then, we discuss the algorithm behind detecting the dead blocks, and how we distribute the blocks over idle cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H: Overview and Organization Figure 3 shows the overview of L2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Without loss of generality, we assume there are 4 cores connected to LLC banks with a shared bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' LLC has MPPP dead block predictor [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H sits between LLC and the memory controller and tracks the writebacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' If a block is detected by the predictor to be not dead, is sent to the load balancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Then, the load balancer decides where this block can be written to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' If there is any idle core that can lend its L2 cache, the load balancer pushes the block up to the lender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Otherwise, if the block is dead, or if there is no free L2, the block is written back to main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Thus, in the next reference to this block, there might be a private L2 cache that responds to the request and thereby saves one off-chip transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H needs four pieces of information to perform prediction and load balancing: (1) L2 MPKIs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (2) Critical Task Map (CTM): a bit mask that determines if the application being run on a core is critical, “1” determines the application being run at core n is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This bit mask is provided by the user or system administrator and is updated as soon as a new application is assigned to cores;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (3) Idle Core Map (ICM): a bit mask that determines if a core is idle and can lend its L2 cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This is updated by the cores if core has nothing 3 Harvester Load Balancer Predictor Eviction MPKIs Writeup L2 Memory Controller Writeback Dead/Alive?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Core 0 Core 1 Core 2 Core 3 Critical Task Map LLC Bank 0 LLC Bank 1 LLC Bank 2 LLC Bank 3 LLC C0 C1 C2 C3 C0 C1 C2 C3 Idle Core Map MPPP Dead/Alive?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2 harvester architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' MPPP [18] is the state-of-the-art dead block predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' to execute;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (4) The output of the MPPP [18] dead block predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H Structures Predictor The purpose of the predictor is to determine if a block is dead, and thus it is not worth keeping on chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This is particularly important for streaming applications because redirecting all cache blocks to upper levels will waste power, increase traffic, and elevate congestion on coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Figure 4 shows the structure of our predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We combine two predictors to find dead blocks: (1) a bloom filter-based predictor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (2) the multi-perspective perceptron predictor (MPPP) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The functionality of the bloom filter-based predictor is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We insert the missed addresses into the bloom filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' To make a prediction, we just need to look up the address, if the address was not found in the filter, we conclude the block is dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Because we have not seen a reference to this block recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We reset the bloom filter periodically to make sure the false positive rate stays low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Unfortunately, after each reset, the bloom filter starts declaring all blocks dead as they have not been seen, thus we need to address this shortcoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Morpheus [11] uses a bloom filter for hit/miss prediction, and addresses this problem by using two separate bloom filters with different reset intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' So, when one of them is being warmed up, the other one services the requests, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' However, we found that we get better accuracy if we combine our bloom filter with another type of dead block predictor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=', perceptron-based dead block predictor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The two predictors complement each other: MPPP covers the bloom filter when it is not warmed up, and the bloom filter makes up for the MPPP sensitivity to thresholds when the system load is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We use MPPP to solve the reset problem of the bloom filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' MPPP [18] is a perceptron-based technique that predicts the future reuse of cache blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' MPPP combines several features including program counter and address to form weight tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Then taking summations of entries from each table, it predicts if a block is: (a) not dead, (b) dead on arrival, and can bypass the cache, and (c) dead, and can be evicted from the cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' MPPP uses three thresholds to make the prediction based on the aggregated values taken from the weight tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Predictor Bloom Filter Block Address From MPPP Seen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Dead?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Average L2 MPKI if RC < Warmup_TH Prediction = MPPP_Dead // 1- MPPP else if Avg L2 MPKI > MPKI_TH Prediction = Seen & !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='MPPP_Dead // 2 else Prediction = Seen | !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='MPPP_Dead // 3 Rest Counter (RC) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2 harvester predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Our experiments show that MPPP works well when MPKI in the system is not very high, but it becomes very sensitive to the thresholds when MPKI is very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The issue is that when there are many misses, the MPPP tables are updated more frequently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' we increase the value for one entry and decrement for the rest (usually cache associativity -1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This lead to a situation where MPPP observes smaller aggregated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Hence, differentiating dead blocks becomes more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' However, this is a situation where the bloom filter works well, because it warms up faster, and can help to detect the addresses that have been evicted recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Hence, while the bloom filter is being warmed up, we use MPPP to find the dead blocks, and we rely on the bloom filter when the load is high and MPPP becomes sensitive to the thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The second advantage is that for challenging applications, we can refer to both predictors to decide if a block is dead to increase the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' As can be seen from Figure 4, when a block arrives, and if the bloom filter is not warmed up (RC < Warmup TH), then we have no options other than relying on MPPP for prediction (Prediction = MPPP Dead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Otherwise (if the bloom is warmed up RC > Warmup TH), then we have both predictors available to make a predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' In such a case, if the load is high (L2 MPKI > MPKIT H) both predictors should agree on the outcome (Prediction = Seen & !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='MPPP Dead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Otherwise, the block is not dead, if either predictor predicts so (Prediction = Seen | !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='MPPP Dead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Load Balancer The purpose of the load balancer is two-fold: (1) find a lender and make decision if a block must be sent up;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (2) redirect dead blocks, and non-critical live blocks to the main memory if the system load is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Figure 6 shows the structure and algorithm of the load balancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The load balancer takes as the input five pieces of in- formation: (1) output of the predictor as a boolean signal called Dead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (2) average L2 MPKI of caches running user- facing applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (3) first idle L2 cache obtained from Idle Core Map (ICM) using a round-robin scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (4) a boolean signal named Critical if this block belongs to the core running critical applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (5) total number of critical 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='8 1 0 20 40 60 80 100 120 Sendup Likelihood Critical L2 MPKI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Probability of sending a non-critical block to a private L2 cache as a function of critical applications MPKI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Load Balancer if Num Idle == 0 or Dead Write to DRAM // no option other than DRAM else if Critical Send to First Idle // All critical blocks are sent up else Chance = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='95𝑀𝑃𝐾𝐼 𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙𝑠 if Chance < Rand(0, 1) // Chance based on MPKI Send non-critical to First Idle else Send non-critical to DRAM Dead?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Criticals L2 MPKI Idle Core Map Critical Task Map First Idle Num Idles Critical?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The overview of the load balancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' applications running at the moment in the system obtained from Critical Task Map (CTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The intuition behind the load balancer algorithm is to give critical applications with maximum L2 capacity and provide the non-critical applications with as much as the capacity that will not negatively impact the critical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The algorithm works as follows: if there is no idle core, or if the block is dead, we must write the block back to main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' If there is an idle core, and if the block belongs to critical applications, it will be pushed to the first idle resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' On the other hand, if the block is not critical, we probabilis- tically send the block to a private L2 cache with a probability that decays as critical L2 MPKI grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The intuition is that requests should not be sent up when L2 MPKI is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We arbitrarily choose an exponentially decaying probability density function (Chance = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='95MP KI) as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Hence, if the MPKI is low for critical applications, we give a fraction of the capacity to the non-critical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' As the MPKI for critical applications increases, the chance for non- critical applications decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For example, if the MPKI=20, the chance of sending a non-critical application reduces to 30%, while for MPKIs > 40, non-critical blocks will be barely sent to the private caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2 Harvester Operation Figure 7 shows how the harvester works in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' As Figure 7 (a) shows, in Step 1 a cache block is evicted from L2-0 (Borrower) L2-1 (Lender) LLC Bus 1 2 5 4 Snoop Filter 6 7 3 L2-0 (Borrower) L2-1 (Lender) LLC Bus 2 1 Snoop Filter 3 4 8 5 6 (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (a) L2H operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (b) Circular problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' its private L2 cache, sends over the bus and checks the snoop filter in Step 2 to find its destination port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The snoop filter directs the block to the LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This block stays in the LLC until it is evicted in Step 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The L2 harvester decides to send it to L2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The block lookups the snoop filter in Step 5 , updates its location to be L2-1, and is filled in the lender in Step 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Later, when a request to this block arrives, the snoop filter redirects the request to the lender (L2-1), and the response is sent back by the lender to the borrower in Step 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Note that we do not change the functionality of the snoop filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' this operation is treated as a normal transfer to L2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Possible Circular Harvesting L2H may create a circular situation where a block stays on the chip and never gets evicted despite not being useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Figure 7 (b) shows such a scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Similar to the previous example, assume that in Step 1 a block is redirected to a lender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Thus, it updates the snoop filter (Step 2 ) and fills in the cache (Step 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Eventually, this block gets evicted and is sent to the LLC in Step 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Upon eviction from the LLC, it may again be redirected to a private L2 cache based on a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This loop can happen infinitely, and this cache block will never depart the chip, even though it is not touched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' To address this problem, we add one extra bit to the L2 cache tag store indicating if a block has been redirected to the upper-level cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Then, when we are evicting this cache from the private cache, instead of writing it back to the LLC, we bypass the LLC in Step 6 and write it directly to the main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We find this approach to help because this block has been given a second chance already and can be evicted from the cache to avoid creating circular harvesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' EVALUATION METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Simulator Configuration We use the gem5 full-system cycle-level simulator to con- duct the experiments [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We model a 3-level cache hierarchy where L1 and L2 are inclusive and private and L3 (the LLC) is non-inclusive and shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L1, L2, and L3 are parallel caches where tag and data stores are accessed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L1 is 16- way 48KB/core with 1-cycle access latency, L2 is 16-way 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='25MB/core with 12-cycle access latency, and the LLC is 16-way 2MB/core at 25-cycle access latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We use one prefetcher per level: L1 uses AMPM [16], L2 runs DCPT [13], and LLC uses STeMS [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L1, L2, and the LLC have 16, 32, 64 MSHR entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 5 TABLE III FEATURES USED TO FORM MPPP TABLES [18]: FEATURE(LRU STACK POSITION, START BIT, END BIT, [nthaccess], [XOR WITH PC]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' bias(6,0) addr(9,9,14,5,1) addr(9,12,29,0) addr(13,21,29,0) addr(14,17,25,0) lastmiss(6,0) lastmiss(18,0) offset(13,0,4,0) offset(14,0,6,0) offset(16,0,1,0) pc(6,13,31,4,0) pc(9,11,7,16,0) pc(13,16,24,17,0) pc(16,2,10,2,0) pc(16,4,46,9,0) pc(17,0,13,5,0) We also find that always enabling these prefetchers signif- icantly degrades system performance for some applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=', 505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='mcf) because the prefetchers contend too strongly with demand requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We, therefore, implement two prefetch throttling mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' In the first scheme, we reserve 25% of MSHR entries for demand accesses, which decreases the prefetch rate and maintains some minimum demand request service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The second throttling mechanism is that we monitor the performance of the prefetcher periodically and disable a prefetcher when its accuracy drops below 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Specifically, in each epoch of 10 million accesses, the prefetchers operate for the first 1 million accesses, then the prefetcher accuracy determines if the prefetcher remains enabled for the following 9 million accesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We use the MPPP [18] dead block predictor for the LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Similarly to the original design, we use all 16 features sug- gested by the authors for multicores as listed in Table IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We use 256 randomly selected sets to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' When a block is accessed in the cache, all features are extracted from the address, and program counters and used to index the weight tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Then, we sum up all weights and if it exceeds a threshold, the block is declared dead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' To train the model, when any of the sampled sets are accessed (fills or hits in the sampler as suggested in the paper), we extract the features from the access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Then, we use the features to look up the tables and increase the counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Also, we decrement the counters associated for those blocks that are impacted by sampled access’s promotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' MPPP [18] does not explicitly provide the thresholds in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Hence, in order to find the threshold to declare a block dead, we ran 10 experiments, each running 4 randomly chosen applications and swept the thresholds comparing the MPPP suggestions with those of the bloom filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We found that if the summation of features is greater than 320, MPPP exhibits the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We refer the reader to MPPP [18] for more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For the bloom filter, we use the structure proposed by Sanchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The bloom filter has 4096 entries and 4 hash functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This bloom filter uses a high-quality hash functions (H3 [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Given that redirecting evictions is not on a critical path, we do not use parallel bloom filter lookup, and instead use a single-port structure to save power and area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The main memory is DDR4-3200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' There is one command and address bus, with timings based on a DDR4-3200 8Gbit device (Micron MT40A1G8) in an 8 × 8 configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The total channel capacity is 16GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This maintains a reasonable core-to-memory ratio for the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The core has 320, 128, and 128 ROB, LQ, and SQ entries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The core frequency is set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='66GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Fetch-, commit-, and writeback-widths are all set to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The branch predictor is TAGE SC L [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The TLB has 128 entries, and there are 8 page-table walkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Benchmarks We evaluate the applications of: (1) Tailbench [19] rep- resenting user-facing jobs in datacenters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (2) SPEC CPU 2017 [5] representing background jobs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (3) gapbs graph analytics benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We mainly choose applications that are memory-bound and benefit from larger cache capacity, but also include some compute-bound applications to show how the proposed solution behaves in such scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We choose 2 memory-bound applications from Tailbnech (moses and img-dnn) and one compute-bound application (massstree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' moses is a statistical machine translation (SMT) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The input is randomly-chosen dialogue snippets from the opensubtitles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='org English-Spanish corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' moses has high L2 and LLC MPKIs of 26, and 22, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' img-dnn is a handwriting recognition that uses OpenCV under the hood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The input to this application is chosen randomly from MNIST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' img-dnn shows L2 and LLC MPKIs of 20, and 18, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We also evaluate masstree fast key-value store applications written in C++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This application has MPKIs of 6 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' masstree is driven with the Yahoo Cloud Serving Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We choose 5 memory-bound applications from SPEC CPU 2017: 502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='gcc, 505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='mcf, 519.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='lbm, 520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='omnetpp, and 549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='fo- tonik3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We also run 3 compute-bound applications: 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='perl- bench, 531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='deepsjeng, and 521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='wrf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' From gapbs, we choose 3 applications: the page rank algorithm to find the web page ranking (pr), the betweenness centrality score for approximate calculations all vertices in a graph by only computing the shortest paths from a subset of the vertices (bc);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and single- source shortest paths that computes the distances of the short- est paths from a given source vertex to every other reachable vertex (sssp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We drive pr, bc, and sssp with synthetic graphs: (1) u: a synthetically generated graph by the Erddos–Reyni model (Uniform Random);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (2) g: a synthetically generated graph by the Kronecker synthetic graph generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We set the input size to be 220 and 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Note that all applications of gapbs are memory-bound, and thus we do not have any compute-bound representative application from this suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Single-Application Runs We run moses, masstree, and img-dnn for 250 requests on gem5: We launch Tailbnech in integrated mode, where both client and server are running within one process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Then, we warm up the internal data structures by running 1000 requests in fast-simulation mode via KVM CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' After the warm-up is finished, we switch the simulator CPU model to the most accurate version (detailed OOO), and continue the simulation until 250 requests are serviced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Due to the fact that clients and the server are run in one process, architectural statistics are not accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Hence, we record request timestamps while 6 TABLE IV EVALUATED SYSTEM CONFIGURATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Processor Single and Quad-core, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='66 GHz, Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='04 OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' ROB:320, LQ:128, SQ:128, Fetch-width=8 L1 Cache 48kB 8-way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' LRU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 1 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Prefetcher: AMPM [16] L2 Cache 1MB 8-way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' LRU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 12 cycles, Prefetcher: DCPT [13] L3 Cache 2MB/core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 16-way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' LRU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 12 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Prefetcher: STeMS [34] Main Memory 16 GB: DDR4-3200 x64, 8x8 Micron MT40A1G8 TABLE V MULTI-PROGRAM APPLICATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' User-facing img-dnn qps=200, 300, 400 masstree qps=200, 300, 500 Background bc u20, pr u20, sssp u20, sjeng, omnet, lbm, mcf, perl the applications are running on top of the simulator, and copy them back to the host, and calculate the P99 of simulated 250 requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For SPEC CPU, we use the SimPoint methodology [15] to find representative regions of each application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We use 2 SimPoints of 250 million instructions each and 250 million instructions for warmup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For gapbs, we run each application 10 times after the graph was generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Multi-Applications Runs We use Tailbench to represent the user-facing latency- critical applications, and SPEC CPU 17 and gapbs applications as background tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Due to gem5 limitations, simulating more than 4 cores is very slow and difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Hence, we limit our study to 4 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For user-facing applications we choose one application from img-dnn, masstree, and moses, and one application from SPEC CPU 2017, or gapbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We leave two cores idle each can provide 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='25MB L2 cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Similar to the single-application scenario, we run the user-facing applications for 250 requests and make sure the background job continues to run until the simulation is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We create 50 random mixes out of the applications listed in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Systems We compare three 4-core systems: (1) baseline with an 8MB LLC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (2) the baseline configuration but with a 12MB LLC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (3) L2H with an 8MB LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Depending on the number of applications running, L2H can borrow 3, 2, or 1 L2 caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Hence, the total L2 and L3 capacity for L2H is 8MB LLC + 3×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='25MB=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='75MB at most when it borrows three L2 caches, and 8MB LLC+ 1×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='25MB=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='25MB, when it borrows one L2 cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' EVALUATION RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Single Application with Three Lenders Performance In this section, we analyze a scenario where one application is running, and there are three idle cores (25% utilization) lending their private L2 caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Figure 8 shows the impact of LLC configuration on the latency-throughput curves in terms of P99 latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We compare three LLC configurations (8MB, 12MB, or 8MB+L2H with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='75MB of borrowed L2 capacity) on three user-facing applications (img-dnn, moses, and masstree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' img-dnn benefits from the larger cache the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H closely follows the 12MB LLC, while the gap between these two and the 8MB LLC stays fairly constant (2X better P99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The reason for such a large performance improvement can stems from the large reduction in MPKI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' As shown in Figure 10, the img-dnn MPKI decreases from 26 to 2 when the LLC size reaches 12MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This implies that img- dnn working set size fits in the larger LLC, and thus a huge P99 improvement is realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H could provide the needed capacity for such applications almost for free with a 33% smaller LLC size (8MB vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 12MB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' moses performance is shown in Figure 8 (middle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' At the lowest qps (100), moses shows 7% and 5% lower P99 for a 12MB LLC and L2H compared to the baseline with 8MB LLC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H closely tracks the 12MB LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Figure 8 (right) shows the performance of masstree, whose MPKI is very low (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This application is not memory bound, so we do not expect to see improvement in P99 when the LLC grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We aldo expect L2H to not negatively impact the P99 latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' As expected, all three systems show very similar P99 latency, meaning L2H does not interfere with compute-bound applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We observe similar behavior (not shown) across other compue-bound applications as well (shore, xapian, specJBB, and silo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' In addition to lowering the P99 latency, extra cache space can increase the maximum supported load: the qps after which the P99 latency increases sharply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For img-dnn the saturation point is pushed to higher qps by the 12 MB LCC and L2H: baseline with 8MB LLC has a rapid increase in P99 for qps>200, but the saturation point occurs at qps=500 for both L2H and the 12MB LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Figure 9 shows system performance on the gapbs and SPEC CPU 2017 benchmark suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The harmonic mean speedups for L2H are 15% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='7% for gapbs, and SPECU CPU 2017, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Among gapbs application, page rank with the u:21 input exhibits the largest speedup (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='77× for the 12MB LLC and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='26 × for L2H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' As with img-dnn case, the MPKI of pr u21 decreases from 36 to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' SPEC CPU applications also benefit from larger caches, but to a lesser extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We found that only 3 applications somewhat benefit from larger caches in this benchmark suite: omnet 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='2%, 505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='mcf 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='5%, and lbm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' However, the majority of applications do not significantly benefit from the larger caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We found two reasons for this behavior: (1) some applications are cache-friendly, but an 8MB LLC is sufficient for them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (2) other applications such as perl and wrf are not memory- bound, and their MPKIs are less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' MPKI Figure 10 shows the MPKI for the three LLC configu- rations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The normalized geo-mean performance of the 12MB LLC and L2H are 15% and 12% better than the baseline with an 8MB LLC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Note that L2H achieves this 12% better MPKI with 33% less LLC size (8MB vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 12MB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This brings a substantial saving in terms of area, power and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 7 0 50 100 150 200 0 200 400 600 800 1000 P99 (ms) qps img-dnn LLC=8MB LLC=12MB L2H 100 120 140 160 180 200 220 240 0 50 100 150 200 250 300 350 P99 (ms) qps moses LLC=8MB LLC=12MB L2H 0 10 20 30 40 50 60 70 80 0 200 400 600 800 P99 (ms) qps masstree LLC=8MB LLC=12MB L2H Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Impact of LLC size, and QPS on applications performance for (1) baseline with 8MB LLC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (2) baseline with 12MB LLC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (3) L2H with 8MB LLC, and 3 lenders each 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='25MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='5 bc_u20 bc_u21 bfs_u20 bfs_u21 pr_u20 pr_u21 sssp_u20 sssp_u21 tc_u20 tc_u21 H-mean Speedup Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' to LLC=8MB LLC=12MB L2H 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='77 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='08 lbm mcf omnet gcc fotonik perl sjeng wrf H-mean Speedup Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' to LLC=8MB LLC=12MB L2H Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Impact of LLC size on applications performance for (1) baseline with 8MB LLC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (2) baseline with 12MB LLC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (3) L2H with 8MB LLC, and 3 lenders each 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='25MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 0 10 20 30 40 50 60 MPKI LLC=8MB LLC=12MB L2H Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' MPKI for 3 systems: (1) an 8MB LLC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (2) a 12 MB LLC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (3) L2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='8 1 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='perl 502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='gcc 505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='mcf 519.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='lbm 520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='omnet 521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='wrf 531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='sjeng 549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='fotonik bc_u20 bc_u21 bfs_u20 bfs_u21 pr_u20 pr_u21 sssp_u20 sssp_u21 tc_u20 tc_u21 img_200 img_300 img_425 img_575 img_750 masstree_200 masstree_400 masstree_600 masstree_750 moses_125 moses_200 moses_250 Accuracy Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Fraction of blocks that the load balancer sends to private L2 caches that satisfy a request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We make two observations: (1) there are applications such as pr and img-dnn whose MPKIs are reduced significantly due to fitting the whole working set in the cache;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (2) there are applications with various MPKI ranging from 1 to 55 in our evaluation, stressing the load balancer properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Prediction Accuracy Figure 11 shows the prediction accuracy of L2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We calculate the accuracy by counting how many blocks are sent up and what fraction of those are requested by the borrower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The average prediction accuracy for memory- bound applications is 89%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The averages are 96%, 75%, and 70% for gapbs (applications with the highest MPKIs), Tailbench, and SPEC CPU 2017, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' There are some applications with low prediction accuracy such as perl, gcc, and wrf, but given that their MPKIs are very low (< 2), the mispredictions have insignificant impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Traffic Analysis L2H sends data blocks to upper-level caches based on a heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Although the prediction accuracy is high, we need to carefully study any increased traffic on the sahred interconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Figure 12 shows the traffic for the 12MB cache and L2H normalized to the baseline traffic of the 8MB LLC: the First bar is the 12MB LLC and the second bar is L2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We also separate the actual packets from the snoop packets, as they usually have different sizes and purposes: the dark blue represents actual packets and the light blue represents the snoop packet seen on the interconnect connecting L2 caches to LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' As can be seen from Figure 12, the 12MB LLC has consistently lower or equal traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This is expected because the larger cache keeps more data blocks on the chip than the 8MB LLC, so it does not generate more traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' In terms of packet count, we can see that the majority of packets are data packets and not snoop, as there is only one application running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Given that we are running in full-system mode, the OS processes are running on the cores and may share data blocks, but this is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Hence, overall, the 12MB LLC has less traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' On the other hand, the geo-mean for L2H is 24% increase in traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This increase in traffic is expected as the blocks are sent up and distributed over private L2 caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' However, the behavior of L2H is very dynamic: some applications, such as mcf generate more traffic (42% more), while others, like img- dnn generate less traffic (-20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Compute-bound applications (those applications for which L2H has no impact) exhibit no change in traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' To understand this behavior better, we show the breakdown of packets for two applications in Figure 13 and Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The increase in traffic comes from two sources: (1) sending blocks up to a private cache, indicated as WriteUp requests in Figure 13 and Figure 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (2) evicting a block that has been sent to a private cache (without first reusing it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The load balancer and the predictor accuracy determine how many 8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='75 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='5 Normliazed to LLC=8MB Packet Snoop First Bar: LLC=12MB Second Bar: L2H Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Traffic increase on the bus between L2s and LLC normalized to a baseline with 8MB LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The first bar is a baseline with 12MB, and the second bar is L2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The dark color is the data packets, and the light blue is the snoop packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 Millions of Transactions 505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='mcf 8MB 12MB L2H Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Example of high prediction accuracy (96%), and high traffic (42%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Breakdown of packets seen on the bus between L2s and LLC for 505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='mcf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H is the only one that has WriteUp packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' WriteUpRequest are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Given that prediction accuracy is high in L2H, we believe that extra traffic generated by WriteUps will actually help performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H increases snoop traffic because it first checks the snoop filter before sending up a block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This ensures that data is not needlessly replicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Depending on the data block status (clean or writeback clean), this snoop request is either CleanEvict or WritebackClean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This snoop check is the main reason why we see an increase in WritebackClean and CleanEvict in Figure 13 and Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We observe that for mcf (prediction accuracy=96%, traffic increase=42%), WritbackClean, and CleanEvict are substan- tially higher than the baselines, leading to a situation where the total traffic increases by 42%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' On the other hand, for img- dnn because the larger cache can fit the working set size, the CleanEvict for L2H stays very close to that of the 12MB LLC, helping to reduce the total traffic by 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Two Applications with Two Lenders Performance We now focus on a more complex scenario, where there are two applications running: core 0 runs a user- facing application and core 1 runs a background job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Thus, there are two idle cores (50% utilization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Figure 15 shows the reduction in P99 for the user-facing application (top) and speedup for the background job (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We normalize both to the baseline with an 8MB LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We sort the workloads in ascending order to yield S-curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For the P99, the lower is the better, while for the speedup the higher is the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We observe that P99 decreases to almost 60%, while the 0 2 4 6 8 10 12 14 16 18 20 Millions of Transactions img-dnn-qps=200 8MB 12MB L2H Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Example of mediocre prediction accuracy (75%), and low bus traffic (-20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Breakdown of packets seen on the bus between L2s and LLC for img-dnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H is the only one that has WriteUp packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='2 0 10 20 30 40 50 P99 Reduction over LLC=8MB Mixes Core 0 (User-facing) LLC=12MB L2H 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='5 2 0 10 20 30 40 50 Speedup Over LLC=8MB Mixes Core 1 (Background Job) LLC=12MB L2H Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' System performance s-curve, normalized to the baseline with an 8MB LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (Top) Normalized P99 latency of user-facing jobs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' the lower, the better;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' (b) Background job speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' background job is sped up by up to 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We also show the 12MB LLC configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' As can be seen, L2H closely follows the behavior of the larger 12MB LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' To better understand the results, we take a deeper look at two mixes shown in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The first mix has img-dnn as the user-facing job and omnet as the background job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' From the single-application experiments (Figure 10), we expect these 9 0 50 100 150 200 250 150 200 250 300 350 400 P99 (ms) qps img-dnn, omnet LLC=8MB LLC=12MB L2H 0 20 40 60 80 100 200 300 400 500 P99 (ms) qps masstree, lbm LLC=8MB LLC=12MB L2H Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' P99 latency of two pairs of appilications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='8 1 1 5 10 15 20 25 30 35 40 45 50 Prediction Accuracy Mixes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Multi-application prediction accuray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' two applications to be very sensitive to cache size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' In this experiment, we vary the request rate from 200 to 400 qps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We make two observations: (1) as expected the absolute P99 latency increases compared to a single-application run (from 50ms to 126ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' However, the server is not saturated and (2) L2H helps P99 stay very close to that with the 12MB LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For example, at qps=400, the P99 latency for the baseline with an 8MB LLC is around 200ms while the L2H keeps it very close to that of the 12MB LLC at 150ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This is significantly given that our result shows that omnet IPC also improves by 4% at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' It is evident that the load balancer has helped both applications to share the extra space provided by the idle cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Prediction Accuracy Figure 17 shows the prediction accuracy for all 50 mixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The average prediction accuracy is 86% and ranges from 62% to 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Overall, the high prediction accuracy carries from the single-application experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We also measure how often the bloom filter is not warmed up and we need to refer or MPPP (15%), the load is high and we must get the same output from both predictors (40%), and finally how often we need one predictor to send a block up (45%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We observe that all three situations are serviced well, given the high prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='6 0 25 50 75 100 125 150 175 IPC Time Core 0 - bc (Foreground Job) LLC=8MB LLC=12MB L2H MPKI=5 MPKI=15 MPKI=41 t0 t1 t2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='4 0 25 50 75 100 125 150 175 IPC Time Core 1 - mcf (Background Job) LLC=8MB LLC=12MB L2H MPKI=5 MPKI=15 MPKI=41 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Load balancer analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' When MPKI is low, L2H approaches the baseline with 12MB LLC as the foreground job yields extra space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' When MPKI is high, the background job approaches the baseline with 8MB LLC as the load balancer gives extra space to the background job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Load Balancer Analysis One major benefit of L2H is software transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The load balancer plays an important role to achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' To better understand how the load balancer works in practice, we designed a simple experiment where we varied the critical application MPKI to reveal how the load balancer works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Figure 18 shows the absolute IPC for two applications and 3 systems (baseline with an 8MB LLC, a 12MB LLC, and L2H) over time: foreground job bc, and background job mcf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We pick bc to be the foreground job because the input to this workload can be changed such that the MPKI changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We call this workload foreground, and not user-facing because this is not a usual user-facing application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We could not find any Tailbnech applications whose MPKI changes easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We use bc with input g19 to have the foreground job show MPKI=5, input u20 to reach the MPKI to 15, and input u21 to increase the MPKI to 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We annotate the figure to show these three MPKI regions over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Based on the load balancer algorithm and Figure 5, we ex- pect that in this first region (MPKI=5), the background job gets the majority of the extra space as the critical application has very low MPKI and is driven with a small graph (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='955 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='77 of alive mcf blocks are sent up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We observe that in this region, all three system show very close IPC for bc, and all provide enough cache for this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Interestingly, for MPKI=5 and mcf, we notice that L2H is very close to the baseline with 12MB LLC, and 13% better than the baseline with 8MB LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Hence, the load balancer has redirected data blocks properly and fairly to private L2 caches in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' In the second region (MPKI=15), we expect that all bc alive blocks and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='9515 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='46 of mcf alive blocks get the chance to stay on the chip because now the foreground MPKI has 10 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We make two observations in this region: (1) bc gets more space allowing it to follow the baseline with 21MB cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Also, this extra space allows L2H and the 12 MB LLC to execute faster;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' the peaks are shifting to the right for the 8MB LLC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' and (2) now the mcf sits between the 8MB and 12MB LLCs because it now must yield the extra space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Finally, in the third region (MPKI=41), baseline the 12MB LLC and L2H continue to execute faster than baseline LLC=8MB for bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The difference between the peaks is now more visible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The peak at t0 for the 12MB LLC arrives earlier than L2H (t1), and the baseline 8MB (t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' In this region, the background job approaches the baseline with 8MB LLC, mainly because the load balancer does not allow it to send the blocks up (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='9541 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Storage Overhead Analysis L2H uses two predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The bloom filter can store 4096 entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' It has 4 tables, each 4K, summing up to a total of 16KB storage overhead per processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' MPPP uses 256 sampled sets, adding up to 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='63KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Other components in L2H are fairly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Idle Core Map (ICM) and Critical Task Map each requires n bits, where n is the number of cores (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=', 128 bits = 16B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We store the sendup likelihood in a lookup table to avoid computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This needs 100×2B=200B storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Overall, L2H needs 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='85KB storge for a 128-core processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' RELATED WORK The insight behind Morpheus [11] is similar to that of L2H, but for GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' The authors observe that increasing the number of SMs is not always useful and system performance stays constant after a certain number of SMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' They propose to not activate several SMs, and instead borrow some resources such as cache or register files from idle SMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Apart from applying this idea to a different context (GPU vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' CPU in L2H), the differences are two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' First, idleness in L2H comes from natural underutilization in the cloud, while Morpheus needs to deactivate SMs to be able to borrow resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' This requires Morpheus to run profiling to find the optimal number of SMs for each application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Second, GPUs lack coherent caches, substantially increasing complexity and requiring extensive changes to the GPU microarchitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' In contrast, L2H relies on existing mechanisms and adds off-the-critical path predic- tors at the LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Overall, both techniques address important underutilization scenarios, but very different ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Jenga [36], and Eva [8] address underutilization in caches by redesigning a new reconfigurable virtual cache hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Jenga defines a pool of caches where a run-time decides how each of them should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' They propose an adaptive hierarchy allocation which finds the exact number of cache banks as well as the right cache level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' They also propose a placement strategy called Bandwidth-aware data placement, where they try to put data blocks in the hierarchy where it makes more sense in terms of bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Jenga breaks the rigid hierarchy in the interest of reconfigurability, where L2H keeps the classic memory hierarchy but opens the path to use all levels automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Jenga requires OS and run-time support, while L2H is completely transparent to software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' D2D [31] split data hierarchy from metadata hierarchy allowing the data blocks to be found in the memory hierarchy with a single lookup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Separating metadata from data allows the authors to propose optimizations for data placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' However, D2D cannot utilize the unused cache, instead helps to find the block faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' IBM Z16 [3], the latest generation of IBM mainframe pro- cessors, has 4 levels of caches L1=128KB, L2=32 MB, L3=up to 256 MB, and L4=2048 MB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L3 and L4 are called virtual caches similar to Jenga’s definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' They can be allocated on any of the share part of any L2 cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Hence, with proper run-time management, the L2 waste can be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' For IBM z16 to work, the IBM Processor Resource/Systems Manager (PR/SM) scheduler and the z/OS WLM and dispatcher must work together to enable and use the large caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' IBM also optimizes the lithography to reduce the cache access latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Z16 also needs a translation layer to be able to find the data block in banked caches scattered throughout the chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We believe that the classic hierarchy offers a simpler design, and can be fixed to make better use of the caches with L2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' CATCH [24] proposes a criticality-aware tiered cache hier- archy, where the authors argue that having a large L2 cache is not an efficient design choice as L2 is not large enough to capture the working set completely, nor as fast as the L1 cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Instead, CATCH proposes to remove the L2 cache and compensate for its loss with new inter-level prefetchers to move data in a timely manner between a larger LLC and the L1 caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We argue that the L2 is still very valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' First, it is very effective for some applications [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Second, L2 is very effective in reducing the number of coherence requests as it is usually inclusive of L1 cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Thus, keeping L2 is a good design choice, and its low hit ratio can be compensated for by borrowing/lending space from/to neighboring cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Dead block prediction is another way to increase LLC utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' A cache block is dead if it has exhausted its useful lifetime in the cache, and can be evicted to make space for other blocks [18], [20], [21], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Using perceptron-based prediction proposed in some prior work [18], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Using sampling to detect dead blocks suggested by authors of [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Cache partitioning is a strategy to provide quality of service for over-provisioning datacenters cores [22], [25]–[28], [30], [33], [37], [39], [40], [42], [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' They use Intel Cache Allo- cation Technology to partition LLC on a real machine or a cluster of machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H is orthogonal to cache partitioning, although try to provide fairness for datacenter applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' CONCLUSION We propose L2 Harvester (L2H), a simple approach to harvest unused L2 caches in low-utilization beefy server processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' We make this observation that number of cores and cache sizes (both L2 and LLC) are constantly increasing while the core utilization struggles to catch up in public clouds (mostly <40% in Azure, and around 20-50% in Alibaba).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' To address this shortcoming, we devise a mechanism to detect 11 LLC evictions that are not dead, and redirect them to upper L2 caches, if the system load permits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' L2H is implemented with minimal changes to the current architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Our experimental results show that L2H improves system performance by up to 2×, and 32% for single-application and multiple-application, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' REFERENCES [1] “Chameleon@tacc.” [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content=' Available: https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='tacc.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} +page_content='2872394 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE2T4oBgHgl3EQf9gki/content/2301.04228v1.pdf'} diff --git a/HNE1T4oBgHgl3EQf_Qa5/content/tmp_files/2301.03579v1.pdf.txt b/HNE1T4oBgHgl3EQf_Qa5/content/tmp_files/2301.03579v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..70a95a77f28bc6bab51fb51200c8118c1e4e2c9d --- /dev/null +++ b/HNE1T4oBgHgl3EQf_Qa5/content/tmp_files/2301.03579v1.pdf.txt @@ -0,0 +1,1151 @@ +Proper evaluation of spatially correlated noise in interferometric +images. +Takafumi Tsukuia,b,c,d, Satoru Iguchic,d, Ikki Mitsuhashic,e, Kenichi Tadakic,d +aResearch School of Astronomy and Astrophysics, Australian National University, Cotter Road, Weston Creek, ACT +2611, Australia +bARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D) +cNational Astronomical Observatory of Japan, National Institute of Natural Sciences, 2-21-1 Osawa, Mitaka, Tokyo, +Japan. +dDepartment of Astronomical Science, SOKENDAI (The Graduate University for Advanced Studies) 2-21-1 Osawa, +Mitaka, Tokyo, Japan. +eDepartment of Astronomy, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan +Abstract. +Interferometers (e.g. ALMA and NOEMA) allow us to obtain the detailed brightness distribution of +astronomical sources in 3 dimensions (R.A., Dec., frequency). However, the spatial correlation of the noise makes it +difficult to evaluate the statistical uncertainty of the measured quantities and the statistical significance of the results +obtained. The noise correlation properties in the interferometric image are fully characterized and easily measured by +the noise autocorrelation function (ACF). We present the method for (1) estimating the statistical uncertainty due to +the correlated noise in the spatially integrated flux and spectra directly, (2) simulating the correlated noise to perform +a Monte Carlo simulation in image analyses, and (3) constructing the covariance matrix and chi-square χ2 distribution +to be used when fitting a model to an image with spatially correlated noise, based on the measured noise ACF. We +demonstrate example applications to scientific data showing that ignoring noise correlation can lead to significant +underestimation of statistical uncertainty of the results and false detections/interpretations. +Keywords: Interferometric imaging, Correlated noise, Image analysis, Astronomy, Monte Carlo Methods. +*Takafumi Tsukui, tsukuitk23@gmail.com +1 Introduction +Recent developments in large interferometers (e.g., ALMA and NOEMA) have made it possible +to spatially resolve the brightness distribution of many more astronomical objects. These obser- +vations have enabled us to obtain a three-dimensional (R.A., Dec., and the line-of-sight velocity) +structure of the gas emission and two-dimensional images of the continuum within galaxies with +high spatial resolution and sensitivity. As a result, the data allow for detailed image analysis; for +example, investigating spectral features of spatially resolved regions, characterizing faint and ex- +tended structures, performing Fourier analysis of the image, etc. However, the spatial correlation +of the noise in interferometric images makes it difficult to evaluate the uncertainty of the results. +1 +arXiv:2301.03579v1 [astro-ph.IM] 9 Jan 2023 + +There has been a lack of quantitative understanding of the spatial correlation of noise and meth- +ods to evaluate the statistical uncertainty of measured quantities and the significance of scientific +results, such as signal detection and image analysis. +To estimate the statistical uncertainty of integrated fluxes or spectra under correlated noise, both +variance and covariance of the pixel pairs in the integrated region need to be taken into account +for the uncertainty propagation. Sun et al.1 proposed a method based on an approximation that +the covariance of noise between pixels is proportional to the synthesized beam. However, the +covariance is actually proportional to the autocorrelation of the synthesized beam.2 Also, more +importantly, the method approximates the synthesized beam with a single Gaussian. In contrast, the +true synthesized beam has a complex structure with a main lobe inducing short-range strong noise +correlation and side lobes inducing long-range weak noise correlation. Such an oversimplified +assumption can lead to underestimation of the uncertainty. More widely used method to estimate +the statistical uncertainty of integrated fluxes is based on an intuitive interpretation that the noise +can be regarded as independent across beam-sized regions as described by Alatalo et al.3 The +noise variance in the integrated values is evaluated by scaling the noise variance of individual +pixels by the number of the beam area in the integrating aperture. This method also implicitly +assumes the Gaussian beam to estimate the beam area of the synthesized beam. To evaluate the +statistical uncertainty of the best-fitting parameters in a model fitting to the interferometric data +with the correlated noise, Davis et al.4 proposed a method to construct a covariance matrix from +the synthesized beam, which describes the covariance of the noise and uses it to compute the χ2 +value. Again, they also assumed a simple Gaussian function for the synthesized beam. +Several studies have used workarounds to avoid the need to characterize the correlated noise +using Monte Carlo methods. Harikane et al.5 measure the statistical uncertainty of the integrated +2 + +flux by randomly placing identical apertures to the emission-free region and adoptting the root +mean square of the summed values. Boizelle et al.6 estimate the statistical uncertainty of the fitting +parameters by Monte Carlo resampling of the best-fitting parameters: adding noise extracted from +the emission-free regions to the original data and refitting models. Another common technique +is to fit the model in the visibility plane to measure the source size and shape where the noise in +the visibility measurements is independent. This method is particularly beneficial if the source +is small compared to the resolution of the interferometer since the model needs to be simple and +axisymmetric for computational efficiency. However, in many cases, analysis in the image plane +is necessary (e.g. complex structures such as spiral arms, bar, clumpy structures).7 +There have yet to be any attempts to evaluate the statistical uncertainty for the general mea- +surements using interferometric images (etc., integrated flux, spectra, fitting, etc.) by fully char- +acterizing the detailed noise correlation. Refregier and Brown2 proposed to use the noise ACF to +characterize the correlated noise of the Very large array (VLA) FIRST radio survey data. They +used the noise ACF to explore the effect of the spatially correlated noise in the signal of the ellip- +ticity correlation function, which encodes the imprint of the weak lensing signal by the large-scale +structure of the universe. The noise ACF fully characterizes the noise correlation properties of in- +terferometric images and provides the covariance of noise between different pixels, which allows +us to measure statistical uncertainty under the noise correlation. +In this paper, we present a method and associate Python code to characterize the spatial cor- +relation of noise in interferometric images by measuring the noise autocorrelation function (ACF) +and evaluating its effect on the measured quantities and the analysis results. +This paper is organized as follows. In Sec. 2, we present the noise correlation properties of +ALMA data characterized by the autocorrelation function (ACF) and show that the noise correla- +3 + +tion originates from the synthesized beam (dirty beam) structures, which remain even in the CLEAN +image and cannot be removed by any deconvolution algorithm. In Sec. 3, we introduce methods +for (1) estimating the statistical uncertainties associated with spatially integrated flux or spectra; +(2) generating simulated noise maps from the measured noise ACF, which are useful to estimate +the statistical significance of the result obtained by any image analysis; and (3) constructing the +covariance matrix from the noise ACF which can be used in the χ2 formalism of the model fitting +to the observed image, with example applications to real scientific data from Ref. 8. +Throughout the paper, we use the noise map from emission line cube and continuum image +data taken by ALMA Band 7 (2017.1.00394.S; PI: Gonz´alez L´opez, Jorge) as an example, but +the method proposed by this paper can be applied to other interferometric images. A Python +package for easy application of the methods described in this paper, Evaluating Statistical Signif- +icance undEr Noise CorrElation (ESSENCE), is publicly available at https://github.com/ +takafumi291/ESSENCE. +2 NOISE CHARACTERIZATION OF INTERFEROMETRIC IMAGE +2.1 The characterization of spatially correlated noise +First, we consider a two-dimensional noise map N(x), where x denotes the position of the pixels. +Pixel regions with signal from the object of interest are excluded. The statistical properties of the +noise are assumed to be uniform in the noise image, which appears to be valid in the interferometric +image*. In most of the literature, the noise in the radio interferometric image is quantified and +*We discuss this in Sec. 2.2 +4 + +reported with the root mean square (rms) of the noise map N(x), +� +⟨N(x)2⟩ ≡ σN, +(1) +where the brackets denote the expected value for each pixel, which is practically estimated by +averaging over the noise map. The mean of the noise in the image µ ≡ ⟨N(x)⟩ ≈ 0, since most +of the noise represented by the system temperature Tsys is not correlated in a pair of antennas and +the power of the noise does not appear in the correlator output of interferometers such as ALMA. +Extended background emission such as the cosmic microwave background (CMB) is resolved out +without total power observation. However, these sources of noise contribute to the random noise +associated with visibility measurements that propagate into the noise on the image by the Fourier +transform. In the rest of the paper, we assume the mean of the noise map to be zero or already +have been subtracted in other cases, and thus the root mean square and the standard deviation of +the noise can be used interchangeably. Figure 1 shows the example ALMA Band 7 noise map and +its histogram from the observation targeting the hyper luminous infrared galaxy BRI 1335-0417 at +redshift of 4.4, which will be used in the later sections. The noise map is created by eliminating +astronomical sources by 4 sigma clipping as well as removing pixels adjacent to these clipped +regions out to 3 times the full width of half maximum (FWHM) of the synthesized beam. The +histogram of the pixel values in the noise map is well fitted by a Gaussian function. +When noise can be assumed to be Gaussian, the statistical and correlation properties of noise +are fully quantified by the noise autocorrelation function (ACF),2 +ξ(xi,j) ≡ ⟨N(x + xi,j)N(x)⟩, +(2) +5 + +Fig 1 Left: Example ALMA Band 7 noise map. The source emission region is eliminated with the 4 sigma clipping; +see text. Right: The histogram of the pixel values of the noise map. The red dashed line indicates the best-fit Gaussian +with the mean µ = 0.000 and the standard deviation σ = 0.036 (mJy beam−1). +where the expected value is estimated by averaging all pairs of pixels with the relative distance +xi,j = (i, j) in the noise image. The value of the ACF noise at zero lag, xi,j = 0, is equal to the +variance of the noise as ξ(0) = ⟨N(x)2⟩ = σ2 +N. When the noise has no inter-pixel correlations, the +noise ACF becomes +ξ(xi,j) = +� +� +� +� +� +� +� +σ2 +N +if xi,j = 0 +0 +otherwise +(3) +To evaluate the statistical uncertainty of the derived noise ACF, we first considered the number +of independent pixel pairs Npair in the number of all available pairs N ′ +pair used to evaluate the +bracket in Eq. 2, since the pixels within a beam area are expected to be strongly correlated and not +independent. We estimated the number of independent pixel pairs Npair as the ratio of the number +of all pixel pairs N ′ +pair and the number of pixels in the beam (beam area in pixels)† Abeam, +Npair = N ′ +pair/Abeam. +(4) +†The beam area in pixels is typically estimated by 2πbmajbmin/8ln2, where bmaj and bmin are the major and minor +FWHMs of the mainlobe of the synthesized beam (the “CLEAN” beam). +6 + +Then, the associated statistical uncertainty of the noise ACF ∆ξ(xi,j) is calculated as the usual +standard error of the mean but with an independent sample size Npair, that is, the standard deviation +of the multiplication of the values across all pairs of pixels with separation xi,j divided by the root +of the number of independent pixel pairs Npair, +∆ξ(xi,j) = +� +⟨N(x + xi,j)2N(x)2⟩/Npair. +(5) +Figure 2 shows the results of the noise ACF (Eq. 2) computed for the noise map shown in Fig. 1, +and the synthesized beam of the observation, both of which are normalized so that the central value +is one. The noise ACF shows a pattern similar to that of the synthesized beam, with a correlation +signal near the center and a correlation signal away from the center corresponding to the main lobe +and side lobe of the synthesized beam, respectively. This suggests that most of the correlation of +the noise originates from the discrete Fourier transform involved in the interferometric imaging, +which is illustrated in the following subsection. +Note that the noise ACF are measured for the noise maps of the Band 7 continuum image +(shown in Fig. 2), [CII] line (velocity integrated over the velocity range of -400 to 400 km s−1, +where the velocity is computed with respect to the redshifted [CII] line frequency with the galaxy’s +redshift of 4.40749) moment 0 map, and the [CII] line cubes (each velocity channel map). These +maps are primary beam uncorrected and CLEANed images produced by the CLEAN algorithm in +CASA (see details in Ref. 8). +7 + +Fig 2 The noise ACF computed for the ALMA Band 7 noise map (left), showing a similar pattern in the synthesized +beam of the observation (right). +2.2 Origin of the noise correlation +In interferometric observations, measurable quantities are visibility (Fourier amplitude and phase) +of the astronomical image at the given spatial frequencies (u, v) = D/λ, which are related to the +antenna baseline vector D‡ projected onto the plane of the sky and the observed wavelength λ. The +image is then computed by the Fourier transform of the measured visibility. +To explore the origin of the noise correlation in the image, seen in (Fig. 2), we start with the +ideal case in which the observation measures visibilities at all spatial frequencies (u, v). The visi- +bility of the source of interest is V (u, v), which is the Fourier transform of the true flux distribution +of the source in the image, ˆS(x, y) = FT[V (u, v)], where FT denotes the Fourier transform. A +measurement of V (u, v) usually involves uncorrelated random noise, which we describe with the +random variable ˆNvis(u, v) with zero mean. We assume that the statistical property of the random +variable ˆNvis(u, v) is uniform as a function of u and v, that is, the system noise temperature is the +same for all antennas. The image obtained I(x, y) is the Fourier transform of the measurement +‡separation vector of pairs of antennas +8 + +V (u, v) + ˆNvis(u, v), +I(x, y) = ˆS(x, y) + ˆN(x, y) += FT[V (u, v) + ˆNvis(u, v)] +(6) +where ˆN(x, y) = FT( ˆNvis(u, v)) is the noise component of the image, which is a random variable +with zero mean§. The noise component of the image ˆN(x, y) is due to the random noise associated +with visibility measurements ˆNvis(u, v), and the resulting noise map N(x, y) = ˆN(x, y) is not +spatially correlated in the ideal case where all spatial frequencies are measured. +In practice, visibilities are measured only at the limited spatial frequencies {(u1, v1), (u2, v2),..., +(uM, vM)} (uv coverage). The spatial transfer function, W(u, v), is used to describe the spatial +frequencies (u, v) at which we measure the visibility. This function W(u, v) is non-zero if the +visibility at (u, v) is actually measured, which can be expressed as, +W(u, v) = +M +� +i=0 +δ(u − ui, v − vi) + δ(u + ui, v + vi), +(7) +where δ is the Dirac delta function. The synthesized beam b(x, y) is the Fourier transform of the +spatial transfer function W(u, v), b(x, y) = FT[W(u, v)]. The resulting image I(x, y), decom- +posed as the signal S(x, y) from the source and noise map N(x, y), is +I(x, y) = S(x, y) + N(x, y) += ˆS(x, y) ∗ b(x, y) + ˆN(x, y) ∗ b(x, y) += FT[(V (u, v) + ˆNvis(u, v))W(u, v)], +(8) +§The Fourier transform of the random variable with zero mean is also random variable with zero mean. +9 + +where ∗ represents convolution. As the noise correlation pattern (noise ACF) and the synthesized +beam show a similar pattern in Fig. 2, the noise component of the image N(x, y) is the convolution +product of the random variable ˆN(x, y) and the synthesized beam b(x, y). Because of this, the noise +in the image is well behaved; in particular, its statistical properties are uniform over the image, as +assumed in Sec. 2.1, when measuring the noise ACF . +For convenience, by replacing the sky position of (x, y) with the pixel position x, the noise map +of the image in Eq. (8) is written as +N(x) = b(x) ∗ ˆN(x) = +� +i,j +b(xi,j) ˆN(x + xi,j). +(9) +The autocorrelation of the noise map then becomes2 +ξ(xi,j) = ⟨N(x + xi,j)N(x)⟩ += ⟨ +� +i′,j′ +b(xi′,j′) ˆN(x + xi,j + xi′,j′) +� +i′′,j′′ +b(xi′′,j′′) ˆN(x + xi′′,j′′)⟩ += +� +i′,j′ +� +i′′,j′′ +b(xi′,j′)b(xi′′,j′′)⟨ ˆN(x + xi,j + xi′,j′) ˆN(x + xi′′,j′′)⟩ += σ2 +Nα(xi,j), +(10) +where we used the noise ACF property of uncorrelated noise ˆN (Eq. 3) for the fourth equality and +we have defined α(xi,j) as beam autocorrelation, +α(xi,j) = +� +i′,j′ +b(xi′,j′)b(xi,j + xi′,j′). +(11) +Equation 10 implies that the noise ACF is related to the ACF of the synthesized beam with a con- +10 + +stant multiplicative factor, which is the variance of the noise. In Fig. 3 we compare the ACF of +noise and that of the synthesized beam, along with the residuals (noise ACF minus beam ACF). +Although the noise ACF and beam ACF show a common characteristic pattern, they do not com- +pletely coincide. The difference of the two ACF shows an extended weak positive correlation and +a relatively large negative around the main beam in the residual. This disagreement is likely due +to not only (1) the remaining contamination by the emission from the sources, but also (2) the +process involved in the imaging of the visibility measurements. These are discussed in detail in +the Appendix A comparing Fig. 3 obtained from the actual data with the one (Fig. 13) obtained +from the simulated data with a similar observational setup and realistic noise in the visibilities, but +without emission in the sky. Note that our interest is on the statistical property of the noise in the +image plane, which we characterize by the noise ACF including effects of contamination from the +source and the imaging process. +Due to the limited spatial frequency coverage, the synthesized beam b(x, y) has a complex +structure with sidelobes that extend from the center to a large radius. The flux from the source +is spread out by the side lobes to the distant pixels in the image. The CLEAN algorithm, which +is most commonly used in radio imaging, deconvolves the beam pattern b(x, y) for signals with +high S/N (S(x, y)/σN) > 3) and replaces it with a CLEAN beam without side lobes (a Gaussian +that approximates the mainlobe of the synthesized beam). The CLEAN algorithm successfully +suppresses the influence of the sidelobe and produces a high-fidelity image, but cannot remove +the spatial correlations that exist in stochastic noise N(x, y). Therefore, it is important to evaluate +their effects on image analysis and signal detection, which we will describe in Sec. 3. +11 + +Fig 3 Top left: the same noise ACF shown in Fig. 2, top right: the ACF of the synthesized beam, bottom: The residual +of the noise ACF minus the ACF of the synthesized beam +3 EXAMPLE APPLICATION TO SCIENTIC DATA +3.1 Contribution of the correlated noise to the statistical uncertainty in the measured flux +The most fundamental measurement of astronomy is the total flux distributing over some sky region +in the images, which are measured by summing the pixel values over the region of interest (i.e., +aperture photometry in optical astronomy). In particular, at the submillimeter band of ALMA, +the flux of the continuum emission arising primarily from thermal dust, and line emission and +absorption by the various atomic and molecular gases are used to estimate the physical properties of +12 + +the interstellar medium (e.g., dust mass, gas mass, the energy source of the ionization or excitation, +etc.). It is important to estimate the uncertainty of the measured quantities. As shown in Fig. 2, the +noise in interferometric images correlates significantly between pixels, making the estimation of +the noise in the integrated flux difficult. In previous literature, statistical uncertainties of integrated +fluxes were estimated by one of two methods: (1) randomly placing identical apertures in the noise +region of the image, measuring the sum within each aperture and then adopting the rms as the noise +in the sum of pixels in the aperture;5 and (2) assuming that the regions in the image separated with +a beam size do not correlate and adopting σNAbeam +√Nbeam, where Nbeam is the number of beams +(independent regions) in the aperture.3 Nbeam is estimated as Aaperture/Abeam, where Aaperture and +Abeam are the aperture area and the CLEAN beam area in pixels, respectively. For convenience, we +call methods (1) and (2) ”random aperture method” and ”independent beam method,” respectively, +in this paper. +In the ”independent beam method” (σNAbeam +√Nbeam), the factor σNAbeam is the standard de- +viation of the sum of noise in individual pixels within a beam assuming that the noise perfectly +correlates within a beam. Then the standard deviation of the sum of the noise of each independent +beam area in the aperture is computed by scaling by the square root of the number of independent +beams Nbeam within the aperture. The terms Abeam and Nbeam in the σNAbeam +√Nbeam denote just +the number of data points to be summed. Therefore, we caution readers that σNAbeam +√Nbeam has +the same unit with σN. Most interferometric maps and measured σN are in brightness units e.g., +Jy beam−1 km s−1 or Jy beam−1. So we need to divide Abeam to compare with the integrated flux +or spectral flux density, e.g., Jy km s−1 or Jy. σNAbeam +√Nbeam is a factor of Abeam different from +σN +√Nbeam described in Alatalo et al.3 due to the unit difference where they assume the quantity +in the unit of flux. +13 + +This section introduces how to derive the statistical uncertainty associated with the spatially +integrated flux directly from the computed noise ACF. We consider adding all the pixel values +at pixel positions x within the sky region of interest S. The random noise N(x) in the map is +characterized by the noise ACF, ξ(xi,j). The 1σ statistical uncertainty associated with the summed +value within the pixel region S, σint, can be estimated as +σ2 +int = Var( +� +x +30 +2 +20 +1 +A +10 +0 +0 +30 +60 +90 +120 +Vioc (km s-1)Mass-Velocity Density Plot, Emission Stars +6 +60 +EB & SB2 +HMXB +5 +Number of Stars In Bin +50 +4 +(M +40 +Mass +3 +30 +2 +20 +1 +10 +0 +0 +30 +60 +90 +120 +Vioc (km s-1)6 +M. S. Oey et al. +References +Blaauw, A. 1961, Bull. Astron. Inst. Netherlands, 15, 265 +de Wit, W. J., Testi, L., Palla, F., Vanzi, L., & Zinnecker, H. 2004, A&A, 425, 937 +Dorigo Jones, J., Oey, M. S., Paggeot, K., Castro, N., & Moe, M. 2020, ApJ, 903, 43 +Hoogerwerf, R., de Bruijne, J. H. J., & de Zeeuw, P. T. 2000, ApJL, 544, L133 +Kriz, S., & Harmanec, P. 1975, Bulletin of the Astronomical Institutes of Czechoslovakia, 26, 65 +Lamb, J. 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M., & Faerber, +T. A. 2020, ApJ, 903, 42 +Discussion +Shenar: +Some later Be stars could originate from mass donors that are less massive +and would not have exploded. Have you considered this in your velocity distributions? +Oey: No, we have not included this effect. Given that binaries tend to have high mass +ratios, the contribution from such objects should be relatively small. There are also others +that we ignored, for example, not all stars that undergo mass transfer will be seen as +Be stars and similarly, there may also be a significant number of Be stars that originate +from a completely unrelated mechanism. So we stress that our results are simply a first +rough estimate of the relative contributions of the acceleration mechanisms. + diff --git a/IdFJT4oBgHgl3EQfFywB/content/tmp_files/load_file.txt b/IdFJT4oBgHgl3EQfFywB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1670214708613cf97e107852f59c63a98211717 --- /dev/null +++ b/IdFJT4oBgHgl3EQfFywB/content/tmp_files/load_file.txt @@ -0,0 +1,287 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf,len=286 +page_content='Massive Stars Near and Far Proceedings IAU Symposium No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 361, 2022 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' St-Louis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Vink & J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Mackey, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' © 2022 International Astronomical Union DOI: 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='0000/X000000000000000X Dynamical vs Supernova Acceleration of OB Stars in the Small Magellanic Cloud M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Oey1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Dorigo Jones2, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Phillips1, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Castro3, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Dallas1,4, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Moe5 1University of Michigan, Astronomy Department, Ann Arbor, MI, 48109-1107, USA 2University of Colorado, Department of Astrophysical and Planetary Sciences, 2000 Colorado Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=', Boulder, CO 80309, USA 3Leibniz-Institut f¨ur Astrophysik, An der Sternwarte, 16 D-14482, Potsdam, Germany 4Current address: Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 5University of Arizona, Astronomy Department, Tucson, AZ, 85721, USA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' We use the RIOTS4 sample of SMC field OB stars to determine the origin of massive runaways in this low-metallicity galaxy using Gaia proper motions, together with stellar masses obtained from RIOTS4 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' These data allow us to estimate the relative contributions of stars accelerated by the dynamical ejection vs binary supernova mechanisms, since dynamical ejection favors faster, more massive runaways, while SN ejection favors the opposite trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' In addition, we use the frequencies of classical OBe stars, high-mass X-ray binaries, and non- compact binaries to discriminate between the mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Our results show that the dynamical mechanism dominates by a factor of 2 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' This also implies a significant contribution from two-step acceleration that occurs when dynamically ejected binaries are followed by SN kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' We update our published quantitative results from Gaia DR2 proper motions with new data from DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' massive stars — Be stars — runaway stars — interacting binary stars — field stars — Small Magellanic Cloud — open star clusters — multiple star evolution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Introduction We know that at least 70% of most massive stars become interacting binaries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=', Sana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Moe & Di Stefano 2017), and that close binaries and multiples generate runaway stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Therefore, field massive stars offer an important probe of the massive binary population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' In particular, the field OB stars in the Small Magellanic Cloud (SMC) are beautifully accessible, since they are bright and easily identified in this Milky Way satellite galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Since the SMC is a dwarf galaxy, the complete OB population can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Moreover, the SMC is metal-poor, allowing us to quantitatively characterize conditions and processes at low metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Our group carried out the Runaways and Isolated O-Type Star Spectroscopic Survey of the SMC (RIOTS4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Lamb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 2016), which identified and spectroscopically confirmed a uniform sample of field OB stars in the SMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' The target stars are those at least 28 pc from other OB stars in this galaxy, photometrically identified as having B ⩽ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='21 and reddening-free Q-parameter ⩽ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='84 (Oey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Vargas-Salazar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' (2020, and these proceedings) find evidence that ∼ 5% of the sample may have formed in situ (see also Oey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' de Wit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Pflamm-Altenburg & Kroupa 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Therefore, ∼ 95% of these field OB stars are ejected from clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' There are two ejection mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Following the nomenclature of Hoogerwerf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='11444v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='SR] 26 Jan 2023 2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Oey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' (2000), they are the dynamical ejection scenario (DES;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=', Poveda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 1967) and the binary supernova scenario (BSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=', Blaauw 1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' The DES mechanism relies heavily on the interaction of binaries with other stars and other binaries or multiples to accelerate stars above the escape velocities of their parent clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' This includes also ejecting some tight binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Meanwhile, the BSS mechanism requires only a single close binary system, in which the SN of the primary unbinds the remaining star, which may also receive a kick from the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' On average, the DES produces faster, more massive runaways since it leverages the gravitational energy of multiple stars, while the BSS generates lower- velocity ejections of lower-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' In binaries ejected by the DES, the explosion of the primary will re-accelerate the companion, resulting in a two-step ejection (Pflamm- Altenburg & Kroupa 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' OBe stars as BSS products Classical OBe stars are known to be fast rotators that have generated decretion disks responsible for their characteristic Balmer emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=', Rivinius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Dorigo Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' These stars most likely acquired their high rotation velocities during binary mass exchange, which also transfers angular momentum (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=', Kriz & Harmanec 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Pols et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Therefore, many, if not most, of these stars are likely to be post- SN systems and thus BSS products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Following Smith & Tombleson (2015), we compare the spatial distributions of various populations using the distance of each star from the nearest O-star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Cumulative distribution function of projected distances from nearest O-stars for different populations as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' From Dallas, Oey & Castro (2022), in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' (A color version of this figure is available in the electronic version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=') Projected distance to nearest O-type star (pc) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='0 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='0 1i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='0 330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='0 Early0stars(03-07) Late0stars(08-09) B stars (B0-B2) Oe stars 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='8 Be stars O,B I/lI stars total Oe/BeI/ll stars All HMXB stars 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='6 Fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='2 a EarlyOstaruncertainties 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='0 Bestaruncertainties 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='5 Log (Projected distance to nearest O-type star)/degDynamical vs Supernova Runaways in the SMC 3 Figure 1 shows the cumulative distribution functions of nearest O-star distances for early and late O-stars, B-stars, Oe stars, Be stars, supergiants (luminosity class I and II), and high-mass X-ray binaries (HMXBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' As expected, early O-stars are closest to other O-stars, while late O-stars and B-stars have larger median distances since these longer-lived populations can disperse farther into the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' The Oe and Be stars have even farther median distances, supporting the scenario that they are dominated by BSS products ejected from clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' In addition, the OBe distributions have a similar locus to the HMXBs, which are bona fide BSS products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' We also see that the Oe and Be distributions are indistinguishable, within the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' This is again consistent with their mass-transfer origin, implying that their current masses are independent of their initial masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Thus, several lines of evidence in our data point to OBe stars largely corresponding to BSS products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' These results are discussed in more detail in Dallas & Oey (2022, in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Thus, to first order, we can distinguish the DES and BSS products by using the field OBe stars to represent BSS products and the remaining stars to represent DES products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' We caution that these are unlikely to be exact representations of these populations, since some BSS systems may produce non-OBe stars, and the OBe phenomenon may also result from other, unrelated processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' We also noted above that some OB stars may have formed as in-situ field stars, which would not be ejected from clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' However, we can use these observations to make a rough first estimate of the relative contributions of the two acceleration mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Kinematics of DES versus BSS ejections Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Map of the SMC with vectors showing residual transverse velocities Oey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' (2018) relative to the local velocity fields within 5′ of each RIOTS4 target star, based on Gaia DR2 data for 304 RIOTS4 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' X Kinematic Center 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='km/s 71°30\'00" SMC\'Systemic: 435 km/s Piatek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' (2008) 72°00\'00" (2000) 30\'00" Dec 73°00\'00" 30\'00" 20°00\'00" 15°00\'00" 10°00\'00" RA (J2000)4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Oey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Adopting OBe stars as tracers of the BSS population, we use Gaia proper motions of the RIOTS4 sample to evaluate the relative contributions from the DES and BSS mech- anisms, as well as the two-step mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Figure 2 shows the proper motion velocity vectors for the 304 RIOTS4 stars with available DR2 Gaia data (Dorigo Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' These are calculated in the frame of the local velocity fields based on the blue stars having (B − V ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='14 and Mbol < −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='0 from (Massey 2002) that are within 5′ (90 pc) of each RIOTS4 target (Oey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' We show updated velocity distributions from Gaia EDR3 for 299 RIOTS4 stars in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' We see that the OBe stars, which are putative BSS products, are indeed at lower median velocities than the remaining stars, which conversely must be a population dominated by DES ejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Eclipsing binaries (EBs) and double-lined spectroscopic binaries (SB2s) are non-compact, pre-SN binaries that are thus bona fide tracers of the DES mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' These can be compared to the HMXBs, which as mentioned above, are similarly tracers of BSS ejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' We likewise see that the HMXBs are restricted to lower velocities while the EBs and SB2s have distributions extending to much higher velocities, mirroring what we see for the OBe versus non-OBe stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Velocity distribution for RIOTS4 field stars and various subsets, as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' The right panel is a zoom showing the eclipsing binaries (EB), double-lined spectroscopic binaries (SB2), and high-mass X-ray binaries (HMXB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' “Unclassified” objects are those that do not belong to any of the other subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' The dashed vertical line shows a nominal 30 km s−1 value, and the dotted lines show 1 and 2 standard deviations above the median of 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='6 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' (A color version of this figure is available in the electronic version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=') Figure 3 shows that the OBe velocity distribution extends to relatively high velocities, well in excess of 30 km s−1, whereas BSS products are largely expected to remain at “walkaway” velocities below this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' This may result from Gaia errors, which are asymmetric and tend to be positive, rather than negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' We find that the median velocity of the entire distribution decreases about 25%, from 39 km s−1 to 29 km s−1, between DR2 and EDR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' However, another, real effect is that some OBe stars have been doubly accelerated, originating as non-compact DES ejections that have subsequently 70 6 All EB 60 SB2 Unclassified 5 HMXB Oe/Be 50 EB of Stars 4 SB2 40 Number of HMXB 30 20 10 1 0 0+ 50 100 150 0 0 255075100 125 Velocity relative to field (km/s) Velocity relative to field (km/s)Dynamical vs Supernova Runaways in the SMC 5 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Distributions of mass vs transverse velocity for 283 stars in our RIOTS4 field star sample with mass determinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' The left panel shows non-OBe stars and the right panel shows OBe stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Individual EBs, SB2s, and HMXBs are overplotted as shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' experienced a SN explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' These are the “two-step” ejections (Pflamm-Altenburg & Kroupa 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Figure 4 shows density plots of the non-OBe and OBe populations for stellar mass vs velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Their tracers, the EB and SB2 populations (DES), and the HMXBs (BSS), are overplotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' The data are again consistent with the expectation that DES products on average accelerate more massive runaways to faster velocities, while BSS products show opposite trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' The tracer populations are also consistent with these patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Based on the DR2 proper motion data, Dorigo Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' (2020) find that on the order of 26% of the SMC OB population corresponds to DES products, including ∼ 8% non-compact binaries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' while ∼ 9% are BSS products and perhaps an additional ∼ 2% are two-step ejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' These values are model-dependent and rely on parameters from N-body simulations for the DES mechanism by Oh & Kroupa (2016) and binary pop- ulation synthesis models by Renzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' (2019) for the BSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' The total frequencies are larger than the SMC field population because they include ejections that do not meet the criteria for RIOTS4 field stars (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='1 of Dorigo Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 2020, for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' We caution that the measured velocities changed substantially between Gaia DR2 and EDR3, as noted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Thus, our derived frequencies have substantial uncertainties and represent only a first crude estimate of the relative contributions of the different ejection mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' We are currently in the process of updating these calculations (Phillips et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=', in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' However, the ability to make such estimates based on empirical data promises major advances in our understanding of massive binary populations and cluster dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Acknowledgements This work was supported in part by NSF grant AST-1514838 to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} 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Castro, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=', Kratter, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=', & Faerber, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' 2020, ApJ, 903, 42 Discussion Shenar: Some later Be stars could originate from mass donors that are less massive and would not have exploded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Have you considered this in your velocity distributions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Oey: No, we have not included this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' Given that binaries tend to have high mass ratios, the contribution from such objects should be relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' There are also others that we ignored, for example, not all stars that undergo mass transfer will be seen as Be stars and similarly, there may also be a significant number of Be stars that originate from a completely unrelated mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} +page_content=' So we stress that our results are simply a first rough estimate of the relative contributions of the acceleration mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdFJT4oBgHgl3EQfFywB/content/2301.11444v1.pdf'} diff --git a/NdFJT4oBgHgl3EQfzy31/content/tmp_files/2301.11645v1.pdf.txt b/NdFJT4oBgHgl3EQfzy31/content/tmp_files/2301.11645v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..51f441b4eef09c918672d8881ae59b993caa0316 --- /dev/null +++ b/NdFJT4oBgHgl3EQfzy31/content/tmp_files/2301.11645v1.pdf.txt @@ -0,0 +1,1354 @@ +Tailored plasmons in pentacene/graphene heterostructures with interlayer electron +transfer + +F. Hu1,2*, M. Kim1,2*, Y. Zhang3, Y. Luan1,2, K. M. Ho1,2, Y. Shi3, C. Z. Wang1,2†, X. +Wang3†, Z. Fei1,2† + +1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA +2U.S. DOE Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA +3National Laboratory of Solid State Microstructures, School of Electronic Science and +Engineering and Collaborative Innovation Center of Advanced Microstructures, Nanjing +University, Nanjing 210093, China + +* These authors contributed equally to this work. + +† C.Z.W. (wangcz@ameslab.gov); X.W. (xrwang@nju.edu.cn); Z.F. (zfei@iastate.edu) + +Keywords + +graphene, pentacene, vdW heterostructures, s-SNOM, plasmons, electron transfer + +Abstract + +Van der Waals (vdW) heterostructures, which are produced by the precise +assemblies of varieties of two-dimensional (2D) materials, have demonstrated many novel +properties and functionalities. Here we report a nano-plasmonic study of vdW +heterostructures that were produced by depositing ordered molecular layers of pentacene +on top of graphene. We find through nano-infrared (IR) imaging that surface plasmons +formed due to the collective oscillations of Dirac fermions in graphene are highly sensitive +to the adjacent pentacene layers. In particular, the plasmon wavelength declines +systematically but nonlinearly with increasing pentacene thickness. Further analysis and +density functional theory (DFT) calculations indicate that the observed peculiar thickness +dependence is mainly due to the tunneling-type electron transfer from pentacene to +graphene. Our work unveils a new method for tailoring graphene plasmons and deepens +our understanding of the intriguing nano-optical phenomena due to interlayer couplings in +novel vdW heterostructures. + +Main text + +Graphene plasmons are collective oscillations of Dirac quasiparticles in graphene +with many desirable characteristics including high spatial confinement, long lifetime, +broad spectral range, and electrical tunability.1-19 These unique properties make graphene +a good candidate for varieties of plasmonic applications that are not accessible by +conventional plasmonics based on noble metals. Despite the above merits, the plasmonic +properties and functionalities of graphene alone are still limited. One convenient way to +engineer graphene plasmons is by constructing van der Waals (vdW) heterostructures using +atomic layers of graphene and other two-dimensional (2D) materials. Indeed, the 2D nature +of graphene makes it extremely sensitive to interlayer couplings that could modify +dramatically the properties of plasmons. Recent studies have explored a variety of new +plasmonic phenomena in graphene-based vdW materials and heterostructures, where the + +coupling mechanisms are mainly plasmon-phonon interactions20-23 and moiré superlattice +modulations24,25. + +Here we report interlayer electron transfer as a new mechanism that can be used to +tailor surface plasmons in graphene. The samples studied here are pentacene/graphene +vdW heterostructures prepared by physical vapor transport deposition of uniform +pentacene molecular layers on graphene, and they are sitting on the standard SiO2/Si +substrates. Detailed introductions about the sample growth and device fabrication +procedures are introduced in the previous study.26 Note that molecule/graphene vdW +heterostructures have been widely studied in recent years and have demonstrated many +superior electronic and optoelectronic properties.26-28 In this work, we report a +comprehensive experimental and theoretical study of the plasmonic responses of the +pentacene/graphene heterostructures. + +To perform nano-IR studies of the plasmonic responses of the heterostructure +samples, we utilized the scattering-type scanning near-field optical microscope (s-SNOM) +that is built on a tapping-mode atomic force microscope (AFM). As illustrated in Figure +1a, the sharp metalized AFM tip is illuminated by a p-polarized IR laser beam, thus +generating an intense electric field underneath the tip apex due to the so-called ‘lightening- +rod effect’. Such a strong field is highly confined in space and possesses a wide range of +in-plane momenta (q), which enables direct optical excitation and detection of graphene +plasmons. The IR detector collects scattered photon signals off the coupled tip-sample +system. Demodulating the signals at high harmonics of the AFM tapping frequency +strongly suppresses the background signal. Furthermore, we implement a pseudo- +heterodyne interferometer that allows us to extract both the amplitude and phase +components of the IR signal. In the current work, we discuss mainly the IR amplitude (s) +signal that scales monotonically with the plasmon field amplitude right underneath the +tip.29 All our experiments were performed at ambient conditions. + +Figure 1b plots the AFM topography image of a typical pentacene/graphene +heterostructure sample, where clear staircase features are seen in the field of view, +corresponding to pentacene layers with different thicknesses. By measuring the thickness +of different regions of the sample (Figure 1d) based on the AFM line profiles (Figure S1), +we can accurately determine the number of pentacene layers as labeled in Figure 1b,c. +According to the previous study26, the orientation of pentacene molecules shows variations +from layer to layer close to the graphene interface due to the competition between the +molecule-graphene interactions and the intermolecular interactions. More specifically, as +shown in Figure 1a, pentacene molecules first form a sheet of the flat-lying wetting layer +(WL) on graphene, then the inclined one layer (1L) following by the upright-standing two +layers (2L) and few layers (3L, 4L, etc.). The orientation angles of pentacene molecules to +the substrate are 0◦, 61◦ and 82◦ for the flat-lying, inclined and standing layers, respectively. +As discussed in detail below, the molecule orientation plays a critical role in the overall +plasmonic responses of pentacene/graphene heterostructures. Note that bare WL pentacene +could exist in freshly grown samples, but it will soon disappear after exposure to air due to +the dewetting, suggesting that upright-standing molecules are possibly more favorable +energetically in air than the flat-lying molecules30. Thicker pentacene layers with standing +molecules are generally more stable and can last for many days at ambient conditions thus +suitable for systematic nano-IR studies. + + +In Figure 1c, we present the s-SNOM imaging data of the sample shown in Figure +1b where we plot the IR amplitude normalized to that of the SiO2/Si substrate. The laser +energy is set to be E =  meV that is away from the strong optical phonon resonance of +SiO2 (~140 meV), so the IR responses at this energy are predominantly due to graphene +plasmons. As shown in Figure 1c, there is a clear IR signal contrast between different +pentacene layers on graphene. For quantitative analysis, we plot in Figure 1e the IR +amplitude versus the number of pentacene layers, which indicates a systematic decrease of +the IR amplitude with increasing pentacene thickness. Moreover, we found a relatively big +drop of IR amplitude signal from 1L to 2L pentacene (~27%), but only slight declines from +2L to 3L pentacene (~9%) and from 3L to 4L pentacene (~4%). Similar pentacene thickness +dependent signal variation is also seen in other samples (e.g. sample 2 in Figure S4). + +From Figure 1c, we also observed a bright edge feature surrounding the sample. To +reveal the details about the bright edge feature, we performed high-resolution s-SNOM +imaging measurements close to the sample edge (Figure 2a-e), where we observed bright +fringe(s) parallel to the sample edge. According to previous studies15,16, these bright fringes +are generated due to the constructive interference between tip-launched and edge-reflected +surface plasmons of graphene. The plasmonic origin of these fringes is verified by +frequency-dependence studies (Figures S2 and S3). In addition to the bright fringes, we +also occasionally see weak oscillations of signals distributed along the sample edge, for +example in Figure 2c,d. These edge oscillations are generated due to scattering and +interference of one-dimensional edge plasmons and they normally appear at relatively +rough edges (e.g. in the case of Figure 2c,d) or close to sharp corners.31,32 +Now we wish to perform quantitative analysis on the imaged plasmon fringes. For +that purpose, we plot in Figure 2f-j the line profiles (grey curves) extracted perpendicular +to the fringes in Figure 2a-e, respectively. From both the IR amplitude images and profiles, +we found a systematic variation of the plasmon fringes with pentacene thickness. First, the +samples with thicker layers of pentacene show weaker fringe intensity, and the strongest +fringe is observed in bare graphene. In addition, the width of the bright fringe decreases +with increasing pentacene thickness, implying a reduction of plasmon wavelength. +Furthermore, the number of fringes decreases with increasing pentacene thickness. For +example, there are at least 3 bright fringes at the edge of bare graphene, 2 clear bright +fringes in the case of 1L pentacene on graphene, and only 1 clear fringe for 2L, 3L and 4L +pentacene on graphene. The decrease of the fringe number indicates an increase in the +plasmon damping rate. + +The fringe profiles shown in Figure 2f-j allow us to fit quantitatively the complex +plasmon wavevector qp = q1 + iq2 of graphene, based on which we can determine the +plasmon wavelength (p = 2/q1) and damping rate (p = q2/q1). To perform the fit, we +adopted a quantitative s-SNOM model that approximates the s-SNOM tip as a conducting +spheroid (Figure 3a). This model calculates accurately the s-SNOM signals by evaluating +the total radiating dipoles (pz) of the tip-sample system. By computing pz at multiple x and +z coordinates of the tip, we were able to obtain line profiles of s-SNOM signals with +quantitative accuracy. More introductions about the model are given in the Supporting +Information. The same model has been applied to calculate the plasmon fringe profiles of +bare graphene and other graphene-based vdW materials and heterostructures reported in +earlier works.15, 33-35 + + +The modeling profiles are plotted in Figure 2f-j as red dashed curves, which show +good consistency with the experimental data profiles (grey). The p and p parameters +determined through the fitting are given in Figure 3b,c, respectively. Figure 3b indicates +that p decreases systematically with increasing pentacene thickness. For example, from +bare graphene to 4L pentacene on graphene, p drops from 250 nm to 205 nm. More +interestingly, p shows a sharp reduction from 240 nm to 215 nm when the pentacene +thickness changes from 1 layer to 2 layers. This sharp reduction of p also results in an +abrupt drop of the overall IR amplitude signal from 1L to 2L pentacene (Figures 1e and +Figure S4). The plasmon damping rate p, on the other hand, increases systematically with +pentacene thickness, which is consistent with the decrease of the number of plasmon +fringes shown in Figure 2. As shown in Figure 3c, the extracted p by fitting the plasmon +fringe profiles increases from 0.14 for bare graphene to 0.17, 0.24, 0.27 and 0.3 when +adding 1L, 2L, 3L and 4L pentacene on top of graphene, respectively. Like p, p also +undergoes a larger change from 1L to 2L pentacene (~0.07) compared to that between other +adjacent layers (~0.03). +We now elaborate on the possible causes of the observed thickness dependence of +the plasmonic parameters, Under the Drude and long-wavelength approximations, the +plasmon wavevector qp can be written as 15,16,25 + +0 +1 +2 +2 +2 +( +) +p +F +q +q +iq +E E +iE +e E +  + + ++ + ++ +, (1) +where e is the elementary charge, EF is the Fermi energy of graphene, E is scattering +energy of Dirac Fermions in graphene, and  = 1 + i2 is the effective dielectric constant +of the environment of graphene (1 and 2 are the real and imaginary parts of ). For bare +graphene,  is an average value from the dielectric constants of air and SiO2:  = (1+ s)/2 +(s ≈ 4.4 + 0.3i at E = 116 meV). In the case of pentacene/graphene heterostructures, +dielectric constants of pentacene also contribute to . Note that our graphene samples are +highly doped at ambient conditions with EF above 0.4 eV (see discussions below), so +contributions from interband transitions at our energy regime are negligible thus not +considered here. From eq 1, one can obtain the plasmon wavelength p = 2/q1: + +2 +2 +1 +0 +1 +2 / +/ ( +) +p +F +q +e E +E + + +  += + +. (2) +Therefore, the observed layer dependence of p (Figure 3b) is possibly due to the change +of EF of graphene and/or the dielectric constants of pentacene. Note that eqs 1 and 2 are +mainly for discussions of general physics of graphene plasmons. We used the transfer +matrix method to compute numerically the plasmon dispersion and plasmon wavelength of +the entire pentacene/graphene/substrate system (Supporting Information). +We first evaluate the effects solely due to the dielectric screening of pentacene +layers with a fixed EF of 0.47 eV ─ the Fermi energy of bare graphene accurately +determined by fringe profile fitting (Figure 2f). The large EF indicates the high hole doping +of graphene on SiO2, which is originated from the vacuum annealing during the pentacene +growth process followed by days of air exposure.36,37 The anisotropic dielectric constants +of pentacene with different thicknesses were calculated from density functional theory +(DFT) calculations (Supporting Information), which vary from 2.1 to 2.7 in the ab plane +and from 1.3 to 2.6 along the c-axis for different pentacene thicknesses. Note that our +excitation laser energy (116 meV) is away from the strong vibrational resonances of +pentacene (the nearest strong resonance is at 112 meV with a resonance width of about 0.4 + +meV)38,39, so the vibrational modes of pentacene do not affect graphene plasmons. The +calculated p of graphene with a fixed EF under various pentacene layers is plotted in Figure +3b as blue triangles, which show a gentle and systematic decline with layer number (p ≈ +-4 nm on average per layer). Based on Figure 3b, we know that dielectric screening of +pentacene alone cannot explain the sharp drop of p as pentacene thickness increases from +1 to 2 layers. The inconsistency between experimental and calculated p assuming a fixed +EF indicates that the layer dependence of doping must be taken into consideration. Indeed, +layer-dependent EF can be obtained accurately by fitting the experimental p. As shown in +Figure 3d, graphene under 1L pentacene has slightly smaller EF (~ 0.46 eV) compared to +that of bare graphene (~ 0.47 eV). The EF of graphene under 2 to 4 layers pentacene is +much lower, down to ~ 0.42 eV. +The unique pentacene layer dependence of EF (Figure 3d) is, in fact, originated +from the charge transfer between graphene and pentacene. Charge transfer phenomena +have also been observed at the interfaces between graphene and other types of molecules +(e.g. C60, CNT, etc.).40-42 To understand the transfer process here, we plot in Figure 4 the +energy alignment diagrams between the graphene Fermi level (dashed line) and the highest +occupied molecular orbits (HOMO) level of pentacene layers. The lowest unoccupied +molecule orbits (LUMO) are ~2 eV above the HOMO level (not shown in Figure 4), so +there are no unoccupied states available in pentacene close to the Fermi level of graphene. +In Figure 4, we label the ionization potential (IP) values of graphene and pentacene, which +is the energy difference from the Fermi level of graphene or HOMO energy of pentacene +to the vacuum energy. Considering that graphene on SiO2 is hole doped at ambient +conditions36,37, the IP can be calculated to be around 5.03 eV by adding the Fermi energy +(~0.47 eV, Fermi level to Dirac point) and the work function of neutral graphene (~4.56 +eV, Dirac point to vacuum energy).42,43 +The IP of pentacene layers is sensitively dependent on the molecule orientation.44- +46 To obtain the IP values of pentacene layers, we performed first-principles electronic +structure calculations based on DFT using the Vienna ab initio simulation package.47,48 The +atomic structures of the pentacene layers (Figure S6) are adopted from the previous study.26 +Such DFT calculations tend to underestimate the value of IP49, so we considered the GW +correction (GW). Detailed introductions about the IP calculations are given in the +Supporting Information. The final IP values of WL, 1L, 2L, and 3L pentacene without and +with GW corrections are summarized in Table 1, where one can see a big drop (~0.77 eV) +of IP from WL to 1L pentacene, followed by a small drop (~0.18 eV) from 1L to 2L +pentacene. Starting from 2L pentacene and above, IP stays constant at 4.78 eV. Such a +layer dependence is originated from the difference of the orientation angles of pentacene +molecules (0◦, 61◦ and 82◦ for the WL, 1L, and 2L or above, respectively). Our IP +calculations are consistent with previous experimental results.45,46 Note that the interface +dipole between graphene and pentacene induces a shift of pentacene vacuum level by ∆ ≈ +0.1 eV.46 + +Pentacene layers +IPDFT (eV) +IPDFT +GW (eV) +WL +4.76 +5.72 +1L +3.99 +4.95 + +2L +3.81 +4.77 +3L +3.82 +4.78 +Table 1. The calculated ionization potential (IP) of pentacene with different +thicknesses. + +Based on Figure 4, one can see that the Fermi level of graphene is much higher than +the HOMO energy level of WL pentacene, so charge transfer between graphene and WL +pentacene is forbidden. For 1L to 4L pentacene, HOMO energy level rises above EF of +graphene, so electron transfer from pentacene to graphene is enabled. The amount of +electron transfer from 1L pentacene to graphene is much less compared to that from thicker +pentacene layers. For 1L pentacene, the reduction of graphene EF due to the charge transfer +(EF) is about 0.013 eV, corresponding to the change of carrier density (n) of about +0.9×1012 cm-2. For 2L to 4L pentacene, the resulting EF is about 0.047 eV, corresponding +to the n ≈ 3.1×1012 cm-2. The size of EF is mainly due to the potential difference between +graphene and pentacene (EIP) (Figure 4). The amount of EIP for 1L pentacene (~0.08 +eV) is much smaller than those of 2L to 4L pentacene (~0.25 eV). Another relevant factor +is the density of states of pentacene layers. In principle, few-layer pentacene should have +more electrons to offer compared to 1L pentacene. Note that the electron transfer discussed +here is a tunneling process due to the presence of the WL pentacene that acts as a tunneling +barrier (Figure 4) with a thickness of about 0.5 nm.26 Effects of electron tunneling on +surface plasmons have been studied in metal-molecule junctions, where unique quantum +plasmonic responses were observed.50 It is also proposed that electron tunneling can be +utilized to generate graphene plasmons.51,52 Therefore, the molecule/graphene +heterostructure with interlayer electron tunneling studied here provides a unique platform +to explore further the role of electron tunneling on graphene plasmons. +Finally, we wish to discuss the dependence of plasmon damping rate p on +pentacene thickness. As discussed above (Figure 3c), p increases with pentacene thickness, +and the increment of p is larger from 1L to 2L pentacene (~0.07) compared to that between +other adjacent layers (~0.03), which implies a possible link between electron transfer and +plasmon damping. Based on eq 1, we know that p can be written approximately as: + +2 +1 +2 +1 +/ +/ +/ +p +q +q +E +E + + + + += + ++ +, (3) +which indicates that p originates from both the loss due to the dielectric environment and +the scattering of Dirac fermions in graphene. As discussed above,  ≈ 2.7 + 0.15i for +graphene sitting directly on SiO2 at E = 116 meV. As a semiconductor, pentacene behaves +like a good dielectric with a negligible imaginary part of permittivity at the mid-IR region +if it is away from the vibrational modes39, so pentacene itself has little contribution to  at +our excitation energy. Therefore, the enhanced p when adding pentacene layers is most +likely due to the scattering of graphene carriers by impurities or localized charges in +pentacene. With electron transfer, additional localized charges could be introduced to +pentacene, which cause higher damping to graphene plasmons (Figure 3c). Increased +charge scattering leading to a lower carrier mobility has been observed previously in +transport studies of C60/graphene heterostructures, where charge transfer was also +involved.53 + + +In summary, we have performed the first nanoplasmonic study of vdW +heterostructures formed by organic 2D materials and graphene. By using the nano-IR +imaging technique, we discovered that the graphene plasmons could be tailored by +depositing molecule layers of pentacene on graphene. Unlike electrical gating that requires +a constant bias voltage, the molecular deposition method is suitable for creating +heterostructure samples or devices with tailored permanent properties for long-term +applications. Through quantitative analysis and DFT calculations, we proved that the +pentacene-layer dependence of graphene plasmons is mainly due to tunneling-type electron +transfer from pentacene to graphene. Moreover, we found the electron transfer process is +determined by the molecule orientation of each pentacene layer. Such a unique sensitivity +to molecular orientations is highly desired for structural characterizations of molecules and +bio-nanoparticles. Of course, the studies should not be limited to pentacene/graphene +heterostructures. We expect more interesting nano-optical properties and functionalities to +be discovered in heterostructures formed by graphene with other types of molecules. Our +work broadens the understanding of the interlayer interactions of graphene with +biomolecules and opens the door to future studies and applications of molecule/graphene +heterostructures in nanophotonics and optoelectronics. + +Acknowledgments + +Work done at Ames Lab was supported by the U.S. Department of Energy, Office +of Basic Energy Science, Division of Materials Sciences and Engineering. Ames +Laboratory is operated for the U.S. Department of Energy by Iowa State University under +Contract No. DE-AC02-07CH11358. The nano-optical imaging set-up was partially +supported by the W. M. Keck Foundation. X.W. acknowledges the funding support from +the National Natural Science Foundation of China 61734003 and the National Key Basic +Research Program of China 2013CBA01604. + +References +1. Ryzhii, V.; Satou, A.; Otsuji, T. J. Appl. Phys. 2007, 101, 024509. +2. Jablan, M.; Buljan, H.; Soljačić, M. Phys. Rev. B 2009, 80, 245435. +3. Koppens, F. H. L.; Chang, D. 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W.; Bai, P.; Bosman, M.; Nijhuis, C. A. Science 2014, +343, 1496-1499. +51. De Vega, S.; Javier Garcia de Abajo, F. ACS Photonics 2017, 4, 2367-2375. +52. Ooi, K. J. A.; Chu, H. S.; Hsieh, C. Y.; Tan, D. T. H. Ang, L. K. Phys. Rev. Applied +2015, 3, 054001. + +53. Ojeda-Aristizabal, C.; Santos, E. J. G.; Onishi, S.; Yan, A.; Rasool, H. I.; Kahn, S.; Lv, +Y.; Latzke, D. W.; Jairo Velasco, Jr.; Crommie, M. F.; Sorensen, M.; Gotlieb, K.; Lin, +C.-Y.; Watanabe, K.; Taniguchi, T.; Lanzara, A.; Zettl, A. ACS Nano 2017, 11, 4686- +4693. + +Figures + + +TOC figure + + +Figure 1. a, Illustration of the s-SNOM study of a pentacene/graphene heterostructure. The +arrows sketch the incident laser and back-scattered photons, respectively. b, The AFM +topography image of a typical pentacene/graphene heterostructure sample. c, The IR +amplitude (s) image of the pentacene/graphene heterostructure taken at a photon energy of +E = 116 meV. Here we normalized the amplitude signal to that of SiO2. The labeling ‘WL’ +represents the wetting layer on graphene, ‘1L’-‘4L’ represent 1-layer to 4-layer pentacene +on graphene, and ‘G’ represents bare graphene. The scale bars: 1 m. d, The thickness (d) +of sample relative to the SiO2 surface versus the number of pentacene layers. e, The IR +amplitude signal versus the number of pentacene layers measured at locations away from +the edge of the sample in Figure 1c. In d and e, the 0 pentacene layer corresponds to bare +graphene. + + +pentacene/graphene +1L +IR laser +graphene +2L +3L +2L +S +11 +3L +WL +graphene +SiO2 +4L +SiO2 +2um +pentacene +5LAFM +Infrared +a +b +C +d +6 +1L +d(nm) +11 +2L +2L +0 +1 +3 +Pentacenelayers +2L +e +1L +3L +3L +(rel. unit) +WL +G +sio- +4L +SiO2 +40 +克克 +5L +51 +SiO2 +s +0 +-7 +17 +0 +72 +0 +2 +3 +4 +Pentacenelayers +height (nm) +s (norm.) +Figure 2. a-e, High-resolution IR amplitude images of bare graphene (G) and +pentacene/graphene heterostructures (1L to 4L) with various pentacene thicknesses taken +at E = 116 meV. The blue dashed curves mark the sample edges. The scale bars: 200 nm. +f-j, The IR amplitude line profiles from both experiments (grey) and simulations (red). The +experimental profiles were taken along the black dash lines in a-e. In all the images and +profiles, the IR amplitude signal is normalized to that of the SiO2/Si substrate. + + +Figure 3. a, Illustration of the spheroid model with different parameters that we used to +model the plasmon fringes profiles in Fig. 2f-j. b, The plasmon wavelength (p) versus the +number of pentacene layers from fringe profile fitting (black squares) and theoretical +calculations (blue triangles) assuming a fixed Fermi energy of 0.47 eV. c, The plasmon +damping rate (p) versus the number of pentacene layers from fringe profile fitting. d, The +Fermi energy of graphene versus the number of pentacene layers calculated based on the +fitted p data (squares) in panel b. In panels b-d, the 0 pentacene layer corresponds to bare +graphene. + + +- +b +C +d +2 +G +1L +Sio2 +SiO2 +SiO2 +Sio +SiO2 +3 +g +h +(norm.) +2 +G +1L +2L +3L +4L +S +- +0 +200 +400 +0 +200 +400 +0 +200 +400 +0 +200 +400 +0 +200 +400 +x (nm) +x (nm) +x (nm) +x (nm) +x (nm)a +b +280 +spheroid tip +2L +240 +p:t +(wu) +200 +pentacene (cab, c) +graphene +81 +(g) +160 +82 +0 +1 +2 +4 +Pentacene layers +c 0.4 +d +0.5 +0.3 +(eV) +xp 0.2 +0.1 +0.0 +0.3 +0 +1 +2 +3 +4 +0 +1 +2 +3 +4 +Pentacene layers +Pentacene layers +Figure 4. Illustration of the graphene/pentacene heterostructure (top) and the energy level +alignment diagram (bottom). As thickness increases, pentacene changes from flat-laying +wetting layer (WL), inclined one layer (1L) to standing few layers (2L, 3L, and 4L) relative +to graphene, resulting in a lifting of pentacene HOMO energy level. The energy values +listed in the alignment diagram are the ionization potentials (IP) of graphene and pentacene +layers before electron transfer. + +Supporting Information for + +Tailor plasmons in pentacene/graphene heterostructures with interlayer electron +transfer + +F. Hu1,2*, M. Kim1,2*, Y. Zhang3, Y. Luan1,2, K. M. Ho1,2, Y. Shi3, C. Z. Wang1,2†, X. +Wang3†, Z. Fei1,2† + +1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA +2U.S. DOE Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA +3National Laboratory of Solid State Microstructures, School of Electronic Science and +Engineering and Collaborative Innovation Center of Advanced Microstructures, Nanjing +University, Nanjing 210093, China + +* These authors contributed equally to this work. + +† C.Z.W. (wangcz@ameslab.gov); X.W. (xrwang@nju.edu.cn); Z.F. (zfei@iastate.edu) + + +List of contents + +Graphene +WL +1L +2L,3L,4L +C:0-0 +△-0.1eV +VacuumLevel +~4.78eV +IP~5.03eV +-5.72 eV +~4.95eV +Fermilevel +HOMO +HOMO +HOMO +graphene +WL +1L +2L, 3L,4L + +1. Experimental details + + +2. Additional s-SNOM imaging data + + +3. Numerical modeling of the plasmon fringe profiles + + +4. Calculations of the plasmon wavelength + + +5. DFT calculation methods and results + + +6. Dielectric constants of pentacene + + + Figures S1 – S9 + + +1. +Experimental details + +To perform nano-infrared (IR) imaging studies of the pentacene/graphene +heterostructures, we employed the scattering-type scanning near-field optical microscopy +(s-SNOM). Our s-SNOM apparatus (Neaspec GmbH) is based on an atomic force +microscope (AFM) operating in the tapping mode. Measurements were acquired at an AFM +tapping frequency of  = 270 kHz and a tapping amplitude of about 60 nm. As illustrated +in Figure 1a, we utilized a metalized AFM probe, which is illuminated by a p-polarized +mid-IR beam from a continuous-wave CO2 laser (Access Laser). In our s-SNOM +measurements, we used Arrow-NCPt probes from NanoAndMore. The radius of tip apex +of these probes is about 25 nm that defines the spatial resolution of the s-SNOM. The +standard observable of an s-SNOM experiment is complex scattering signal demodulated +at the nth (n = 3 in the current work) harmonics of the AFM tip oscillation. We discuss +mainly the amplitude part of the signal that is enough to describe the plasmonic responses +of the samples. + +Our pentacene/graphene vdW heterostructures were prepared by physical vapor +transport deposition of uniform pentacene molecular layers on graphene. The +heterostructure samples are sitting on the standard silicon wafers with 300-nm-thick +thermal oxide on the top. The samples that we studied in this work include bare graphene, +and one-layer (1L), two-layer (2L), three-layer (3L) and four-layer (4L) pentacene on +graphene, determined by accurate AFM measurements (Figure S1). + +2. +Additional s-SNOM imaging data + +In Figures S2 and S3, we present the excitation energy (E) dependent nano-IR +amplitude images of graphene, 1L pentacene on graphene, and 2L pentacene on graphene. +Here the IR amplitude is normalized to that of the SiO2 substrate. From Figures S2 and S3, +one can see that the IR contrast between the samples and the SiO2 substrate shows a clear +evolution with energy. This is mainly due to the increase of the substrate signal as E +approaches the surface phonon resonance of SiO2 at around 140 meV. Moreover, the fringe +period or the fringe width of the samples shrinks with increasing laser energy, indicating +smaller plasmon wavelength. This is consistent with the dispersion properties of graphene + +plasmons (Figure S5). In all laser energies, there is a small signal difference between bare +graphene and 1L pentacene on graphene and a larger contrast between 1L and 2L pentacene +on graphene. This is consistent with the results discussed in the main text (Figures 1-3). + +Figure S4a,b present the nano-IR imaging data of two heterostructure samples at an +excitation energy of E = 110 meV. Sample 1 is the one that we extensively studied in the +main text. Sample 2 is from a different wafer and it also contains bare graphene and 1L to +4L pentacene on graphene. Note that the excitation energy used here is slightly lower than +that used in Figures 1 and 2 in the main text (116 meV). Here we compare the general +signal contrast of different sample areas, which show good consistency among the two +samples. For quantitative comparison, we plot in Figure S4c the average IR amplitude at +the sample interior versus the number of pentacene layers (0 layer corresponds to bare +graphene). Here one can see that the general trend of the signal evolution with layer +thickness is consistent in the two heterostructure samples. There is a slight difference in 1L +pentacene on graphene, which is possibly due to the degradation of 1L pentacene on sample +2 that leads to nonuniform signal distributions (Figure S4b). + +3. Numerical modeling of the plasmon fringe profiles + +To model the fringes profiles of plasmons confined inside graphene or +pentacene/graphene heterostructures, we model our AFM tip as an elongated metallic +spheroid (see Figure 3a in the main text): the length of the spheroid is 2L and the radius of +curvature at the tip ends is a. Here, a is set to be 25 nm according to the manufacturer and +L is set to be 500 nm and it is not a very sensitive parameter so long as L >> a. The scattering +amplitude s (before demodulation) scales with the total radiating dipole pz of the spheroid. +Therefore, to fit the line profiles perpendicular to the fringes inside samples, we need to +calculate pz at different spatial coordinates (x, z) of the lower end of the AFM tip. Here, x +is the in-plane coordinate perpendicular to samples and z is the out-of-plane coordinate +perpendicular to the sample surface. By calculating pz at different z, we can perform +‘demodulation’ of the scattering amplitude s and get different harmonics of the scattering +signal and calculating pz at different x allows us to plot the modeling profiles of IR +amplitude. In all our simulations, we assume no position dependence in the y-direction for +simplicity. The dielectric constants of SiO2 used in the calculations are adopted from +literature.1 The dielectric constants of pentacene layers (ab, c) used in the calculations +given in Section 6 below. The key modeling parameters for graphene are the plasmon +wavelength (p) and damping rate (p). By fitting the experimental plasmon fringe profiles +(Figure 2f-j in the main text), we can determine accurately p and p based on the +experimental data (Figure 3b,c in the main text). + + +4. Calculations of the plasmon wavelength + +In order to determine the Fermi energy (EF) of graphene, we need to calculate p +theoretically and then compared to the experimental result obtained from fringe profile +fitting (see the section above). For that purpose, we first compute the plasmon dispersion +colormaps (Figure S5) by evaluating numerically the imaginary part of the reflection +coefficient Im(rp) for the entire pentacene/graphene/substrate heterostructure system by +using the transfer matrix method. These colormaps reveal the photonic density of states +(DOS), and the plasmonic mode appears as a bright curve revealed by the colormaps. Such +a dispersion calculation method has been widely applied in the studies of graphene + +plasmons and other types of polaritons. The optical conductivity of graphene is obtained +by the Radom phase approximation methods (see Ref. 11 in the main text). The dielectric +constants of pentacene layers used in the calculations given in Section 6 below. The +dielectric constants of SiO2 used in the calculations are adopted from literature.1 Based on +the calculated dispersion colormaps, we can determine the plasmon wavevector qp and +hence the plasmon wavelength p = 2/qp at any excitation energy. The dispersion +colormaps shown in Figure S5 are calculated with different choices of EF values shown in +Figure 3d in the main text. The p values read out from these colormaps match well the p +results determined through fringe profile fitting (Figure 2 in the main text), which confirms +the validity of EF values shown in Figure 3d. + +5. DFT calculation methods and results + +We performed first-principles electronic structure calculations based on density +functional theory (DFT) using Vienna ab initio simulation package (VASP).2,3 We +employed the projector augmented wave method4 and a plane-wave basis set with 400 eV +energy cutoff. For the exchange-correlation functional, we used Perdew-Burke-Ernzerhof +(PBE) functional5, and a total of 521 k point meshes were used. We included the van der +Waals energy using the DFT-D3 method.6 In our slab calculations for the pentacene thin +films, we used sufficiently thick vacuum regions (> 23 Å) to prevent the unwanted +interactions between periodic images. We considered the structural model of pentacene +layers that consist flat-lying wetting layer (WL) on graphene, the inclined one layer (1L) +and the standing two (2L) and three layers (3L) (Figure S6).7 The dielectric matrix was +determined using density functional perturbation theory. In Figures S7 and S8, we +presented the calculated band structures of pentacene layers and their potential energy line +profiles taken along the vertical direction. From these calculations, we can determine the +ionization potential (IP) of all pentacene layers (Table 1 in the main text). + +Due to the self-interaction error, the conventional DFT calculations with local +density approximation (LDA) or generalized gradient approximation (GGA) are not +supposed to give accurate valence band energies, which in turn results in underestimated +IP values.8 Therefore, we performed the GW calculation (in the GW0 level) that is expected +to give a reasonable value for the IP. Due to the large computational cost, we used the +approximation where we estimate the shift of the valence band maximum by that of the +highest valence band energy at the  point of the Brillouin zone (VB,) in the 3D bulk +pentacene crystal (Figure S9a ). In the GW calculation of the bulk pentacene, we used +221 k point meshes. We considered further correction coming from limited number of +empty bands (eb) Specifically, we estimated the valence band energy with the infinite +number of the empty bands (Neb = ) included in the calculation by fitting the results of a +series of different Neb to the formula A/Neb + B. + +The results of both corrections are shown in Figure S9b,c, respectively. We denoted +the sum of the two correction terms by GW (i.e., GW = VB, + eb). The total GW +correction is as large as  0.96 eV. After corrections, the IPs of thin pentacene layers are +calculated to be 5.72 eV, 4.95 eV, 4.78 eV, and 4.78 eV for WL, 1L, 2L and 3L pentacene, +respectively (Table 1 in the main text). These results are consistent with previous +experiments.9 Based on the calculation results, we conclude that the IP of pentacene is +solely dependent on the orientation angle of pentacene molecules. Therefore, we expect +that IPs of thicker pentacene layers (4L or above) are all about 4.78 eV. + + +6. Dielectric constants of pentacene + +The dielectric constants of pentacene layers we used for the fringe profile modeling +and dispersion calculations are also from DFT calculations. With DFT, we calculated the +static dielectric constants of pentacene layers with different thicknesses. It is known from +the previous study10 that pentacene has a flat dielectric response up to the visible region +when it is away from the strong vibrational modes of pentacene (the nearest strong +resonance is at about 112 meV with a resonance width of 0.4 meV). Therefore, it is +appropriate to use the static dielectric constants for calculations in the mid-infrared region. +The calculated in-plane dielectric constants (ab) for 1L, 2L and 3L pentacene are about +2.1, 2.6 and 2.7, respectively. The out-of-plane dielectric constants (c) for 1L, 2L and 3L +pentacene are about 1.3, 2.0 and 2.6, respectively. Both ab and c increase with pentacene +thickness and they are trending towards the value for bulk pentacene films: bulk ≈ 3.0.10 +For 4L pentacene, we used dielectric constants of 3L pentacene as an approximation. The +calculated p only varies a little (~1.5%) even using bulk ≈ 3.0 for 4L pentacene. There is +in fact a small ab-plane anisotropy (about 1%, 5% and 12% for 1L, 2L and 3L pentacene, +respectively) in the dielectric constants according to our calculations, but it only causes +tiny variations to p (about 0.05%, 0.3% and 0.8% for 1L, 2L and 3L pentacene, +respectively) due to the nanoscale thicknesses of the pentacene layers. Therefore, we used +averaged ab-plane dielectric constants in our calculations (aa + bb)/2. + +References for the Supporting Information +1. +Palik, E. D. Handbook of Optical Constants of Solids. Academic Press (2012). +2. +Kresse, G.; Hafner, J. Phys. Rev. B 1993, 47, 558. +3. +Kresse, G.; Furthmüller, J. Phys. Rev. B 1996, 54, 11169. +4. +Blöchl, P. E. Phys. Rev. B 1994, 50, 17953. +5. +Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865. +6. +Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. J. Chem. Phys. 2010, 132, 154104. +7. +Zhang, Y.; Qiao, J.; Gao, S.; Hu, F.; He, D.; Wu, B.; Yang, Z.; Xu, B.; Li, Y.; Shi, +Y.; Ji, W.; Wang, P.; Wang, X.; Xiao, M.; Xu, H.; Xu, J.-B.; Wang, X. Phys. Rev. Lett. +2016, 116, 016602. +8. +Jiang, H.; Shen, Y.-C. J. Chem. Phys. 2013, 139, 164114. +9. +Yoshida, H.; Yamada, K.; Tsutsumi, J.; Sato, N. Phys. Rev. B 2015, 92, 075145. +10. +Schubert, M.; Bundesmann, C.; Jacopic, G.; Maresch, H.; Arwin. H. Appl. Phys. +Lett. 2004, 84, 200-202. + + + +Figure S1. The AFM topography profiles of graphene (G) and 1L to 4L pentacene on +graphene extracted from Figure 1b in the main text. Here the 0 pentacene layer corresponds +to bare graphene. + + +Figure S2. Excitation laser energy dependent nano-IR imaging data of bare graphene (G) +and 1L pentacene on graphene. Here we plot the IR amplitude normalized to that of the +SiO2 substrate. + + +8 +G +1 L +2L +9 +3L +Height (nm) +41 +4 +0 +-100 +0 +100 +200 +300 +(wu) xa +E = 110 meV +b +E = 112 meV +c +E= 115 meV +1L +SiO2 +G +1L +SiO2 +G +1L +SiO2 +G +3 +S +d +E = 121 meV +e +E = 128 meV +t +E=134 meV +0 +1L +SiO2 +G +1L +SiO2 +G +SiO2 +G +400nm +Figure S3. Excitation laser energy dependent nano-IR imaging data of 1L and 2L +pentacene on graphene. Here we plot the IR amplitude normalized to that of the SiO2 +substrate. + + + +Figure S4. a,b, Nano-IR imaging of two samples at an excitation energy of 110 meV +(slightly lower energy compared to that used in Figure 1 and 2 in the main text). Sample +1 is the sample we studied extensively in the main text. Sample 2 is a different sample on +a different wafer. Scale bars: 2 m. c, The IR amplitude signals of the two samples taken +from a,b versus the number of pentacene layers. + + +E=110meV +b +a +E= 112 meV +C +E = 115 meV +1L +1L +1L +SiO2 +SiO2 +SiO2 +2L +2L +2L +73 +d +E= 121 meV +e +E=128meV +h +E=134meV +S +1L +0 +1L +1 +SiO2 +SiO2 +SiO2 +2L +2L +2L +400nmSample 1 +G +3L 4L +Sample 2 +a +b +C +3.0 +--Sample1 +2.5 +-- Sample 2 +8 +2L +2.0 +G +('n: +1.5 +2L +S +1.0 +口 +3L +0.5 +SiO2 +4L +SiO2 +0.0 +1 +1 +1 +L +0 +1 +2 +3 +4 +0 +13 +s (norm.) +Layer number +Figure S5. Calculated dispersion colormaps of bare graphene (a) and pentacene layers with +different thicknesses on graphene (b-e). The white dashed curves mark the dispersion +relation of graphene plasmons revealed by the color maps. The horizontal and vertical blue +dashed lines mark the excitation energy (E = 116 meV) and corresponding plasmon +wavevector determined by the dispersion diagrams. + + +Figure S6. Atomic structures of different pentacene layers that we constructed for DFT +calculations. + + +a +b +Graphene +1Lpentaceneongraphene +125 +125 +120 +120 +(Maw) +115 +115 +110 +110 +2 = 250 nm +p = 240 nm +105 +105 +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +q(cm-1) +×105 +q(cm-1) +×105 +c +2Lpentaceneongraphene +d +3Lpentaceneongraphene +e +4Lpentaceneongraphene +125 +125 +125 +120 +120 +120 +(Maw) +115 +115 +115 +E +E +110 +110 +110 +2p = 215 nm +2p = 210 nm +2p = 205 nm +105 +105 +105 +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +q(cm-1) +×105 +q(cm-1) +×105 +q(cm-1) +×105e +WLpentacene +C +2Lpentacene +p +3Lpentacene +b +1Lpentacene + +Figure S7. The DFT calculations of band structures of WL, 1L, 2L and 3L pentacene. In +all the plots, the band structures are shifted on purpose to set the valence band maximum +right at 0 eV. + + + +Figure S8. The DFT calculations of potential energy profiles of WL, 1L, 2L, and 3L +pentacene along the c axis (perpendicular to the pentacene layers). + + + + +a +b +WL pentacene +1L pentacene +1.0 +1.0 +0.5 +0.5 +Energy (eV) +Energy (eV) +0.0 +0.0 +-0.5 +-0.5 +-1.0 +-1.0 +X +S +Y +S +X +S +Y +s +c +2L pentacene +d +3L pentacene +1.0 +1.0 +0.5 +0.5 +Energy (eV) +Energy (eV) +0.0 +0.0 +0.5 +-0.5 +-1.0 +-1.0 +X +S +┌Y +S +X +S +Y +Sa +WL pentacene +b +1L pentacene +3.0 +3.0 +2.0 +2.0 +1.0 +1.0 +0.0 +@-1.0 +@-1.0 +0 -2.0 +6 -2.0 +iers +-3.0 +-3.0 +-4.0 +卤-4.0 +-5.0 +-5.0 +-6.0 +-6.0 +-7.0 +-7.0 +2L pentacene +d +3L pentacene +c +4.0 +5.0 +3.0 +4.0 +2.0 +3.0 +1.0 +2.0 +e +0.0 +e +1.0 +o-1.0 +60 +0.0 +-2.0 +-1.0 +卤-3.0 +-2.0 +-4.0 +-3.0 +-5.0 +-4.0 +-6.0 +-5.0 +Figure S9. a, Unit cell of bulk pentacene that we used for GW calculations. b, Shift of the +valence band maximum by that of the highest valence band energy at the  point of the +Brillouin zone (VB,). c, Correction coming from limited number of empty bands (eb) + + + + + + + + + + + + + + + + + + + +a +b +4.0 +C +0.6 +CB +3.5 +VB +0.5 +3.0 +(eV) +0.4 +2.5 +(eV) +0.3 +2.0 +0.2 +1.5 +at +0.1 +1.0 +B +0.5 +0.0 +AvB,r = 0.78 +Aeb = 0.18 +0.0 +-0.1 +-0.5 +-0.2 +DFT +GoWo +GWo +0 +0.0003 +0.0006 +0.0009 +Neb \ No newline at end of file diff --git a/NdFJT4oBgHgl3EQfzy31/content/tmp_files/load_file.txt b/NdFJT4oBgHgl3EQfzy31/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cbccac225c2897820ed936d554a15c6aff46a887 --- /dev/null +++ b/NdFJT4oBgHgl3EQfzy31/content/tmp_files/load_file.txt @@ -0,0 +1,1926 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf,len=1925 +page_content='Tailored plasmons in pentacene/graphene heterostructures with interlayer electron transfer F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Hu1,2*, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Kim1,2*, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Zhang3, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Luan1,2, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Ho1,2, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Shi3, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Wang1,2†, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Wang3†, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Fei1,2† 1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA 2U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' DOE Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA 3National Laboratory of Solid State Microstructures, School of Electronic Science and Engineering and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' † C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' (wangcz@ameslab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='gov);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' (xrwang@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='cn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' (zfei@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='edu) Keywords graphene, pentacene, vdW heterostructures, s-SNOM, plasmons, electron transfer Abstract Van der Waals (vdW) heterostructures, which are produced by the precise assemblies of varieties of two-dimensional (2D) materials, have demonstrated many novel properties and functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Here we report a nano-plasmonic study of vdW heterostructures that were produced by depositing ordered molecular layers of pentacene on top of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' We find through nano-infrared (IR) imaging that surface plasmons formed due to the collective oscillations of Dirac fermions in graphene are highly sensitive to the adjacent pentacene layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In particular, the plasmon wavelength declines systematically but nonlinearly with increasing pentacene thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Further analysis and density functional theory (DFT) calculations indicate that the observed peculiar thickness dependence is mainly due to the tunneling-type electron transfer from pentacene to graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Our work unveils a new method for tailoring graphene plasmons and deepens our understanding of the intriguing nano-optical phenomena due to interlayer couplings in novel vdW heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Main text Graphene plasmons are collective oscillations of Dirac quasiparticles in graphene with many desirable characteristics including high spatial confinement, long lifetime, broad spectral range, and electrical tunability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='1-19 These unique properties make graphene a good candidate for varieties of plasmonic applications that are not accessible by conventional plasmonics based on noble metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Despite the above merits, the plasmonic properties and functionalities of graphene alone are still limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' One convenient way to engineer graphene plasmons is by constructing van der Waals (vdW) heterostructures using atomic layers of graphene and other two-dimensional (2D) materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Indeed, the 2D nature of graphene makes it extremely sensitive to interlayer couplings that could modify dramatically the properties of plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Recent studies have explored a variety of new plasmonic phenomena in graphene-based vdW materials and heterostructures, where the coupling mechanisms are mainly plasmon-phonon interactions20-23 and moiré superlattice modulations24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Here we report interlayer electron transfer as a new mechanism that can be used to tailor surface plasmons in graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The samples studied here are pentacene/graphene vdW heterostructures prepared by physical vapor transport deposition of uniform pentacene molecular layers on graphene, and they are sitting on the standard SiO2/Si substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Detailed introductions about the sample growth and device fabrication procedures are introduced in the previous study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='26 Note that molecule/graphene vdW heterostructures have been widely studied in recent years and have demonstrated many superior electronic and optoelectronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='26-28 In this work, we report a comprehensive experimental and theoretical study of the plasmonic responses of the pentacene/graphene heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' To perform nano-IR studies of the plasmonic responses of the heterostructure samples, we utilized the scattering-type scanning near-field optical microscope (s-SNOM) that is built on a tapping-mode atomic force microscope (AFM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' As illustrated in Figure 1a, the sharp metalized AFM tip is illuminated by a p-polarized IR laser beam, thus generating an intense electric field underneath the tip apex due to the so-called ‘lightening- rod effect’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Such a strong field is highly confined in space and possesses a wide range of in-plane momenta (q), which enables direct optical excitation and detection of graphene plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The IR detector collects scattered photon signals off the coupled tip-sample system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Demodulating the signals at high harmonics of the AFM tapping frequency strongly suppresses the background signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Furthermore, we implement a pseudo- heterodyne interferometer that allows us to extract both the amplitude and phase components of the IR signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In the current work, we discuss mainly the IR amplitude (s) signal that scales monotonically with the plasmon field amplitude right underneath the tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='29 All our experiments were performed at ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Figure 1b plots the AFM topography image of a typical pentacene/graphene heterostructure sample, where clear staircase features are seen in the field of view, corresponding to pentacene layers with different thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' By measuring the thickness of different regions of the sample (Figure 1d) based on the AFM line profiles (Figure S1), we can accurately determine the number of pentacene layers as labeled in Figure 1b,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' According to the previous study26, the orientation of pentacene molecules shows variations from layer to layer close to the graphene interface due to the competition between the molecule-graphene interactions and the intermolecular interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' More specifically, as shown in Figure 1a, pentacene molecules first form a sheet of the flat-lying wetting layer (WL) on graphene, then the inclined one layer (1L) following by the upright-standing two layers (2L) and few layers (3L, 4L, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The orientation angles of pentacene molecules to the substrate are 0◦, 61◦ and 82◦ for the flat-lying, inclined and standing layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' As discussed in detail below, the molecule orientation plays a critical role in the overall plasmonic responses of pentacene/graphene heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Note that bare WL pentacene could exist in freshly grown samples, but it will soon disappear after exposure to air due to the dewetting, suggesting that upright-standing molecules are possibly more favorable energetically in air than the flat-lying molecules30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Thicker pentacene layers with standing molecules are generally more stable and can last for many days at ambient conditions thus suitable for systematic nano-IR studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In Figure 1c, we present the s-SNOM imaging data of the sample shown in Figure 1b\uf02c where we plot the IR amplitude normalized to that of the SiO2/Si substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The laser energy is set to be E = \uf031\uf031\uf036 meV that is away from the strong optical phonon resonance of SiO2 (~140 meV), so the IR responses at this energy are predominantly due to graphene plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' As shown in Figure 1c, there is a clear IR signal contrast between different pentacene layers on graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' For quantitative analysis, we plot in Figure 1e the IR amplitude versus the number of pentacene layers, which indicates a systematic decrease of the IR amplitude with increasing pentacene thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Moreover, we found a relatively big drop of IR amplitude signal from 1L to 2L pentacene (~27%), but only slight declines from 2L to 3L pentacene (~9%) and from 3L to 4L pentacene (~4%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Similar pentacene thickness dependent signal variation is also seen in other samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' sample 2 in Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' From Figure 1c, we also observed a bright edge feature surrounding the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' To reveal the details about the bright edge feature, we performed high-resolution s-SNOM imaging measurements close to the sample edge (Figure 2a-e), where we observed bright fringe(s) parallel to the sample edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' According to previous studies15,16, these bright fringes are generated due to the constructive interference between tip-launched and edge-reflected surface plasmons of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The plasmonic origin of these fringes is verified by frequency-dependence studies (Figures S2 and S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In addition to the bright fringes, we also occasionally see weak oscillations of signals distributed along the sample edge, for example in Figure 2c,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' These edge oscillations are generated due to scattering and interference of one-dimensional edge plasmons and they normally appear at relatively rough edges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' in the case of Figure 2c,d) or close to sharp corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='31,32 Now we wish to perform quantitative analysis on the imaged plasmon fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' For that purpose, we plot in Figure 2f-j the line profiles (grey curves) extracted perpendicular to the fringes in Figure 2a-e, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' From both the IR amplitude images and profiles, we found a systematic variation of the plasmon fringes with pentacene thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' First, the samples with thicker layers of pentacene show weaker fringe intensity, and the strongest fringe is observed in bare graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In addition, the width of the bright fringe decreases with increasing pentacene thickness, implying a reduction of plasmon wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Furthermore, the number of fringes decreases with increasing pentacene thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' For example, there are at least 3 bright fringes at the edge of bare graphene, 2 clear bright fringes in the case of 1L pentacene on graphene, and only 1 clear fringe for 2L, 3L and 4L pentacene on graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The decrease of the fringe number indicates an increase in the plasmon damping rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The fringe profiles shown in Figure 2f-j allow us to fit quantitatively the complex plasmon wavevector qp = q1 + iq2 of graphene, based on which we can determine the plasmon wavelength (\uf06cp = 2\uf070/q1) and damping rate (\uf067p = q2/q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' To perform the fit, we adopted a quantitative s-SNOM model that approximates the s-SNOM tip as a conducting spheroid (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' This model calculates accurately the s-SNOM signals by evaluating the total radiating dipoles (pz) of the tip-sample system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' By computing pz at multiple x and z coordinates of the tip, we were able to obtain line profiles of s-SNOM signals with quantitative accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' More introductions about the model are given in the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The same model has been applied to calculate the plasmon fringe profiles of bare graphene and other graphene-based vdW materials and heterostructures reported in earlier works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='15, 33-35 The modeling profiles are plotted in Figure 2f-j as red dashed curves, which show good consistency with the experimental data profiles (grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The \uf06cp and \uf067p parameters determined through the fitting are given in Figure 3b,c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Figure 3b indicates that \uf06cp decreases systematically with increasing pentacene thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' For example, from bare graphene to 4L pentacene on graphene, \uf06cp drops from 250 nm to 205 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' More interestingly, \uf06cp shows a sharp reduction from 240 nm to 215 nm when the pentacene thickness changes from 1 layer to 2 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' This sharp reduction of \uf06cp also results in an abrupt drop of the overall IR amplitude signal from 1L to 2L pentacene (Figures 1e and Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The plasmon damping rate \uf067p, on the other hand, increases systematically with pentacene thickness, which is consistent with the decrease of the number of plasmon fringes shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' As shown in Figure 3c, the extracted \uf067p by fitting the plasmon fringe profiles increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='14 for bare graphene to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='17, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='24, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='27 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='3 when adding 1L, 2L, 3L and 4L pentacene on top of graphene, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Like \uf06cp, \uf067p also undergoes a larger change from 1L to 2L pentacene (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='07) compared to that between other adjacent layers (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' We now elaborate on the possible causes of the observed thickness dependence of the plasmonic parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Under the Drude and long-wavelength approximations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' the plasmon wavevector qp can be written as 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='25 0 1 2 2 2 ( ) p F q q iq E E iE e E \uf070\uf065 \uf06b \uf047 \uf0ba + \uf0bb + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' (1) where e is the elementary charge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' EF is the Fermi energy of graphene,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' E\uf047 is scattering energy of Dirac Fermions in graphene,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' and \uf06b = \uf06b1 + i\uf06b2 is the effective dielectric constant of the environment of graphene (\uf06b1 and \uf06b2 are the real and imaginary parts of \uf06b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' For bare graphene, \uf06b is an average value from the dielectric constants of air and SiO2: \uf06b = (1+ \uf065s)/2 (\uf065s ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='4 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='3i at E = 116 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In the case of pentacene/graphene heterostructures, dielectric constants of pentacene also contribute to \uf06b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Note that our graphene samples are highly doped at ambient conditions with EF above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='4 eV (see discussions below), so contributions from interband transitions at our energy regime are negligible thus not considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' From eq 1, one can obtain the plasmon wavelength \uf06cp = 2\uf070/q1: 2 2 1 0 1 2 / / ( ) p F q e E E \uf06c \uf070 \uf065 \uf06b = \uf0bb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' (2) Therefore, the observed layer dependence of \uf06cp (Figure 3b) is possibly due to the change of EF of graphene and/or the dielectric constants of pentacene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Note that eqs 1 and 2 are mainly for discussions of general physics of graphene plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' We used the transfer matrix method to compute numerically the plasmon dispersion and plasmon wavelength of the entire pentacene/graphene/substrate system (Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' We first evaluate the effects solely due to the dielectric screening of pentacene layers with a fixed EF of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='47 eV ─ the Fermi energy of bare graphene accurately determined by fringe profile fitting (Figure 2f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The large EF indicates the high hole doping of graphene on SiO2, which is originated from the vacuum annealing during the pentacene growth process followed by days of air exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='36,37 The anisotropic dielectric constants of pentacene with different thicknesses were calculated from density functional theory (DFT) calculations (Supporting Information), which vary from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='7 in the ab plane and from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='3 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='6 along the c-axis for different pentacene thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Note that our excitation laser energy (116 meV) is away from the strong vibrational resonances of pentacene (the nearest strong resonance is at 112 meV with a resonance width of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='4 meV)38,39, so the vibrational modes of pentacene do not affect graphene plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The calculated \uf06cp of graphene with a fixed EF under various pentacene layers is plotted in Figure 3b as blue triangles, which show a gentle and systematic decline with layer number (\uf044\uf06cp ≈ 4 nm on average per layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Based on Figure 3b, we know that dielectric screening of pentacene alone cannot explain the sharp drop of \uf06cp as pentacene thickness increases from 1 to 2 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The inconsistency between experimental and calculated \uf06cp assuming a fixed EF indicates that the layer dependence of doping must be taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Indeed, layer-dependent EF can be obtained accurately by fitting the experimental \uf06cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' As shown in Figure 3d, graphene under 1L pentacene has slightly smaller EF (~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='46 eV) compared to that of bare graphene (~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='47 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The EF of graphene under 2 to 4 layers pentacene is much lower, down to ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='42 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The unique pentacene layer dependence of EF (Figure 3d) is, in fact, originated from the charge transfer between graphene and pentacene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Charge transfer phenomena have also been observed at the interfaces between graphene and other types of molecules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' C60, CNT, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='40-42 To understand the transfer process here, we plot in Figure 4 the energy alignment diagrams between the graphene Fermi level (dashed line) and the highest occupied molecular orbits (HOMO) level of pentacene layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The lowest unoccupied molecule orbits (LUMO) are ~2 eV above the HOMO level (not shown in Figure 4), so there are no unoccupied states available in pentacene close to the Fermi level of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In Figure 4, we label the ionization potential (IP) values of graphene and pentacene, which is the energy difference from the Fermi level of graphene or HOMO energy of pentacene to the vacuum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Considering that graphene on SiO2 is hole doped at ambient conditions36,37, the IP can be calculated to be around 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='03 eV by adding the Fermi energy (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='47 eV, Fermi level to Dirac point) and the work function of neutral graphene (~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='56 eV, Dirac point to vacuum energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='42,43 The IP of pentacene layers is sensitively dependent on the molecule orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='44- 46 To obtain the IP values of pentacene layers, we performed first-principles electronic structure calculations based on DFT using the Vienna ab initio simulation package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='47,48 The atomic structures of the pentacene layers (Figure S6) are adopted from the previous study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='26 Such DFT calculations tend to underestimate the value of IP49, so we considered the GW correction (\uf044GW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Detailed introductions about the IP calculations are given in the Supporting Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The final IP values of WL, 1L, 2L, and 3L pentacene without and with GW corrections are summarized in Table 1, where one can see a big drop (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='77 eV) of IP from WL to 1L pentacene, followed by a small drop (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='18 eV) from 1L to 2L pentacene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Starting from 2L pentacene and above, IP stays constant at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='78 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Such a layer dependence is originated from the difference of the orientation angles of pentacene molecules (0◦, 61◦ and 82◦ for the WL, 1L, and 2L or above, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Our IP calculations are consistent with previous experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='45,46 Note that the interface dipole between graphene and pentacene induces a shift of pentacene vacuum level by ∆ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='46 Pentacene layers IPDFT (eV) IPDFT +\uf044GW (eV) WL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='76 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='72 1L 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='99 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='95 2L 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='81 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='77 3L 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='82 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='78 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The calculated ionization potential (IP) of pentacene with different thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Based on Figure 4, one can see that the Fermi level of graphene is much higher than the HOMO energy level of WL pentacene, so charge transfer between graphene and WL pentacene is forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' For 1L to 4L pentacene, HOMO energy level rises above EF of graphene, so electron transfer from pentacene to graphene is enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The amount of electron transfer from 1L pentacene to graphene is much less compared to that from thicker pentacene layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' For 1L pentacene, the reduction of graphene EF due to the charge transfer (\uf044EF) is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='013 eV, corresponding to the change of carrier density (\uf044n) of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='9×1012 cm-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' For 2L to 4L pentacene, the resulting \uf044EF is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='047 eV, corresponding to the \uf044n ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='1×1012 cm-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The size of \uf044EF is mainly due to the potential difference between graphene and pentacene (\uf044EIP) (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The amount of \uf044EIP for 1L pentacene (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='08 eV) is much smaller than those of 2L to 4L pentacene (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='25 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Another relevant factor is the density of states of pentacene layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In principle, few-layer pentacene should have more electrons to offer compared to 1L pentacene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Note that the electron transfer discussed here is a tunneling process due to the presence of the WL pentacene that acts as a tunneling barrier (Figure 4) with a thickness of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='26 Effects of electron tunneling on surface plasmons have been studied in metal-molecule junctions, where unique quantum plasmonic responses were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='50 It is also proposed that electron tunneling can be utilized to generate graphene plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='51,52 Therefore, the molecule/graphene heterostructure with interlayer electron tunneling studied here provides a unique platform to explore further the role of electron tunneling on graphene plasmons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Finally, we wish to discuss the dependence of plasmon damping rate \uf067p on pentacene thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' As discussed above (Figure 3c), \uf067p increases with pentacene thickness, and the increment of \uf067p is larger from 1L to 2L pentacene (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='07) compared to that between other adjacent layers (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='03), which implies a possible link between electron transfer and plasmon damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Based on eq 1, we know that \uf067p can be written approximately as: 2 1 2 1 / / / p q q E E \uf067 \uf06b \uf06b \uf047 = \uf0bb + , (3) which indicates that \uf067p originates from both the loss due to the dielectric environment and the scattering of Dirac fermions in graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' As discussed above, \uf06b ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='7 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='15i for graphene sitting directly on SiO2 at E = 116 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' As a semiconductor, pentacene behaves like a good dielectric with a negligible imaginary part of permittivity at the mid-IR region if it is away from the vibrational modes39, so pentacene itself has little contribution to \uf06b\uf032 at our excitation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Therefore, the enhanced \uf067p when adding pentacene layers is most likely due to the scattering of graphene carriers by impurities or localized charges in pentacene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' With electron transfer, additional localized charges could be introduced to pentacene, which cause higher damping to graphene plasmons (Figure 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Increased charge scattering leading to a lower carrier mobility has been observed previously in transport studies of C60/graphene heterostructures, where charge transfer was also involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='53 In summary, we have performed the first nanoplasmonic study of vdW heterostructures formed by organic 2D materials and graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' By using the nano-IR imaging technique, we discovered that the graphene plasmons could be tailored by depositing molecule layers of pentacene on graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Unlike electrical gating that requires a constant bias voltage, the molecular deposition method is suitable for creating heterostructure samples or devices with tailored permanent properties for long-term applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Through quantitative analysis and DFT calculations, we proved that the pentacene-layer dependence of graphene plasmons is mainly due to tunneling-type electron transfer from pentacene to graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Moreover, we found the electron transfer process is determined by the molecule orientation of each pentacene layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Such a unique sensitivity to molecular orientations is highly desired for structural characterizations of molecules and bio-nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Of course, the studies should not be limited to pentacene/graphene heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' We expect more interesting nano-optical properties and functionalities to be discovered in heterostructures formed by graphene with other types of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Our work broadens the understanding of the interlayer interactions of graphene with biomolecules and opens the door to future studies and applications of molecule/graphene heterostructures in nanophotonics and optoelectronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Acknowledgments Work done at Ames Lab was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Department of Energy, Office of Basic Energy Science, Division of Materials Sciences and Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Ames Laboratory is operated for the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Department of Energy by Iowa State University under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' DE-AC02-07CH11358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The nano-optical imaging set-up was partially supported by the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Keck Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' acknowledges the funding support from the National Natural Science Foundation of China 61734003 and the National Key Basic Research Program of China 2013CBA01604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Ryzhii, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Satou, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Lanzara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Zettl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' ACS Nano 2017, 11, 4686- 4693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Figures TOC figure Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' a, Illustration of the s-SNOM study of a pentacene/graphene heterostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The arrows sketch the incident laser and back-scattered photons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' b, The AFM topography image of a typical pentacene/graphene heterostructure sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' c, The IR amplitude (s) image of the pentacene/graphene heterostructure taken at a photon energy of E = 116 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Here we normalized the amplitude signal to that of SiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The labeling ‘WL’ represents the wetting layer on graphene, ‘1L’-‘4L’ represent 1-layer to 4-layer pentacene on graphene, and ‘G’ represents bare graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The scale bars: 1 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' d, The thickness (d) of sample relative to the SiO2 surface versus the number of pentacene layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' e, The IR amplitude signal versus the number of pentacene layers measured at locations away from the edge of the sample in Figure 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In d and e, the 0 pentacene layer corresponds to bare graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' pentacene/graphene 1L IR laser graphene 2L 3L 2L S 11 3L WL graphene SiO2 4L SiO2 2um pentacene 5LAFM Infrared a b C d 6 1L d(nm) 11 2L 2L 0 1 3 Pentacenelayers 2L e 1L 3L 3L (rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' unit) WL G sio- 4L SiO2 40 克克 5L 51 SiO2 s 0 7 17 0 72 0 2 3 4 Pentacenelayers height (nm) s (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=') Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' a-e, High-resolution IR amplitude images of bare graphene (G) and pentacene/graphene heterostructures (1L to 4L) with various pentacene thicknesses taken at E = 116 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The blue dashed curves mark the sample edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The scale bars: 200 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' f-j, The IR amplitude line profiles from both experiments (grey) and simulations (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The experimental profiles were taken along the black dash lines in a-e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In all the images and profiles, the IR amplitude signal is normalized to that of the SiO2/Si substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' a, Illustration of the spheroid model with different parameters that we used to model the plasmon fringes profiles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' 2f-j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' b, The plasmon wavelength (\uf06cp) versus the number of pentacene layers from fringe profile fitting (black squares) and theoretical calculations (blue triangles) assuming a fixed Fermi energy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='47 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' c, The plasmon damping rate (\uf067p) versus the number of pentacene layers from fringe profile fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' d, The Fermi energy of graphene versus the number of pentacene layers calculated based on the fitted \uf06cp data (squares) in panel b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In panels b-d, the 0 pentacene layer corresponds to bare graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' b C d 2 G 1L Sio2 SiO2 SiO2 Sio SiO2 3 g h (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=') 2 G 1L 2L 3L 4L S 0 200 400 0 200 400 0 200 400 0 200 400 0 200 400 x (nm) x (nm) x (nm) x (nm) x (nm)a b 280 spheroid tip 2L 240 p:t (wu) 200 pentacene (cab, c) graphene 81 (g) 160 82 0 1 2 4 Pentacene layers c 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='4 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='3 (eV) xp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='3 0 1 2 3 4 0 1 2 3 4 Pentacene layers Pentacene layers Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Illustration of the graphene/pentacene heterostructure (top) and the energy level alignment diagram (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' As thickness increases, pentacene changes from flat-laying wetting layer (WL), inclined one layer (1L) to standing few layers (2L, 3L, and 4L) relative to graphene, resulting in a lifting of pentacene HOMO energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The energy values listed in the alignment diagram are the ionization potentials (IP) of graphene and pentacene layers before electron transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Supporting Information for Tailor plasmons in pentacene/graphene heterostructures with interlayer electron transfer F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Hu1,2*, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Kim1,2*, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Zhang3, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Luan1,2, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Ho1,2, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Shi3, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Wang1,2†, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Wang3†, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Fei1,2† 1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA 2U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' DOE Ames Laboratory, Iowa State University, Ames, Iowa 50011, USA 3National Laboratory of Solid State Microstructures, School of Electronic Science and Engineering and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' † C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' (wangcz@ameslab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='gov);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' (xrwang@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='cn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' (zfei@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='edu) List of contents Graphene WL 1L 2L,3L,4L C:0-0 △-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='1eV VacuumLevel ~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='78eV IP~5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='03eV 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='72 eV ~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='95eV Fermilevel HOMO HOMO HOMO graphene WL 1L 2L, 3L,4L 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Experimental details 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Additional s-SNOM imaging data 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Numerical modeling of the plasmon fringe profiles 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Calculations of the plasmon wavelength 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' DFT calculation methods and results 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Dielectric constants of pentacene Figures S1 – S9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Experimental details To perform nano-infrared (IR) imaging studies of the pentacene/graphene heterostructures, we employed the scattering-type scanning near-field optical microscopy (s-SNOM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Our s-SNOM apparatus (Neaspec GmbH) is based on an atomic force microscope (AFM) operating in the tapping mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Measurements were acquired at an AFM tapping frequency of \uf057 = 270 kHz and a tapping amplitude of about 60 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' As illustrated in Figure 1a, we utilized a metalized AFM probe, which is illuminated by a p-polarized mid-IR beam from a continuous-wave CO2 laser (Access Laser).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In our s-SNOM measurements, we used Arrow-NCPt probes from NanoAndMore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The radius of tip apex of these probes is about 25 nm that defines the spatial resolution of the s-SNOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The standard observable of an s-SNOM experiment is complex scattering signal demodulated at the nth (n = 3 in the current work) harmonics of the AFM tip oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' We discuss mainly the amplitude part of the signal that is enough to describe the plasmonic responses of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Our pentacene/graphene vdW heterostructures were prepared by physical vapor transport deposition of uniform pentacene molecular layers on graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The heterostructure samples are sitting on the standard silicon wafers with 300-nm-thick thermal oxide on the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The samples that we studied in this work include bare graphene, and one-layer (1L), two-layer (2L), three-layer (3L) and four-layer (4L) pentacene on graphene, determined by accurate AFM measurements (Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Additional s-SNOM imaging data In Figures S2 and S3, we present the excitation energy (E) dependent nano-IR amplitude images of graphene, 1L pentacene on graphene, and 2L pentacene on graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Here the IR amplitude is normalized to that of the SiO2 substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' From Figures S2 and S3, one can see that the IR contrast between the samples and the SiO2 substrate shows a clear evolution with energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' This is mainly due to the increase of the substrate signal as E approaches the surface phonon resonance of SiO2 at around 140 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Moreover, the fringe period or the fringe width of the samples shrinks with increasing laser energy, indicating smaller plasmon wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' This is consistent with the dispersion properties of graphene plasmons (Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In all laser energies, there is a small signal difference between bare graphene and 1L pentacene on graphene and a larger contrast between 1L and 2L pentacene on graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' This is consistent with the results discussed in the main text (Figures 1-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Figure S4a,b present the nano-IR imaging data of two heterostructure samples at an excitation energy of E = 110 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Sample 1 is the one that we extensively studied in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Sample 2 is from a different wafer and it also contains bare graphene and 1L to 4L pentacene on graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Note that the excitation energy used here is slightly lower than that used in Figures 1 and 2 in the main text (116 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Here we compare the general signal contrast of different sample areas, which show good consistency among the two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' For quantitative comparison, we plot in Figure S4c the average IR amplitude at the sample interior versus the number of pentacene layers (0 layer corresponds to bare graphene).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Here one can see that the general trend of the signal evolution with layer thickness is consistent in the two heterostructure samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' There is a slight difference in 1L pentacene on graphene, which is possibly due to the degradation of 1L pentacene on sample 2 that leads to nonuniform signal distributions (Figure S4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Numerical modeling of the plasmon fringe profiles To model the fringes profiles of plasmons confined inside graphene or pentacene/graphene heterostructures, we model our AFM tip as an elongated metallic spheroid (see Figure 3a in the main text): the length of the spheroid is 2L and the radius of curvature at the tip ends is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Here, a is set to be 25 nm according to the manufacturer and L is set to be 500 nm and it is not a very sensitive parameter so long as L >> a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The scattering amplitude s (before demodulation) scales with the total radiating dipole pz of the spheroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Therefore, to fit the line profiles perpendicular to the fringes inside samples, we need to calculate pz at different spatial coordinates (x, z) of the lower end of the AFM tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Here, x is the in-plane coordinate perpendicular to samples and z is the out-of-plane coordinate perpendicular to the sample surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' By calculating pz at different z, we can perform ‘demodulation’ of the scattering amplitude s and get different harmonics of the scattering signal and calculating pz at different x allows us to plot the modeling profiles of IR amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In all our simulations, we assume no position dependence in the y-direction for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The dielectric constants of SiO2 used in the calculations are adopted from literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='1 The dielectric constants of pentacene layers (\uf065ab, \uf065c) used in the calculations given in Section 6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The key modeling parameters for graphene are the plasmon wavelength (\uf06cp) and damping rate (\uf067p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' By fitting the experimental plasmon fringe profiles (Figure 2f-j in the main text), we can determine accurately \uf06cp and \uf067p based on the experimental data (Figure 3b,c in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Calculations of the plasmon wavelength In order to determine the Fermi energy (EF) of graphene, we need to calculate \uf06cp theoretically and then compared to the experimental result obtained from fringe profile fitting (see the section above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' For that purpose, we first compute the plasmon dispersion colormaps (Figure S5) by evaluating numerically the imaginary part of the reflection coefficient Im(rp) for the entire pentacene/graphene/substrate heterostructure system by using the transfer matrix method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' These colormaps reveal the photonic density of states (DOS), and the plasmonic mode appears as a bright curve revealed by the colormaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Such a dispersion calculation method has been widely applied in the studies of graphene plasmons and other types of polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The optical conductivity of graphene is obtained by the Radom phase approximation methods (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' 11 in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The dielectric constants of pentacene layers used in the calculations given in Section 6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The dielectric constants of SiO2 used in the calculations are adopted from literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='1 Based on the calculated dispersion colormaps, we can determine the plasmon wavevector qp and hence the plasmon wavelength \uf06cp = 2\uf070/qp at any excitation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The dispersion colormaps shown in Figure S5 are calculated with different choices of EF values shown in Figure 3d in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The \uf06cp values read out from these colormaps match well the \uf06cp results determined through fringe profile fitting (Figure 2 in the main text), which confirms the validity of EF values shown in Figure 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' DFT calculation methods and results We performed first-principles electronic structure calculations based on density functional theory (DFT) using Vienna ab initio simulation package (VASP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='2,3 We employed the projector augmented wave method4 and a plane-wave basis set with 400 eV energy cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' For the exchange-correlation functional, we used Perdew-Burke-Ernzerhof (PBE) functional5, and a total of 5\uf0b42\uf0b41 k point meshes were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' We included the van der Waals energy using the DFT-D3 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='6 In our slab calculations for the pentacene thin films, we used sufficiently thick vacuum regions (> 23 Å) to prevent the unwanted interactions between periodic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' We considered the structural model of pentacene layers that consist flat-lying wetting layer (WL) on graphene, the inclined one layer (1L) and the standing two (2L) and three layers (3L) (Figure S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='7 The dielectric matrix was determined using density functional perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In Figures S7 and S8, we presented the calculated band structures of pentacene layers and their potential energy line profiles taken along the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' From these calculations, we can determine the ionization potential (IP) of all pentacene layers (Table 1 in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Due to the self-interaction error, the conventional DFT calculations with local density approximation (LDA) or generalized gradient approximation (GGA) are not supposed to give accurate valence band energies, which in turn results in underestimated IP values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='8 Therefore, we performed the GW calculation (in the GW0 level) that is expected to give a reasonable value for the IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Due to the large computational cost, we used the approximation where we estimate the shift of the valence band maximum by that of the highest valence band energy at the \uf047 point of the Brillouin zone (\uf044VB,\uf047) in the 3D bulk pentacene crystal (Figure S9a ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In the GW calculation of the bulk pentacene, we used 2\uf0b42\uf0b41 k point meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' We considered further correction coming from limited number of empty bands (\uf044eb)\uf02e Specifically, we estimated the valence band energy with the infinite number of the empty bands (Neb = \uf0a5) included in the calculation by fitting the results of a series of different Neb to the formula A/Neb + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The results of both corrections are shown in Figure S9b,c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' We denoted the sum of the two correction terms by \uf044GW (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=', \uf044GW = \uf044VB,\uf047 + \uf044eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The total GW correction is as large as \uf040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='96 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' After corrections, the IPs of thin pentacene layers are calculated to be 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='72 eV, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='95 eV, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='78 eV, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='78 eV for WL, 1L, 2L and 3L pentacene, respectively (Table 1 in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' These results are consistent with previous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='9 Based on the calculation results, we conclude that the IP of pentacene is solely dependent on the orientation angle of pentacene molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Therefore, we expect that IPs of thicker pentacene layers (4L or above) are all about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='78 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Dielectric constants of pentacene The dielectric constants of pentacene layers we used for the fringe profile modeling and dispersion calculations are also from DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' With DFT, we calculated the static dielectric constants of pentacene layers with different thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' It is known from the previous study10 that pentacene has a flat dielectric response up to the visible region when it is away from the strong vibrational modes of pentacene (the nearest strong resonance is at about 112 meV with a resonance width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='4 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Therefore, it is appropriate to use the static dielectric constants for calculations in the mid-infrared region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The calculated in-plane dielectric constants (\uf065ab) for 1L, 2L and 3L pentacene are about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The out-of-plane dielectric constants (\uf065c) for 1L, 2L and 3L pentacene are about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Both \uf065ab and \uf065c increase with pentacene thickness and they are trending towards the value for bulk pentacene films: \uf065bulk ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='10 For 4L pentacene, we used dielectric constants of 3L pentacene as an approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The calculated \uf06cp only varies a little (~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='5%) even using \uf065bulk ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0 for 4L pentacene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' There is in fact a small ab-plane anisotropy (about 1%, 5% and 12% for 1L, 2L and 3L pentacene, respectively) in the dielectric constants according to our calculations, but it only causes tiny variations to \uf06cp (about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='05%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='3% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='8% for 1L, 2L and 3L pentacene, respectively) due to the nanoscale thicknesses of the pentacene layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Therefore, we used averaged ab-plane dielectric constants in our calculations (\uf065aa + \uf065bb)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' References for the Supporting Information 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Palik, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Handbook of Optical Constants of Solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Academic Press (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Kresse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Hafner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Arwin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' 2004, 84, 200-202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The AFM topography profiles of graphene (G) and 1L to 4L pentacene on graphene extracted from Figure 1b in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Here the 0 pentacene layer corresponds to bare graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Excitation laser energy dependent nano-IR imaging data of bare graphene (G) and 1L pentacene on graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Here we plot the IR amplitude normalized to that of the SiO2 substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' 8 G 1 L 2L 9 3L Height (nm) 41 4 0 100 0 100 200 300 (wu) xa E = 110 meV b E = 112 meV c E= 115 meV 1L SiO2 G 1L SiO2 G 1L SiO2 G 3 S d E = 121 meV e E = 128 meV t E=134 meV 0 1L SiO2 G 1L SiO2 G SiO2 G 400nm Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Excitation laser energy dependent nano-IR imaging data of 1L and 2L pentacene on graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Here we plot the IR amplitude normalized to that of the SiO2 substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' a,b, Nano-IR imaging of two samples at an excitation energy of 110 meV (slightly lower energy compared to that used in Figure 1 and 2 in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Sample 1 is the sample we studied extensively in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Sample 2 is a different sample on a different wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Scale bars: 2 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' c, The IR amplitude signals of the two samples taken from a,b versus the number of pentacene layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' E=110meV b a E= 112 meV C E = 115 meV 1L 1L 1L SiO2 SiO2 SiO2 2L 2L 2L 73 d E= 121 meV e E=128meV h E=134meV S 1L 0 1L 1 SiO2 SiO2 SiO2 2L 2L 2L 400nmSample 1 G 3L 4L Sample 2 a b C 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0 --Sample1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='5 -- Sample 2 8 2L 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content="0 G ('n: 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='5 2L S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0 口 3L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='5 SiO2 4L SiO2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0 1 1 1 L 0 1 2 3 4 0 13 s (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=') Layer number Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Calculated dispersion colormaps of bare graphene (a) and pentacene layers with different thicknesses on graphene (b-e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The white dashed curves mark the dispersion relation of graphene plasmons revealed by the color maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The horizontal and vertical blue dashed lines mark the excitation energy (E = 116 meV) and corresponding plasmon wavevector determined by the dispersion diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Atomic structures of different pentacene layers that we constructed for DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='Graphene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='1Lpentaceneongraphene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='(Maw) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='115 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='115 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='2 = 250 nm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='p = 240 nm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='q(cm-1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='×105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='q(cm-1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='×105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='2Lpentaceneongraphene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='3Lpentaceneongraphene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='4Lpentaceneongraphene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='(Maw) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='115 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='115 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='115 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='2p = 215 nm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='2p = 210 nm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='2p = 205 nm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='q(cm-1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='×105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='q(cm-1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='×105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='q(cm-1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='×105e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='WLpentacene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='2Lpentacene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='3Lpentacene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='1Lpentacene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The DFT calculations of band structures of WL, 1L, 2L and 3L pentacene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' In all the plots, the band structures are shifted on purpose to set the valence band maximum right at 0 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' The DFT calculations of potential energy profiles of WL, 1L, 2L, and 3L pentacene along the c axis (perpendicular to the pentacene layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' a b WL pentacene 1L pentacene 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='5 0.' 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+page_content=' b, Shift of the valence band maximum by that of the highest valence band energy at the \uf047 point of the Brillouin zone (\uf044VB,\uf047).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content=' c, Correction coming from limited number of empty bands (\uf044eb)\uf02e a b 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='0 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='6 CB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='5 VB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfzy31/content/2301.11645v1.pdf'} +page_content='5 3.' metadata={'source': 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a/P9AzT4oBgHgl3EQfWvzh/content/tmp_files/2301.01308v1.pdf.txt b/P9AzT4oBgHgl3EQfWvzh/content/tmp_files/2301.01308v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f0131431e3437fcdac9e95776acda45d79a0a06b --- /dev/null +++ b/P9AzT4oBgHgl3EQfWvzh/content/tmp_files/2301.01308v1.pdf.txt @@ -0,0 +1,1160 @@ +Astronomy & Astrophysics manuscript no. main +©ESO 2023 +January 5, 2023 +Letter to the Editor +Strong magnetic fields detected in the cores of 11 red giant stars +using gravity-mode period spacings +S. Deheuvels1, G. Li1, J. Ballot1, F. Lignières1 +IRAP, Université de Toulouse, CNRS, CNES, UPS, 31400 Toulouse, France +January 5, 2023 +ABSTRACT +Despite their importance in stellar evolution, little is known about magnetic fields in the interior of stars. The recent +seismic detection of magnetic fields in the core of several red giant stars has given measurements of their strength and +information on their topology. We revisit the puzzling case of hydrogen-shell burning giants that show deviations from +the expected regular period spacing of gravity modes. These stars also tend to have a too low measured period spacing +compared to their counterparts. We here show that these two features are well accounted for by strong magnetic fields +in the cores of these stars. For 11 Kepler red giants showing these anomalies, we place lower limits on the core field +strengths ranging from 40 to 610 kG. For one star, the measured field exceeds the critical field above which gravity +waves no longer propagate in the core. We find that this star shows mixed mode suppression at low frequency, which +further suggests that this phenomenon might be related to strong core magnetic fields. +Key words. Asteroseismology – Stars: magnetic fields +1. Introduction +Magnetic fields affect stars at all evolutionary stages from +star-forming molecular clouds to white dwarfs and magne- +tars (McKee & Ostriker 2007, Kaspi & Beloborodov 2017, +Ferrario et al. 2020). In particular, they are expected to +play a central role in the redistribution of angular momen- +tum inside stars (Maeder & Meynet 2005, Cantiello et al. +2014, Rüdiger et al. 2015), and thus in the transport of +chemical elements. While surface magnetic fields have been +detected and characterized in stars across the HR diagram +(Landstreet 1992, Donati & Landstreet 2009), internal mag- +netic fields have long remained inaccessible to direct obser- +vations. In red giant stars, the detection of mixed modes – +that is, oscillation modes that behave as gravity (g) modes +in the core and as pressure modes in the envelope – has +given strong evidence that the cores of red giant stars are +rotating slowly (e.g., Deheuvels et al. 2012, Mosser et al. +2012, Gehan et al. 2018). This yielded evidence that angu- +lar momentum is redistributed much more efficiently than +if only purely hydrodynamical processes were at work (e.g., +Marques et al. 2013). Magnetic fields could produce the +additional transport that is needed (Rüdiger et al. 2015, +Jouve et al. 2015, Fuller et al. 2019, Petitdemange et al. +2022). Observational constraints on the properties of inter- +nal magnetic fields are crucially needed to assess the nature +and the efficiency of the magnetic transport of angular mo- +mentum inside stars. +The propagation of magneto-gravity waves is expected +to be suppressed when the magnetic field exceeds a critical +strength Bc above which Alfvén wave frequencies become +comparable to those of gravity waves. This phenomenon +Send offprint requests to: S. Deheuvels +e-mail: sebastien.deheuvels@irap.omp.eu +was invoked by Fuller et al. (2015) to account for the unex- +pectedly low amplitudes of dipole mixed modes in a frac- +tion of red giants. For core fields above Bc, the authors +suggested that the mode energy reaching the magnetized +core would be entirely dissipated and lost, giving rise to +purely p-like dipole modes. This interpretation was ques- +tioned by Mosser et al. (2017), who found that partially +suppressed dipole modes still retain a g-like character. Loi +(2020b) later showed that even with strong fields, a frac- +tion of the incoming waves could remain g-like, which would +allow for partial energy return from the core. The interpre- +tation of suppressed dipole modes remains debated. +Magnetic fields also produce shifts in the oscillation +mode frequencies (Gough 1990). Several studies have re- +cently investigated the impact of internal fields on the fre- +quencies of mixed modes in red giants (Gomes & Lopes +2020, Bugnet et al. 2021, Loi 2021). Very recently, Li et al. +(2022) detected clear asymmetries in the rotational mul- +tiplets of dipole mixed modes in three Kepler red giants. +They showed that these features can only be accounted for +by internal magnetic fields with intensities ranging from 30 +to 130 kG in the vicinity of the hydrogen burning shell. +These findings opened the exciting opportunity to charac- +terize magnetic fields in the cores of red giants. +We here investigate the irregularity of g-mode period +spacings in a group of red giant branch (RGB) stars, which +remains so far unexplained. High-radial order g modes are +expected to be approximately equally spaced in period by +∆Πl, where l is the mode degree. In red giants, dipole +mixed modes can be used to measure ∆Π1 using asymptotic +expressions of the mode frequencies (Mosser et al. 2015). +While ∆Π1 is nearly constant over the frequency range of +observed modes for the vast majority of RGB stars, some +red giants show significant variations of ∆Π1 (Mosser et al. +Article number, page 1 of 10 +arXiv:2301.01308v1 [astro-ph.SR] 3 Jan 2023 + +A&A proofs: manuscript no. main +2018, Deheuvels et al. 2022). We here show that this fea- +ture is the signature of strong magnetic fields in the cores +of these stars. This constitutes a new way of detecting and +characterizing magnetic field in the cores of red giants. +In Sect. 2, we present red giants that exhibit deviations +from the regular period spacing pattern of g modes, and we +find additional such targets in Kepler data. We then show in +Sect. 3 that strong core magnetic fields can account for this +phenomenon. In Sect. 4, we determine the field strengths +that are required to match the seismic observations. We +discuss these measurements in Sect. 5, before concluding in +Sect. 6. +2. Red giants with non-constant ∆Π1 +2.1. Previous detections of non-constant ∆Π1 in RGB stars +To first order, high-radial-order gravity modes are ex- +pected to be equally spaced in period. Among the 160 +RGB stars studied by Mosser et al. (2018), only one shows +clear deviations from a regular period spacing of g modes +(KIC3216736). The authors attributed this irregularity to +a buoyancy glitch (that is, a sharp variation in the Brunt- +Väisälä frequency N), which induces periodic variations in +the asymptotic period spacing ∆Π1. +More recently, Deheuvels et al. (2022) identified addi- +tional RGB stars with non-constant ∆Π1, in a different con- +text. These stars appeared among a peculiar class of RGB +stars that are located below the so-called “degeneracy se- +quence” in the (∆ν, ∆Π1) plane, where RGB stars regroup +when electron degeneracy becomes strong in their core. +Most of the stars in this class are intermediate-mass stars +and are thought to result from mass transfer (Deheuvels +et al. 2022). The only four lower-mass stars with too-low +∆Π1 must have a different origin. Contrary to intermediate- +mass stars, they all show clear departures from a constant +period spacing of g modes. Interestingly, the star identified +by Mosser et al. (2018) (KIC3216736) is among these tar- +gets. This suggests that there might be a link between the +non-constancy of ∆Π1 and the fact that its measured value +is abnormally low. These four stars show only one detected +mode per rotational multiplet. +2.2. Additional targets +We searched for other targets showing non-constant ∆Π1 +among RGB stars with detected oscillations using the cat- +alog of Yu et al. (2018). To estimate the period spacings +of g modes using dipole mixed modes, we computed the +so-called “stretched” periods τ, defined by the differential +equation dτ = dP/ζ (Mosser et al. 2015), where ζ corre- +sponds to the fraction of the mode kinetic energy that is en- +closed in the g-mode cavity (ζ tends to 1 for pure g modes, +and 0 for pure p modes). When building échelle diagrams of +these stretched periods, mixed modes are expected to align +in a vertical ridge if ∆Π1 is constant and deviations from a +regular period spacing induce curvature in this ridge. +We searched for stars with only one curved ridge de- +tected in order to avoid the additional complication com- +ing from rotational effects (these effects will be addressed +in a subsequent work). This can mean that these stars are +seen pole-on, so that only the m = 0 modes can be de- +tected. This could also arise if the core rotation is too weak +to produce detectable rotational splitting in Kepler data. +KIC8560280 +−20 0 +20 +40 +60 +80 100 +Stretched period modulo ∆Π1 (s) +140 +160 +180 +200 +Frequency (µHz) +KIC3216736 +−20 0 +20 40 60 80 100 +Stretched period modulo ∆Π1 (s) +120 +140 +160 +180 +Frequency (µHz) +Fig. 1. Stretched period échelle diagrams of two red giants +showing distortion from the regular g-mode pattern. Blue cir- +cles show detected dipole modes. Red crosses correspond to the +best-fit asymptotic mixed mode frequencies obtained by includ- +ing a magnetic perturbation. +We thus found seven additional targets, bringing the total +of the sample to 11 stars (see Table B.1). Their stretched +period échelle diagrams are shown in Fig. 1 and B.1. They +were folded using an average value of the asymptotic period +spacing over the frequency range of the observations, which +is further referred to as ∆Π(meas) +1 +. +The location of these 11 targets in the (∆ν, ∆Π1) plane +is shown in Fig. 2 (black star symbols), where we have used +the values ∆Π(meas) +1 +as the measured asymptotic period +spacing. Three of the seven additional targets lie well below +the degeneracy sequence of RGB stars, which confirms the +link between non-constant ∆Π1 and low measured values +for these quantities. +2.3. Origin of distortions in g-mode pattern +Deviations from a regular ∆Π1 are generally attributed to +buoyancy glitches (e.g., Mosser et al. 2015). To produce +the observed distortions in ∆Π1, we showed in Appendix +A that a buoyancy glitch needs to have a large amplitude +(the local value of N must be multiplied by a factor of at +least six) and be located either deep inside the inert He +core, or well above the H-burning shell. While this cannot +be excluded, no known process is expected to produce such +strong features in these regions of an RGB star. Secondly, +the shape of the modulation in ∆Π1 that is produced by a +large-amplitude glitch strongly differs from the observations +(see Appendix A). Finally, the hypothesis of a buoyancy +glitch would not explain why the measured values of ∆Π1 +are unexpectedly low for most of our stars. It thus seems +unlikely that the observed deviations in ∆Π1 arise from +buoyancy glitches. In the following sections, we explore the +possibility that the irregularities in ∆Π1 are produced by +internal magnetic fields. +Article number, page 2 of 10 + +S. Deheuvels, G. Li, J. Ballot, F. Lignières: Strong magnetic fields detected in red giant cores +Fig. 2. Location of RGB stars with non-constant ∆Π1 in the +(∆ν, ∆Π1) plane. Black star symbols correspond to the values +of ∆Π1 that produce the best vertical alignment of modes in the +stretched period échelle diagram (Sect. 2). Colored star symbols +correspond to the corrected values of ∆Π1 obtained by taking +into account the magnetic perturbation to the mode frequencies +(Sect. 4). Other RGB stars from Vrard et al. (2016) are show as +grey circles (for clarity, stars flagged by the authors as potential +aliases were omitted). +3. Effects of magnetic fields on g-mode period +spacings +The influence of magnetic fields over oscillation mode fre- +quencies has been studied over the last decades using a +perturbative approach (Unno et al. 1989, Gough 1990). Re- +cently, their effects on mixed modes in red giants have been +addressed in the special case of dipolar fields with specific +radial profiles, either aligned with the rotation axis (Hasan +et al. 2005, Gomes & Lopes 2020, Mathis et al. 2021, Bugnet +et al. 2021) or inclined (Loi 2021). +Li et al. (2022) extended these studies to an arbitrary +magnetic field and obtained a general expression for the +magnetic frequency shift that is valid provided that the +azimuthal component is not much larger than the radial one +(Bφ/Br ≪ ωmax/N, where ωmax is the angular frequency +at the maximum power of oscillations and N is the Brunt- +Väisälä frequency1). Accordingly, the multiplets of l = 1 +pure g modes (that is, g modes that are not coupled to p +modes) undergo an average shift ωB, given by +ωB = +I +µ0ω3 +ˆ ro +ri +K(r)B2rdr, +(1) +where K(r) is a weight function that probes the g-mode +cavity and sharply peaks in the vicinity of the H-burning +shell (HBS), I is a factor that depends on the core struc- +ture (see Eq. 45 and 46 of Li et al. 2022), and B2r = +(4π)−1 ˜ +B2 +r sin θ dθ dφ. The angular frequency shifts of the +components of g-mode dipole multiplets are then given by +δωg(m = 0) = (1 − a) ωB +(2) +δωg(m = ±1) = +� +1 + a +2 +� +ωB, +(3) +1 The ratio ωmax/N typically exceeds 102 for red giants. +where a is a dimensionless coefficient that depends on the +horizontal geometry of B2 +r (a ∝ +˜ +B2 +rP2(cos θ) sin θ dθdφ, +where P2(cos θ) is the second order Legendre polynomial). +The dependency of magnetic shifts with ω−3 shows that +low-frequency (that is, high-radial-order) g modes are more +affected by magnetic fields. For this reason, magnetic shifts +create a deviation from the regular period spacing of pure g +modes, as was already pointed out by Loi (2020a), Bugnet +et al. (2021), and Li et al. (2022). Very recently Bugnet +(2022) proposed a method to detect the signature of mag- +netic fields exploiting this property. +For illustration, Fig. 1 shows stretched échelle diagrams +of mixed modes with magnetic perturbations (see Sect. 4). +The left panel corresponds to a case where the unperturbed +g modes have an asymptotic period spacing of ∆Π1 = 85 s, +and we have added a magnetic perturbation corresponding +to a frequency shift of 3.9 µHz at νmax. The ridge appears +strongly curved, similarly to the observations. To visualize +the ridge properly, the stretched échelle diagram was folded +using a period spacing of ∆Π(meas) +1 += 73.8 s, which is much +lower than the unperturbed period spacing ∆Π1. +Magnetic perturbations thus account for both charac- +teristics of the stars identified in Sect. 2. First, they produce +curved ridges in the period échelle diagram. Since magnetic +shifts are always positive, the period spacings of g modes +decrease with decreasing mode frequency. Thus, the cur- +vature always has the same shape, the low-frequency part +of the ridge being bent to the left direction of the period +échelle diagram. Interestingly, all the targets identified in +Sect. 2 show ridges curved in this direction (see Fig. B.1). +Secondly, magnetic perturbations yield a measured period +spacing that is significantly lower than the asymptotic un- +perturbed period spacing ∆Π1. This can explain why most +of the targets identified in Sect. 2 are located below the +degenerate sequence in the (∆ν, ∆Π1) plane. +4. Measurement of magnetic field strengths +We then estimated the field strengths that are required to +account for the observations. For this purpose, we computed +asymptotic expressions of the mixed mode frequencies in- +cluding magnetic perturbations. We followed the method +that we proposed in Li et al. (2022), which is briefly re- +called here. The effects magnetic fields are taken into ac- +count by adding a magnetic perturbation to the frequen- +cies of pure p and g modes. These perturbed frequencies +are then plugged into the asymptotic expression of mixed +mode frequencies given by Shibahashi (1979). While the +frequencies of p modes are unaffected (Li et al. 2022), the +periods of g modes are expressed as +Pg = Pg,0 +� +1 + δωg +2π Pg,0 +�−1 +(4) +where Pg,0 = (ng + 1/2 + εg)∆Π1 is the first-order asymp- +totic expression of l = 1 g modes without perturbation, and +δωg is the magnetic perturbation to g-mode frequencies. Us- +ing Eq. 1-3, δωg can be written as δωg = δω0 (ωmax/ω)3 , +where δω0 corresponds to the magnetic shift at ωmax. +For the 11 stars of our sample, we optimized the val- +ues of ∆Π1, δω0, εg, and d01 (defined below) to match the +observations at best using a Markov chain Monte Carlo +approach. Based on the measurements of εg for hundreds +Article number, page 3 of 10 + +A&A proofs: manuscript no. main +of Kepler red giants by Mosser et al. (2018), we assumed +a gaussian prior on εg with a mean of 0.28 and a stan- +dard deviation of 0.08, and we considered uniform priors +for the other parameters. The characteristics of pure p +modes were derived from the observed radial modes, with +the exception of d01, defined as the average small separa- +tion νp,l=0 − νp,l=1 + ∆ν/2, which was considered as a free +parameter of the fit. +The optimal parameters of the fit are given in Table +B.1. The corresponding asymptotic frequencies are shown +as red crosses in Fig. 1 and B.1. The agreement with the +observations is very good, the curvature of the ridge being +well reproduced for all the stars. The fit also provides an es- +timate of the unperturbed asymptotic period spacing ∆Π1 +for these stars, which is, as expected, larger than the appar- +ent period spacing ∆Π(meas) +1 +. We used the newly determined +values of ∆Π1 to update the location of the 11 targets in +the (∆ν, ∆Π1) plane in Fig. 2 (colored star symbols). It +is striking to observe that they now lie on the degenerate +sequence, as expected for stars in this mass range and evo- +lutionary state. Thus, there is a body of evidence that the +distortions to the g-mode pattern that are observed in the +11 targets of the sample are indeed produced by internal +magnetic fields. This yields the opportunity to characterize +these fields. +The measurement of δω0 can be used to derive an esti- +mate of ⟨B2 +r⟩ = +´ ro +ri K(r)B2rdr using Eq. 1-3. The obtained +expression depends on the asymmetry parameter a, which +can unfortunately not be measured with only one compo- +nent detected per multiplet. However, we have shown that +−1/2 ⩽ a ⩽ 1 (Li et al. 2022), so that we can place a lower +limit on the value of ⟨B2 +r⟩. We obtain +⟨B2 +r⟩min = 2 +3 +δω0ω3 +maxµ0 +I +, +(5) +where µ0 is the magnetic permeability. This expression is +valid regardless of whether the observed modes have an +azimuthal number of m = 0 or m = ±1 (indeed, the factors +1−a and 1+a/2 appearing in Eq. 2 and 3, respectively, are +both always inferior to 3/2). Only in the very specific case +of a field that is entirely concentrated on the poles (a → 1) +would the measured field be much larger than ⟨B2 +r⟩min. For +instance, if B2 +r has an axisymmetric dipolar configuration +(a = 2/5), we have ⟨B2 +r⟩ = 5/2⟨B2 +r⟩min. +To calculate ⟨B2 +r⟩min, the term I must be known, for +which a model of the stellar internal structure is needed. +For this purpose, we used a pre-computed grid of stellar +models of red giants with various masses, metallicities, and +evolutionary stages, built with the evolution code MESA +(Paxton et al. 2011). For each target, we selected models +from the grid that simultaneously reproduce the asymptotic +large separation of p modes ∆ν and the asymptotic period +spacing of dipole g modes ∆Π1. The models that satisfy +this condition all give similar estimates of I. We thus ob- +tained measurements of ⟨B2 +r⟩0.5 +min ranging from about 40 kG +to about 610 kG (see Table B.1). +5. Discussion +5.1. Magnetic field strength vs evolution +In Fig. 3, we plot the measured field strengths as a func- +tion of the density of mixed modes N = ∆ν/(∆Π1ν2 +max), +Fig. 3. Minimal field strength ⟨B2 +r⟩0.5 +min required to account for +the observed distortions in the g-mode period spacing for the +11 stars of our sample (black circles). They are plotted as func- +tion of the mixed mode density N = ∆ν/(∆Π1ν2 +max), which +is a proxy for evolution along the RGB (Gehan et al. 2018). +The red dashed line indicates the critical field Bc and the grey +long-dashed line shows the minimal field strength Bth required +to detect the distortions in the g-mode pattern. The blue star +symbols show the stars from Li et al. (2022). +which is a good proxy for the evolution along the red giant +branch (Gehan et al. 2018). We observe a clear decrease of +the measured field intensities along the evolution. At first +sight, this trend is surprising. Indeed, assuming conserva- +tion of the magnetic flux, the contraction of the core as red +giants evolve should increase the field intensity, so that one +would have expected the opposite trend. Before interpreting +this trend, we addressed the question of potential observa- +tional biases. In Appendix C, we calculated the threshold +field strength Bth that is required to produce detectable +variations in the g-mode period spacing over the observed +frequency range. As shown in Fig. 3, Bth decreases along +the evolution on the RGB. This explains why we do not de- +tect lower-intensity fields in unevolved red giants. However, +the lack of higher-intensity fields in more evolved stars can- +not be explained by this observational bias, and thus the +decrease in the field strength with evolution seems real. +5.2. Comparison with the critical field +We compared the measured minimal field intensities with +the critical field Bc. We stress that for fields over Bc, a +local analysis shows that gravity waves can no longer prop- +agate (Fuller et al. 2015). While the details of how global +modes are affected remain uncertain, it is clear that they +will be impacted. We used the stellar models selected from +our grid in Sect. 4 to estimate Bc for each star of the sam- +ple. We evaluated Bc in the HBS, where it reaches a sharp +minimum (Fuller et al. 2015), and where our field measure- +ments have the highest sensitivity. Fig. 3 shows that the +value of Bc in the HBS decreases with evolution, as was al- +ready pointed out by Fuller et al. (2015). We observe that +our minimal field strength measurements closely follow the +trend of Bc with evolution. One possible explanation for the +trend observed in Fig. 3 is that the core field increases with +evolution, owing to magnetic flux conservation, and even- +Article number, page 4 of 10 + +S. Deheuvels, G. Li, J. Ballot, F. Lignières: Strong magnetic fields detected in red giant cores +Fig. 4. Power spectrum of KIC 6975038 obtained from Kepler +data. Color-shaded areas indicate the location of l = 0 (blue), +l = 1 (red) and l = 2 (green) modes. +tually reaches the critical field Bc. Above this field, mixed +modes would no longer form, making the seismic detection +of core magnetic fields impossible. +5.3. Link with stars with suppressed dipole mixed modes +The ratio between the minimal measured field strength and +the critical field Bc is maximal for KIC 6975038, where it +reaches a factor of about 1.7. Interestingly, this star shows +clear signs of dipole mixed mode suppression. Fig. 4 shows +the power spectrum of KIC 6975038 built with Kepler data. +The regions of the spectrum where dipole mixed modes are +expected are highlighted in red. While the dipole mixed +mode pattern clearly appears at high frequency, it is nearly +absent for frequencies around νmax and below. +This type of behavior is expected, assuming that field +intensities above the critical field Bc can suppress mixed +modes. Indeed, Bc varies as ω2, so that for a given field +strength, there exists a transition frequency ωc below which +mixed modes should be strongly suppressed and above +which they should be unaffected (Fuller et al. 2015, Loi +2020b). This can be used to estimate the field strength for +stars where the transition between suppressed and normal +modes can be detected. For KIC6975038, the observed tran- +sition frequency ωc yields a radial field intensity of about +180 kG in the HBS. This estimate has the same order of +magnitude as the minimal field strength ⟨B2 +r⟩0.5 +min = 301 kG +that was inferred in an independent way using the perturba- +tions to the g-mode period spacing (Sect. 4). This star thus +combines two different features that have been interpreted +as potential indications of the presence of core magnetic +fields and they both lead to comparable estimates of mag- +netic field strength. While more stars of this type would be +required to draw conclusions, this is further indication that +there might be a link between mixed mode suppression and +strong core fields. +5.4. Origin of the detected fields +One possibility is that the detected fields were produced +by a dynamo in the convective core during the main se- +quence. The stars of our sample have masses ranging from +1.11 to 1.56 M⊙ (see Table B.1). Contrary to the three stars +studied in Li et al. (2022), the lowest-mass stars likely had +a radiative core during most the main sequence. However, +even these stars possessed a small initial convective core at +the beginning of the main sequence, owing to the burning of +3He and 12C outside of equilibrium (Deheuvels et al. 2010). +The ohmic diffusion timescale being longer than the evolu- +tion timescale (Cantiello et al. 2016), these fields can have +survived until the red giant phase and relaxed into stable +configurations (Braithwaite & Spruit 2004). By using the +stellar models introduced in Sect. 4 and assuming a conser- +vation of the magnetic flux, we estimated the main-sequence +field strengths that would be required to produce the de- +tected fields (Appendix D). We found minimal field intensi- +ties ranging from 1 to 26 kG inside the main-sequence con- +vective cores. This is in general lower than the radial mag- +netic field strengths found by the numerical simulation of a +convective core (Brun et al. 2005) or order-of-magnitude es- +timates assuming equipartition with the convective motion +kinetic energy (Cantiello et al. 2016). A dedicated study will +be necessary to determine whether the measured core fields +can be accounted for by this possible origin of the fields, +taking into account the diversity of the dynamo-generated +fields and the dissipation provoked by their relaxation (Be- +cerra et al. 2022) and potential instabilities (Gouhier et al. +2022) in the post-main-sequence phase. +6. Conclusion +We here revisited the puzzling case of H-shell burning red +giants that exhibit strong deviations from the regular pe- +riod spacing that gravity modes should reach in the high- +radial order limit (Deheuvels et al. 2022). We showed that +this peculiarity is unlikely to be produced by buoyancy +glitches, and on the contrary very well accounted for by +strong magnetic fields in the core of these stars. We thus +placed lower limits on the strength of the radial field in the +vicinity of the H-burning shell, ranging from 40 to 610 kG +for the 11 stars of our sample. We also showed that for +one star, the measured field exceeds the critical field Bc +above which gravity waves can no longer propagate in the +core (Fuller et al. 2015). Interestingly, this star shows mixed +mode suppression at low frequency, which further suggests +that this phenomenon might be related to strong core mag- +netic fields, although it should be noted that the mecha- +nisms leading to mode suppression remain uncertain. This +study focused on red giants with one single component de- +tected per multiplet, to avoid the additional complication +arising from rotational effects. We plan to search more gen- +erally for similar behavior in Kepler data in the near future. +Acknowledgements. S.D., J.B. and F.L. acknowledge support from +from the project BEAMING ANR-18-CE31-0001 of the French +National Research Agency (ANR) and from the Centre National +d’Etudes Spatiales (CNES). +References +Becerra, L., Reisenegger, A., Valdivia, J. A., & Gusakov, M. E. 2022, +MNRAS, 511, 732 +Braithwaite, J. & Spruit, H. C. 2004, Nature, 431, 819 +Brun, A. S., Browning, M. 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J. +2015, A&A, 584, A50 +Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3 +Petitdemange, L., Marcotte, F., & Gissinger, C. 2022, arXiv e-prints, +arXiv:2206.13819 +Rüdiger, G., Gellert, M., Spada, F., & Tereshin, I. 2015, A&A, 573, +A80 +Shibahashi, H. 1979, PASJ, 31, 87 +Unno, W., Osaki, Y., Ando, H., Saio, H., & Shibahashi, H. 1989, +Nonradial oscillations of stars (Tokyo: University of Tokyo Press) +Vrard, M., Mosser, B., & Samadi, R. 2016, A&A, 588, A87 +Yu, J., Huber, D., Bedding, T. R., et al. 2018, ApJS, 236, 42 +Article number, page 6 of 10 + +S. Deheuvels, G. Li, J. Ballot, F. Lignières: Strong magnetic fields detected in red giant cores +Appendix A: Can the deviations in g-mode period +spacings be related to buoyancy glitches? +It is well known that buoyancy glitches induce deviations +in the pattern of high-radial-order g modes (e.g., Miglio +et al. 2008). Such deviations were already found by exploit- +ing the mixed modes of core-helium burning giants (Mosser +et al. 2015). Cunha et al. (2015) and Cunha et al. (2019) +provided the appropriate formalism to determine the prop- +erties of buoyancy glitches (location and amplitude) from +their seismic signature. We here investigate what types of +glitches could produce the strong deviations in the period +spacings of g modes that we observed in our sample of Ke- +pler red giants. +A buoyancy glitch produces a periodic modulation in +the period spacings of g modes. The period of this modula- +tion is directly related to the position of the glitch. This po- +sition is generally expressed in terms of its buoyancy radius +or depth. Following the notations of Cunha et al. (2019), +the buoyancy radius ωr +g and depth ˜ωr +g at a radius r are +defined as +ωr +g = +ˆ r +r1 +LN +r +dr +; +˜ωr +g = +ˆ r2 +r +LN +r +dr, +(A.1) +respectively, where L = [l(l + 1)]1/2, and r1 and r2 are the +inner and outer turning points of the g-mode cavity. We also +introduce the total buoyancy radius of the g-mode cavity +ωg ≡ ωr2 +g . For a glitch located at a radius r⋆, one period of +the modulation covers ∆n radial orders, where +∆n = ωg/ωr⋆ +g , +(A.2) +if the glitch is located in the inner half of the cavity +(ωr +g⋆ < 0.5). If it is located in the outer half, then ωr⋆ +g +needs to be replaced by ˜ωr⋆ +g +in Eq. A.2. The amplitude of +the modulation depends on the sharpness of the variations +in N. +A.1. Glitch location +The g-mode period spacings can be obtained from the peri- +ods of mixed modes by applying a stretching (see Sect. 2). +The difference ∆τ between the stretched periods of consecu- +tive mixed modes (shown as an illustration for KIC5180345 +in Fig. A.1) provides an estimate of the g-mode period spac- +ing. For all the stars in our sample, the observed devia- +tions do not show the periodic behavior that is expected +for buoyancy glitches. If the observed deviations arise from +glitches, the period of the modulation needs to be larger +than the range defined by the observed modes. For exam- +ple, for KIC5180345 the glitch period would have to cover at +least 40 radial orders. Using Eq. A.2 and the stellar model +of KIC5180345 obtained in Sect. 4, this means that the +buoyancy glitch would need to be located either very deep +within the g-mode cavity (below a fractional radius of 10−4, +that is, deep within the inert He core) or nearly at the outer +edge of the g-mode cavity (that is, well above the H-burning +shell). +A.2. Glitch amplitude +We also addressed the question of the glitch amplitude +that would be required to reproduce the observations. As +Fig. A.1. Variations in the g-mode period spacings as a function +of mode frequency for KIC5180345. The observed period spac- +ings (filled circles) were computed as the difference ∆τ between +the stretched periods τ of consecutive dipolar mixed modes (see +Sect. 2.2). The blue dashed line indicates the g-mode period +spacings for the best-fit buoyancy glitch perturbation. The red +long-dashed line corresponds to the best-fit magnetic perturba- +tion (see Sect. 4). For the magnetic perturbation, we also show +the differences ∆τ, which are directly comparable to the obser- +vations (black solid line). +shown in Fig. A.1, the observed deviations have an ampli- +tude that reaches about 30% of the average period spacing. +For comparison, Cunha et al. (2019) show the example of a +Gaussian-shaped glitch in an RGB star, with an amplitude +of about twice the local value of N and a width of about +0.001 R (see their Fig. 1). They find that it yields a modu- +lation in the g-mode period spacing corresponding to only +1.5% of the asymptotic period spacing (see their Fig. 5). +To roughly estimate the glitch amplitude that would be +needed in our case, we used the formalism of Cunha et al. +(2019). We assumed a Gaussian-shaped buoyancy glitch +and we used a Markov chain Monte Carlo (MCMC) to opti- +mize the glitch properties (amplitude and width) in order to +reproduce at best the observed g-mode period spacings. Fig. +A.1 shows the best-fit solution (blue dashed line). This fit- +ting problem appears to be highly degenerated: similar pro- +files may be generated by different sets of parameters. How- +ever, some properties of the glitch can be derived from the +MCMC. In particular, we conclude that its amplitude must +be greater than six times the local value of N. Any smaller +value fails to reproduce the amplitude of the deviations ob- +served in g-mode period spacings. However, even with the +appropriate glitch amplitude, Fig. A.1 clearly shows that +the best-fit solution cannot correctly reproduce the shape +of the modulation. Indeed, it is well known (e.g., Miglio +et al. 2008; Cunha et al. 2019) that large-amplitude glitches +yield modulations in the g-mode period spacings that in- +volve sharp localized features (as opposed to small glitches, +which produce sinusoidal modulations), which seem incom- +patible with the smoothly varying period spacings that are +observed. On the contrary, a magnetic perturbation to the +oscillation modes provides a very good agreement with the +observations (red and black lines in Fig. A.1). +Article number, page 7 of 10 + +A&A proofs: manuscript no. main +Appendix B: Fit of asymptotic frequencies +including magnetic perturbation +Fig. B.1 shows the stretched échelle diagrams of the de- +tected dipole mixed modes for 11 red giants in our sam- +ple (blue circles). They were folded using the apparent +(perturbed) period spacing ∆Π(meas) +1 +. In Sect. 4, we fit an +asymptotic expression of the mode frequencies including a +magnetic perturbation to the observed modes. The optimal +solutions are shown in Fig. B.1 as red crosses. The agree- +ment is very good. We give in Table B.1 the parameters of +the best-fit solutions, along with general stellar properties, +for each star. +Appendix C: Minimal field strength required to +detect magnetic distortion in g-mode pattern +We search for the minimal field strength that produces a +detectable deviation in the regular period spacing of pure +gravity modes. For this purpose, we consider typical os- +cillation properties for red giants. More refined estimates +could be obtained on a star-to-star basis, but we are here +interested in deriving broad estimates of magnetic intensity +thresholds in order to investigate observational biases. +For a given red giant with a large separation ∆ν and a +frequency of maximum power of the oscillations νmax, we +consider that the modes can be detected in a frequency +interval ranging from fmin = νmax − 2∆ν and fmax = +νmax + 2∆ν. The asymptotic expression of unperturbed +pure gravity modes is given by Pn = ∆Π1(n + 1/2 + εg), +so that we expect to detect g modes with radial orders +ranging from nmin = 1/(∆Π1fmax) − εg − 1/2 and nmax = +1/(∆Π1fmin) − εg − 1/2. We then consider the asymptotic +periods P ′ +n of perturbed g modes in the presence of a field +that produces a frequency shift δν0 at νmax. Assuming that +the perturbation remains small compared to the mode pe- +riods themselves (this is well verified at the detection limit +for all stars of the sample), we have +P ′ +n ≈ Pn(1 − δν0P 4 +nν3 +max). +(C.1) +When analyzing the seismic data in red giants, high- +radial order gravity modes are assumed to be regularly +spaced in period and they are thus fit by an expression of +the type P ′n = ∆Π(meas) +1 +(n + 1/2 + ε(meas) +g +). Magnetic per- +turbations to the g-mode periods can be detected if the de- +viations compared to a regular spacing in period, expressed +as δPn = P ′n − P ′ +n, are sufficiently large. Since the unper- +turbed periods Pn vary linearly with n, the deviations δPn +can be written as +δPn = +� +αn + β + P 5 +nν3 +max +� +δν0 +(C.2) +where α and β are the parameters of a linear regression of +the term P 5 +nν3 +max as a function of n. Eq. C.2 shows that the +intensity of the deviation from a regular period spacing is +proportional to δν0. This can also be seen in the measured +period spacing ∆Π(meas) +1 +, which here corresponds to ∆Π1 + +αδν0. +Fig. C.1. Departures from a regular period spacing of gravity +modes in the presence of a magnetic field, shown as frequency +differences δνn as a function of the radial order n of gravity +modes. +The period differences can be translated into frequency +differences as δν = −δPn/P 2 +n. Fig. C.1 shows the variations +in δνn as a function of n for an illustration case with ∆ν = +10.6 µHz, ∆Π1 = 79.9 s, εg = 0.3, and δν0 = 0.4 µHz. The +maximal values of |δνn| are reached at the boundaries of +the interval, more particularly for n = nmin. +To determine whether these differences are detectable, +we need to compare them with the frequency resolution of +the measurements of oscillation mode frequencies. At the +edge of the frequency interval where oscillations are de- +tected, typical uncertainties reach several tens of nHz. We +thus considered here that a deviation from a regular pe- +riod spacing can be detected if δνnmin exceeds a threshold +δνth = 100 nHz. We thus obtained the following expression +for the minimal detectable magnetic perturbation +δν0,min = δνth +�αnmin + β +P 2nmin ++ P 3 +nminν3 +max +�−1 +(C.3) +We then used Eq. 5 to translate the minimal magnetic fre- +quency shifts into minimal detectable field intensities Bth, +which are shown in Fig. 3. +Appendix D: Can detected fields result from +dynamo action in previous convective cores? +The stars in which we detected strong core magnetic fields +in this study have masses ranging from 1.11 to 1.56 M⊙ +(see Table B.1). The lowest-mass stars of the sample have a +radiative core during the bulk of their main-sequence evo- +lution, but even these stars possessed a small convective +core at the beginning of the main sequence because of the +burning of 3He and 12C outside of equilibrium. We tried +to determine to what extent the detected fields are com- +patible with dynamo-generated fields in the main-sequence +convective core. +For this purpose, we assumed that a uniform field Br,MS +was produced during the main sequence, over a distance +corresponding to the maximal extent of the convective core. +After the withdrawal of the convective core, we assumed a +conservation of the magnetic flux in each layer, so that the +Article number, page 8 of 10 + +S. Deheuvels, G. Li, J. Ballot, F. Lignières: Strong magnetic fields detected in red giant cores +KIC6182668 +0 +50 +100 +Stretched period modulo ∆Π1 (s) +200 +210 +220 +230 +240 +250 +Frequency (µHz) +KIC9474201 +−20 0 +20 40 60 80 100 +Stretched period modulo ∆Π1 (s) +160 +180 +200 +220 +240 +Frequency (µHz) +KIC6842204 +0 +50 +100 +Stretched period modulo ∆Π1 (s) +120 +140 +160 +180 +200 +220 +Frequency (µHz) +KIC8689270 +0 +50 +100 +Stretched period modulo ∆Π1 (s) +120 +140 +160 +180 +200 +Frequency (µHz) +KIC6975038 +−20 +0 +20 +40 +60 +80 +Stretched period modulo ∆Π1 (s) +130 +140 +150 +160 +170 +Frequency (µHz) +KIC7728945 +−20 +0 +20 +40 +60 +80 100 +Stretched period modulo ∆Π1 (s) +70 +80 +90 +100 +110 +120 +Frequency (µHz) +KIC6614684 +−20 +0 +20 +40 +60 +80 100 +Stretched period modulo ∆Π1 (s) +60 +70 +80 +90 +100 +110 +120 +Frequency (µHz) +KIC3109742 +−20 +0 +20 +40 +60 +80 100 +Stretched period modulo ∆Π1 (s) +80 +90 +100 +110 +120 +Frequency (µHz) +Fig. B.1. Same as Fig. 1 for the remaining stars of the sample. +Article number, page 9 of 10 + +A&A proofs: manuscript no. main +Table B.1. Red giant branch stars showing strong variations in ∆Π1 among the catalog of Yu et al. (2018). +KIC Id +∆ν +νmax +M (a) +N +∆Π(meas) +1 +∆Π1 +δω0/(2π) +⟨B2 +r⟩min +(µHz) +(µHz) +(M⊙) +(s) +(s) +(µHz) +kG +6182668 +16.48 ± 0.12 +213.31 ± 2.69 +1.24 ± 0.09 +4.18 +80.4 +86.72 ± 0.15 +4.49 ± 0.09 +609 +9474201 +14.99 ± 0.02 +197.58 ± 1.41 +1.53 ± 0.09 +4.54 +76.8 +84.65 ± 0.07 +3.71 ± 0.02 +488 +6842204 +14.14 ± 0.04 +179.79 ± 0.63 +1.26 ± 0.06 +5.14 +81.9 +85.01 ± 0.05 +0.87 ± 0.02 +264 +8560280 +13.43 ± 0.07 +165.82 ± 0.75 +1.11 ± 0.06 +5.73 +73.8 +85.23 ± 0.05 +3.85 ± 0.01 +369 +8689270 +13.21 ± 0.03 +164.91 ± 0.55 +1.14 ± 0.05 +5.77 +79.3 +84.21 ± 0.05 +1.83 ± 0.01 +283 +3216736 +12.44 ± 0.02 +150.38 ± 0.71 +1.23 ± 0.08 +6.61 +74.5 +83.29 ± 0.03 +3.03 ± 0.01 +278 +5180345 +11.81 ± 0.03 +140.52 ± 0.50 +1.17 ± 0.07 +7.11 +74.5 +84.11 ± 0.04 +3.07 ± 0.01 +248 +6975038 +10.52 ± 0.02 +127.98 ± 0.80 +1.29 ± 0.08 +7.81 +63.5 +82.22 ± 0.08 +7.18 ± 0.03 +286 +3109742 +9.09 ± 0.01 +101.67 ± 0.55 +1.32 ± 0.09 +11.55 +71.5 +76.14 ± 0.05 +1.07 ± 0.01 +73 +6614684 +8.14 ± 0.01 +92.04 ± 0.45 +1.56 ± 0.10 +13.31 +71.5 +72.19 ± 0.01 +0.29 ± 0.00 +38 +7728945 +8.27 ± 0.02 +91.33 ± 0.44 +1.51 ± 0.09 +13.64 +71.4 +72.71 ± 0.03 +0.35 ± 0.01 +35 +Notes. (a): Masses from Yu et al. (2018). +Table D.1. Minimal main-sequence field intensities required to +account for current measured fields in the cores of our sample +of red giants, assuming conservation of the magnetic flux along +the evolution. +KIC Id +Br,MS (kG) +6182668 +18.1 +9474201 +16.2 +6842204 +18.7 +8560280 +25.8 +8689270 +19.6 +3216736 +18.9 +5180345 +17.8 +6975038 +17.7 +3109742 +5.3 +6614684 +0.8 +7728945 +0.8 +field intensity varies as 1/r(m)2 for a layer at Lagrangian +coordinate m. We could then calculate the average field +strength ⟨B2 +r⟩0.5 = +�´ ro +ri K(r)B2r dr +�0.5 +at each step of the +evolution. For each star of the sample, we calculated the +intensity of the main-sequence radial field Br,MS that is re- +quired to produce the minimal average fields ⟨B2 +r⟩0.5 +min that +we measured in Sect. 4. We thus found main-sequence field +strengths ranging from about 1 to 26 kG (see Table D.1). +These values correspond to lower limits on Br,MS, the actual +value depending on the geometry of the current radial mag- +netic field. For instance, if B2 +r has an axisymmetric dipolar +configuration (asymmetry parameter a = 2/5), the recov- +ered values of Br,MS range from 1 to 40 kG. +Article number, page 10 of 10 + diff --git a/P9AzT4oBgHgl3EQfWvzh/content/tmp_files/load_file.txt b/P9AzT4oBgHgl3EQfWvzh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c8bcf323fe00b7a79ae3f2e4f0139c7afa3a33c --- /dev/null +++ b/P9AzT4oBgHgl3EQfWvzh/content/tmp_files/load_file.txt @@ -0,0 +1,951 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf,len=950 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' main ©ESO 2023 January 5, 2023 Letter to the Editor Strong magnetic fields detected in the cores of 11 red giant stars using gravity-mode period spacings S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Deheuvels1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Li1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Ballot1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Lignières1 IRAP, Université de Toulouse, CNRS, CNES, UPS, 31400 Toulouse, France January 5, 2023 ABSTRACT Despite their importance in stellar evolution, little is known about magnetic fields in the interior of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The recent seismic detection of magnetic fields in the core of several red giant stars has given measurements of their strength and information on their topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We revisit the puzzling case of hydrogen-shell burning giants that show deviations from the expected regular period spacing of gravity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' These stars also tend to have a too low measured period spacing compared to their counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We here show that these two features are well accounted for by strong magnetic fields in the cores of these stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For 11 Kepler red giants showing these anomalies, we place lower limits on the core field strengths ranging from 40 to 610 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For one star, the measured field exceeds the critical field above which gravity waves no longer propagate in the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We find that this star shows mixed mode suppression at low frequency, which further suggests that this phenomenon might be related to strong core magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Asteroseismology – Stars: magnetic fields 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Introduction Magnetic fields affect stars at all evolutionary stages from star-forming molecular clouds to white dwarfs and magne- tars (McKee & Ostriker 2007, Kaspi & Beloborodov 2017, Ferrario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' In particular, they are expected to play a central role in the redistribution of angular momen- tum inside stars (Maeder & Meynet 2005, Cantiello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2014, Rüdiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2015), and thus in the transport of chemical elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' While surface magnetic fields have been detected and characterized in stars across the HR diagram (Landstreet 1992, Donati & Landstreet 2009), internal mag- netic fields have long remained inaccessible to direct obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' In red giant stars, the detection of mixed modes – that is, oscillation modes that behave as gravity (g) modes in the core and as pressure modes in the envelope – has given strong evidence that the cores of red giant stars are rotating slowly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=', Deheuvels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2012, Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2012, Gehan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This yielded evidence that angu- lar momentum is redistributed much more efficiently than if only purely hydrodynamical processes were at work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=', Marques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Magnetic fields could produce the additional transport that is needed (Rüdiger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2015, Jouve et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2015, Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2019, Petitdemange et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Observational constraints on the properties of inter- nal magnetic fields are crucially needed to assess the nature and the efficiency of the magnetic transport of angular mo- mentum inside stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The propagation of magneto-gravity waves is expected to be suppressed when the magnetic field exceeds a critical strength Bc above which Alfvén wave frequencies become comparable to those of gravity waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This phenomenon Send offprint requests to: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Deheuvels e-mail: sebastien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='deheuvels@irap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='omp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='eu was invoked by Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2015) to account for the unex- pectedly low amplitudes of dipole mixed modes in a frac- tion of red giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For core fields above Bc, the authors suggested that the mode energy reaching the magnetized core would be entirely dissipated and lost, giving rise to purely p-like dipole modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This interpretation was ques- tioned by Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2017), who found that partially suppressed dipole modes still retain a g-like character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Loi (2020b) later showed that even with strong fields, a frac- tion of the incoming waves could remain g-like, which would allow for partial energy return from the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The interpre- tation of suppressed dipole modes remains debated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Magnetic fields also produce shifts in the oscillation mode frequencies (Gough 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Several studies have re- cently investigated the impact of internal fields on the fre- quencies of mixed modes in red giants (Gomes & Lopes 2020, Bugnet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2021, Loi 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Very recently, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2022) detected clear asymmetries in the rotational mul- tiplets of dipole mixed modes in three Kepler red giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' They showed that these features can only be accounted for by internal magnetic fields with intensities ranging from 30 to 130 kG in the vicinity of the hydrogen burning shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' These findings opened the exciting opportunity to charac- terize magnetic fields in the cores of red giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We here investigate the irregularity of g-mode period spacings in a group of red giant branch (RGB) stars, which remains so far unexplained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' High-radial order g modes are expected to be approximately equally spaced in period by ∆Πl, where l is the mode degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' In red giants, dipole mixed modes can be used to measure ∆Π1 using asymptotic expressions of the mode frequencies (Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' While ∆Π1 is nearly constant over the frequency range of observed modes for the vast majority of RGB stars, some red giants show significant variations of ∆Π1 (Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Article number, page 1 of 10 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='01308v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='SR] 3 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' main 2018, Deheuvels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We here show that this fea- ture is the signature of strong magnetic fields in the cores of these stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This constitutes a new way of detecting and characterizing magnetic field in the cores of red giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2, we present red giants that exhibit deviations from the regular period spacing pattern of g modes, and we find additional such targets in Kepler data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We then show in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 3 that strong core magnetic fields can account for this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 4, we determine the field strengths that are required to match the seismic observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We discuss these measurements in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 5, before concluding in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Red giants with non-constant ∆Π1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Previous detections of non-constant ∆Π1 in RGB stars To first order, high-radial-order gravity modes are ex- pected to be equally spaced in period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Among the 160 RGB stars studied by Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2018), only one shows clear deviations from a regular period spacing of g modes (KIC3216736).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The authors attributed this irregularity to a buoyancy glitch (that is, a sharp variation in the Brunt- Väisälä frequency N), which induces periodic variations in the asymptotic period spacing ∆Π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' More recently, Deheuvels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2022) identified addi- tional RGB stars with non-constant ∆Π1, in a different con- text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' These stars appeared among a peculiar class of RGB stars that are located below the so-called “degeneracy se- quence” in the (∆ν, ∆Π1) plane, where RGB stars regroup when electron degeneracy becomes strong in their core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Most of the stars in this class are intermediate-mass stars and are thought to result from mass transfer (Deheuvels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The only four lower-mass stars with too-low ∆Π1 must have a different origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Contrary to intermediate- mass stars, they all show clear departures from a constant period spacing of g modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Interestingly, the star identified by Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2018) (KIC3216736) is among these tar- gets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This suggests that there might be a link between the non-constancy of ∆Π1 and the fact that its measured value is abnormally low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' These four stars show only one detected mode per rotational multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Additional targets We searched for other targets showing non-constant ∆Π1 among RGB stars with detected oscillations using the cat- alog of Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' To estimate the period spacings of g modes using dipole mixed modes, we computed the so-called “stretched” periods τ, defined by the differential equation dτ = dP/ζ (Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2015), where ζ corre- sponds to the fraction of the mode kinetic energy that is en- closed in the g-mode cavity (ζ tends to 1 for pure g modes, and 0 for pure p modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' When building échelle diagrams of these stretched periods, mixed modes are expected to align in a vertical ridge if ∆Π1 is constant and deviations from a regular period spacing induce curvature in this ridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We searched for stars with only one curved ridge de- tected in order to avoid the additional complication com- ing from rotational effects (these effects will be addressed in a subsequent work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This can mean that these stars are seen pole-on, so that only the m = 0 modes can be de- tected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This could also arise if the core rotation is too weak to produce detectable rotational splitting in Kepler data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' KIC8560280 −20 0 20 40 60 80 100 Stretched period modulo ∆Π1 (s) 140 160 180 200 Frequency (µHz) KIC3216736 −20 0 20 40 60 80 100 Stretched period modulo ∆Π1 (s) 120 140 160 180 Frequency (µHz) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Stretched period échelle diagrams of two red giants showing distortion from the regular g-mode pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Blue cir- cles show detected dipole modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Red crosses correspond to the best-fit asymptotic mixed mode frequencies obtained by includ- ing a magnetic perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We thus found seven additional targets, bringing the total of the sample to 11 stars (see Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Their stretched period échelle diagrams are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' They were folded using an average value of the asymptotic period spacing over the frequency range of the observations, which is further referred to as ∆Π(meas) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The location of these 11 targets in the (∆ν, ∆Π1) plane is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2 (black star symbols), where we have used the values ∆Π(meas) 1 as the measured asymptotic period spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Three of the seven additional targets lie well below the degeneracy sequence of RGB stars, which confirms the link between non-constant ∆Π1 and low measured values for these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Origin of distortions in g-mode pattern Deviations from a regular ∆Π1 are generally attributed to buoyancy glitches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=', Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' To produce the observed distortions in ∆Π1, we showed in Appendix A that a buoyancy glitch needs to have a large amplitude (the local value of N must be multiplied by a factor of at least six) and be located either deep inside the inert He core, or well above the H-burning shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' While this cannot be excluded, no known process is expected to produce such strong features in these regions of an RGB star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Secondly, the shape of the modulation in ∆Π1 that is produced by a large-amplitude glitch strongly differs from the observations (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Finally, the hypothesis of a buoyancy glitch would not explain why the measured values of ∆Π1 are unexpectedly low for most of our stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' It thus seems unlikely that the observed deviations in ∆Π1 arise from buoyancy glitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' In the following sections, we explore the possibility that the irregularities in ∆Π1 are produced by internal magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Article number, page 2 of 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Deheuvels, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Ballot, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Lignières: Strong magnetic fields detected in red giant cores Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Location of RGB stars with non-constant ∆Π1 in the (∆ν, ∆Π1) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Black star symbols correspond to the values of ∆Π1 that produce the best vertical alignment of modes in the stretched period échelle diagram (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Colored star symbols correspond to the corrected values of ∆Π1 obtained by taking into account the magnetic perturbation to the mode frequencies (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Other RGB stars from Vrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2016) are show as grey circles (for clarity, stars flagged by the authors as potential aliases were omitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Effects of magnetic fields on g-mode period spacings The influence of magnetic fields over oscillation mode fre- quencies has been studied over the last decades using a perturbative approach (Unno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 1989, Gough 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Re- cently, their effects on mixed modes in red giants have been addressed in the special case of dipolar fields with specific radial profiles, either aligned with the rotation axis (Hasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2005, Gomes & Lopes 2020, Mathis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2021, Bugnet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2021) or inclined (Loi 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2022) extended these studies to an arbitrary magnetic field and obtained a general expression for the magnetic frequency shift that is valid provided that the azimuthal component is not much larger than the radial one (Bφ/Br ≪ ωmax/N, where ωmax is the angular frequency at the maximum power of oscillations and N is the Brunt- Väisälä frequency1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Accordingly, the multiplets of l = 1 pure g modes (that is, g modes that are not coupled to p modes) undergo an average shift ωB, given by ωB = I µ0ω3 ˆ ro ri K(r)B2rdr, (1) where K(r) is a weight function that probes the g-mode cavity and sharply peaks in the vicinity of the H-burning shell (HBS), I is a factor that depends on the core struc- ture (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 45 and 46 of Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2022), and B2r = (4π)−1 ˜ B2 r sin θ dθ dφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The angular frequency shifts of the components of g-mode dipole multiplets are then given by δωg(m = 0) = (1 − a) ωB (2) δωg(m = ±1) = � 1 + a 2 � ωB, (3) 1 The ratio ωmax/N typically exceeds 102 for red giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' where a is a dimensionless coefficient that depends on the horizontal geometry of B2 r (a ∝ ˜ B2 rP2(cos θ) sin θ dθdφ, where P2(cos θ) is the second order Legendre polynomial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The dependency of magnetic shifts with ω−3 shows that low-frequency (that is, high-radial-order) g modes are more affected by magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For this reason, magnetic shifts create a deviation from the regular period spacing of pure g modes, as was already pointed out by Loi (2020a), Bugnet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2021), and Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Very recently Bugnet (2022) proposed a method to detect the signature of mag- netic fields exploiting this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For illustration, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 1 shows stretched échelle diagrams of mixed modes with magnetic perturbations (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The left panel corresponds to a case where the unperturbed g modes have an asymptotic period spacing of ∆Π1 = 85 s, and we have added a magnetic perturbation corresponding to a frequency shift of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='9 µHz at νmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The ridge appears strongly curved, similarly to the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' To visualize the ridge properly, the stretched échelle diagram was folded using a period spacing of ∆Π(meas) 1 = 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='8 s, which is much lower than the unperturbed period spacing ∆Π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Magnetic perturbations thus account for both charac- teristics of the stars identified in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' First, they produce curved ridges in the period échelle diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Since magnetic shifts are always positive, the period spacings of g modes decrease with decreasing mode frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Thus, the cur- vature always has the same shape, the low-frequency part of the ridge being bent to the left direction of the period échelle diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Interestingly, all the targets identified in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2 show ridges curved in this direction (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Secondly, magnetic perturbations yield a measured period spacing that is significantly lower than the asymptotic un- perturbed period spacing ∆Π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This can explain why most of the targets identified in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2 are located below the degenerate sequence in the (∆ν, ∆Π1) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Measurement of magnetic field strengths We then estimated the field strengths that are required to account for the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For this purpose, we computed asymptotic expressions of the mixed mode frequencies in- cluding magnetic perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We followed the method that we proposed in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2022), which is briefly re- called here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The effects magnetic fields are taken into ac- count by adding a magnetic perturbation to the frequen- cies of pure p and g modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' These perturbed frequencies are then plugged into the asymptotic expression of mixed mode frequencies given by Shibahashi (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' While the frequencies of p modes are unaffected (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2022), the periods of g modes are expressed as Pg = Pg,0 � 1 + δωg 2π Pg,0 �−1 (4) where Pg,0 = (ng + 1/2 + εg)∆Π1 is the first-order asymp- totic expression of l = 1 g modes without perturbation, and δωg is the magnetic perturbation to g-mode frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Us- ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 1-3, δωg can be written as δωg = δω0 (ωmax/ω)3 , where δω0 corresponds to the magnetic shift at ωmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For the 11 stars of our sample, we optimized the val- ues of ∆Π1, δω0, εg, and d01 (defined below) to match the observations at best using a Markov chain Monte Carlo approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Based on the measurements of εg for hundreds Article number, page 3 of 10 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' main of Kepler red giants by Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2018), we assumed a gaussian prior on εg with a mean of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='28 and a stan- dard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='08, and we considered uniform priors for the other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The characteristics of pure p modes were derived from the observed radial modes, with the exception of d01, defined as the average small separa- tion νp,l=0 − νp,l=1 + ∆ν/2, which was considered as a free parameter of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The optimal parameters of the fit are given in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The corresponding asymptotic frequencies are shown as red crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The agreement with the observations is very good, the curvature of the ridge being well reproduced for all the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The fit also provides an es- timate of the unperturbed asymptotic period spacing ∆Π1 for these stars, which is, as expected, larger than the appar- ent period spacing ∆Π(meas) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We used the newly determined values of ∆Π1 to update the location of the 11 targets in the (∆ν, ∆Π1) plane in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2 (colored star symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' It is striking to observe that they now lie on the degenerate sequence, as expected for stars in this mass range and evo- lutionary state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Thus, there is a body of evidence that the distortions to the g-mode pattern that are observed in the 11 targets of the sample are indeed produced by internal magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This yields the opportunity to characterize these fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The measurement of δω0 can be used to derive an esti- mate of ⟨B2 r⟩ = ´ ro ri K(r)B2rdr using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The obtained expression depends on the asymmetry parameter a, which can unfortunately not be measured with only one compo- nent detected per multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' However, we have shown that −1/2 ⩽ a ⩽ 1 (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2022), so that we can place a lower limit on the value of ⟨B2 r⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We obtain ⟨B2 r⟩min = 2 3 δω0ω3 maxµ0 I , (5) where µ0 is the magnetic permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This expression is valid regardless of whether the observed modes have an azimuthal number of m = 0 or m = ±1 (indeed, the factors 1−a and 1+a/2 appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2 and 3, respectively, are both always inferior to 3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Only in the very specific case of a field that is entirely concentrated on the poles (a → 1) would the measured field be much larger than ⟨B2 r⟩min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For instance, if B2 r has an axisymmetric dipolar configuration (a = 2/5), we have ⟨B2 r⟩ = 5/2⟨B2 r⟩min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' To calculate ⟨B2 r⟩min, the term I must be known, for which a model of the stellar internal structure is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For this purpose, we used a pre-computed grid of stellar models of red giants with various masses, metallicities, and evolutionary stages, built with the evolution code MESA (Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For each target, we selected models from the grid that simultaneously reproduce the asymptotic large separation of p modes ∆ν and the asymptotic period spacing of dipole g modes ∆Π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The models that satisfy this condition all give similar estimates of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We thus ob- tained measurements of ⟨B2 r⟩0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='5 min ranging from about 40 kG to about 610 kG (see Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Discussion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Magnetic field strength vs evolution In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 3, we plot the measured field strengths as a func- tion of the density of mixed modes N = ∆ν/(∆Π1ν2 max), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Minimal field strength ⟨B2 r⟩0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='5 min required to account for the observed distortions in the g-mode period spacing for the 11 stars of our sample (black circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' They are plotted as func- tion of the mixed mode density N = ∆ν/(∆Π1ν2 max), which is a proxy for evolution along the RGB (Gehan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The red dashed line indicates the critical field Bc and the grey long-dashed line shows the minimal field strength Bth required to detect the distortions in the g-mode pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The blue star symbols show the stars from Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' which is a good proxy for the evolution along the red giant branch (Gehan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We observe a clear decrease of the measured field intensities along the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' At first sight, this trend is surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Indeed, assuming conserva- tion of the magnetic flux, the contraction of the core as red giants evolve should increase the field intensity, so that one would have expected the opposite trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Before interpreting this trend, we addressed the question of potential observa- tional biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' In Appendix C, we calculated the threshold field strength Bth that is required to produce detectable variations in the g-mode period spacing over the observed frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 3, Bth decreases along the evolution on the RGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This explains why we do not de- tect lower-intensity fields in unevolved red giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' However, the lack of higher-intensity fields in more evolved stars can- not be explained by this observational bias, and thus the decrease in the field strength with evolution seems real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Comparison with the critical field We compared the measured minimal field intensities with the critical field Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We stress that for fields over Bc, a local analysis shows that gravity waves can no longer prop- agate (Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' While the details of how global modes are affected remain uncertain, it is clear that they will be impacted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We used the stellar models selected from our grid in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 4 to estimate Bc for each star of the sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We evaluated Bc in the HBS, where it reaches a sharp minimum (Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2015), and where our field measure- ments have the highest sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 3 shows that the value of Bc in the HBS decreases with evolution, as was al- ready pointed out by Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We observe that our minimal field strength measurements closely follow the trend of Bc with evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' One possible explanation for the trend observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 3 is that the core field increases with evolution, owing to magnetic flux conservation, and even- Article number, page 4 of 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Deheuvels, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Ballot, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Lignières: Strong magnetic fields detected in red giant cores Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Power spectrum of KIC 6975038 obtained from Kepler data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Color-shaded areas indicate the location of l = 0 (blue), l = 1 (red) and l = 2 (green) modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' tually reaches the critical field Bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Above this field, mixed modes would no longer form, making the seismic detection of core magnetic fields impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Link with stars with suppressed dipole mixed modes The ratio between the minimal measured field strength and the critical field Bc is maximal for KIC 6975038, where it reaches a factor of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Interestingly, this star shows clear signs of dipole mixed mode suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 4 shows the power spectrum of KIC 6975038 built with Kepler data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The regions of the spectrum where dipole mixed modes are expected are highlighted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' While the dipole mixed mode pattern clearly appears at high frequency, it is nearly absent for frequencies around νmax and below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This type of behavior is expected, assuming that field intensities above the critical field Bc can suppress mixed modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Indeed, Bc varies as ω2, so that for a given field strength, there exists a transition frequency ωc below which mixed modes should be strongly suppressed and above which they should be unaffected (Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2015, Loi 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This can be used to estimate the field strength for stars where the transition between suppressed and normal modes can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For KIC6975038, the observed tran- sition frequency ωc yields a radial field intensity of about 180 kG in the HBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This estimate has the same order of magnitude as the minimal field strength ⟨B2 r⟩0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='5 min = 301 kG that was inferred in an independent way using the perturba- tions to the g-mode period spacing (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This star thus combines two different features that have been interpreted as potential indications of the presence of core magnetic fields and they both lead to comparable estimates of mag- netic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' While more stars of this type would be required to draw conclusions, this is further indication that there might be a link between mixed mode suppression and strong core fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Origin of the detected fields One possibility is that the detected fields were produced by a dynamo in the convective core during the main se- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The stars of our sample have masses ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='11 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='56 M⊙ (see Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Contrary to the three stars studied in Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2022), the lowest-mass stars likely had a radiative core during most the main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' However, even these stars possessed a small initial convective core at the beginning of the main sequence, owing to the burning of 3He and 12C outside of equilibrium (Deheuvels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The ohmic diffusion timescale being longer than the evolu- tion timescale (Cantiello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2016), these fields can have survived until the red giant phase and relaxed into stable configurations (Braithwaite & Spruit 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' By using the stellar models introduced in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 4 and assuming a conser- vation of the magnetic flux, we estimated the main-sequence field strengths that would be required to produce the de- tected fields (Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We found minimal field intensi- ties ranging from 1 to 26 kG inside the main-sequence con- vective cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This is in general lower than the radial mag- netic field strengths found by the numerical simulation of a convective core (Brun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2005) or order-of-magnitude es- timates assuming equipartition with the convective motion kinetic energy (Cantiello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' A dedicated study will be necessary to determine whether the measured core fields can be accounted for by this possible origin of the fields, taking into account the diversity of the dynamo-generated fields and the dissipation provoked by their relaxation (Be- cerra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2022) and potential instabilities (Gouhier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2022) in the post-main-sequence phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Conclusion We here revisited the puzzling case of H-shell burning red giants that exhibit strong deviations from the regular pe- riod spacing that gravity modes should reach in the high- radial order limit (Deheuvels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We showed that this peculiarity is unlikely to be produced by buoyancy glitches, and on the contrary very well accounted for by strong magnetic fields in the core of these stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We thus placed lower limits on the strength of the radial field in the vicinity of the H-burning shell, ranging from 40 to 610 kG for the 11 stars of our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We also showed that for one star, the measured field exceeds the critical field Bc above which gravity waves can no longer propagate in the core (Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Interestingly, this star shows mixed mode suppression at low frequency, which further suggests that this phenomenon might be related to strong core mag- netic fields, although it should be noted that the mecha- nisms leading to mode suppression remain uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This study focused on red giants with one single component de- tected per multiplet, to avoid the additional complication arising from rotational effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We plan to search more gen- erally for similar behavior in Kepler data in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' acknowledge support from from the project BEAMING 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 1989, Nonradial oscillations of stars (Tokyo: University of Tokyo Press) Vrard, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=', Mosser, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=', & Samadi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2016, A&A, 588, A87 Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=', Huber, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=', Bedding, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2018, ApJS, 236, 42 Article number, page 6 of 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Deheuvels, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Ballot, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Lignières: Strong magnetic fields detected in red giant cores Appendix A: Can the deviations in g-mode period spacings be related to buoyancy glitches?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' It is well known that buoyancy glitches induce deviations in the pattern of high-radial-order g modes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=', Miglio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Such deviations were already found by exploit- ing the mixed modes of core-helium burning giants (Mosser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2015) and Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2019) provided the appropriate formalism to determine the prop- erties of buoyancy glitches (location and amplitude) from their seismic signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We here investigate what types of glitches could produce the strong deviations in the period spacings of g modes that we observed in our sample of Ke- pler red giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' A buoyancy glitch produces a periodic modulation in the period spacings of g modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The period of this modula- tion is directly related to the position of the glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This po- sition is generally expressed in terms of its buoyancy radius or depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Following the notations of Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2019), the buoyancy radius ωr g and depth ˜ωr g at a radius r are defined as ωr g = ˆ r r1 LN r dr ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' ˜ωr g = ˆ r2 r LN r dr, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1) respectively, where L = [l(l + 1)]1/2, and r1 and r2 are the inner and outer turning points of the g-mode cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We also introduce the total buoyancy radius of the g-mode cavity ωg ≡ ωr2 g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For a glitch located at a radius r⋆, one period of the modulation covers ∆n radial orders, where ∆n = ωg/ωr⋆ g , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='2) if the glitch is located in the inner half of the cavity (ωr g⋆ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' If it is located in the outer half, then ωr⋆ g needs to be replaced by ˜ωr⋆ g in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The amplitude of the modulation depends on the sharpness of the variations in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Glitch location The g-mode period spacings can be obtained from the peri- ods of mixed modes by applying a stretching (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The difference ∆τ between the stretched periods of consecu- tive mixed modes (shown as an illustration for KIC5180345 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1) provides an estimate of the g-mode period spac- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For all the stars in our sample, the observed devia- tions do not show the periodic behavior that is expected for buoyancy glitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' If the observed deviations arise from glitches, the period of the modulation needs to be larger than the range defined by the observed modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For exam- ple, for KIC5180345 the glitch period would have to cover at least 40 radial orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='2 and the stellar model of KIC5180345 obtained in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 4, this means that the buoyancy glitch would need to be located either very deep within the g-mode cavity (below a fractional radius of 10−4, that is, deep within the inert He core) or nearly at the outer edge of the g-mode cavity (that is, well above the H-burning shell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Glitch amplitude We also addressed the question of the glitch amplitude that would be required to reproduce the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Variations in the g-mode period spacings as a function of mode frequency for KIC5180345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The observed period spac- ings (filled circles) were computed as the difference ∆τ between the stretched periods τ of consecutive dipolar mixed modes (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The blue dashed line indicates the g-mode period spacings for the best-fit buoyancy glitch perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The red long-dashed line corresponds to the best-fit magnetic perturba- tion (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For the magnetic perturbation, we also show the differences ∆τ, which are directly comparable to the obser- vations (black solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1, the observed deviations have an ampli- tude that reaches about 30% of the average period spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For comparison, Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2019) show the example of a Gaussian-shaped glitch in an RGB star, with an amplitude of about twice the local value of N and a width of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='001 R (see their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' They find that it yields a modu- lation in the g-mode period spacing corresponding to only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='5% of the asymptotic period spacing (see their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' To roughly estimate the glitch amplitude that would be needed in our case, we used the formalism of Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We assumed a Gaussian-shaped buoyancy glitch and we used a Markov chain Monte Carlo (MCMC) to opti- mize the glitch properties (amplitude and width) in order to reproduce at best the observed g-mode period spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1 shows the best-fit solution (blue dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This fit- ting problem appears to be highly degenerated: similar pro- files may be generated by different sets of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' How- ever, some properties of the glitch can be derived from the MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' In particular, we conclude that its amplitude must be greater than six times the local value of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Any smaller value fails to reproduce the amplitude of the deviations ob- served in g-mode period spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' However, even with the appropriate glitch amplitude, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1 clearly shows that the best-fit solution cannot correctly reproduce the shape of the modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Indeed, it is well known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=', Miglio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Cunha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 2019) that large-amplitude glitches yield modulations in the g-mode period spacings that in- volve sharp localized features (as opposed to small glitches, which produce sinusoidal modulations), which seem incom- patible with the smoothly varying period spacings that are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' On the contrary, a magnetic perturbation to the oscillation modes provides a very good agreement with the observations (red and black lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Article number, page 7 of 10 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' main Appendix B: Fit of asymptotic frequencies including magnetic perturbation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1 shows the stretched échelle diagrams of the de- tected dipole mixed modes for 11 red giants in our sam- ple (blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' They were folded using the apparent (perturbed) period spacing ∆Π(meas) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 4, we fit an asymptotic expression of the mode frequencies including a magnetic perturbation to the observed modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The optimal solutions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1 as red crosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The agree- ment is very good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We give in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1 the parameters of the best-fit solutions, along with general stellar properties, for each star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Appendix C: Minimal field strength required to detect magnetic distortion in g-mode pattern We search for the minimal field strength that produces a detectable deviation in the regular period spacing of pure gravity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For this purpose, we consider typical os- cillation properties for red giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' More refined estimates could be obtained on a star-to-star basis, but we are here interested in deriving broad estimates of magnetic intensity thresholds in order to investigate observational biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For a given red giant with a large separation ∆ν and a frequency of maximum power of the oscillations νmax, we consider that the modes can be detected in a frequency interval ranging from fmin = νmax − 2∆ν and fmax = νmax + 2∆ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The asymptotic expression of unperturbed pure gravity modes is given by Pn = ∆Π1(n + 1/2 + εg), so that we expect to detect g modes with radial orders ranging from nmin = 1/(∆Π1fmax) − εg − 1/2 and nmax = 1/(∆Π1fmin) − εg − 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We then consider the asymptotic periods P ′ n of perturbed g modes in the presence of a field that produces a frequency shift δν0 at νmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Assuming that the perturbation remains small compared to the mode pe- riods themselves (this is well verified at the detection limit for all stars of the sample), we have P ′ n ≈ Pn(1 − δν0P 4 nν3 max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1) When analyzing the seismic data in red giants, high- radial order gravity modes are assumed to be regularly spaced in period and they are thus fit by an expression of the type P ′n = ∆Π(meas) 1 (n + 1/2 + ε(meas) g ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Magnetic per- turbations to the g-mode periods can be detected if the de- viations compared to a regular spacing in period, expressed as δPn = P ′n − P ′ n, are sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Since the unper- turbed periods Pn vary linearly with n, the deviations δPn can be written as δPn = � αn + β + P 5 nν3 max � δν0 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='2) where α and β are the parameters of a linear regression of the term P 5 nν3 max as a function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='2 shows that the intensity of the deviation from a regular period spacing is proportional to δν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' This can also be seen in the measured period spacing ∆Π(meas) 1 , which here corresponds to ∆Π1 + αδν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Departures from a regular period spacing of gravity modes in the presence of a magnetic field, shown as frequency differences δνn as a function of the radial order n of gravity modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The period differences can be translated into frequency differences as δν = −δPn/P 2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1 shows the variations in δνn as a function of n for an illustration case with ∆ν = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='6 µHz, ∆Π1 = 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='9 s, εg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='3, and δν0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='4 µHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The maximal values of |δνn| are reached at the boundaries of the interval, more particularly for n = nmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' To determine whether these differences are detectable, we need to compare them with the frequency resolution of the measurements of oscillation mode frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' At the edge of the frequency interval where oscillations are de- tected, typical uncertainties reach several tens of nHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We thus considered here that a deviation from a regular pe- riod spacing can be detected if δνnmin exceeds a threshold δνth = 100 nHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We thus obtained the following expression for the minimal detectable magnetic perturbation δν0,min = δνth �αnmin + β P 2nmin + P 3 nminν3 max �−1 (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='3) We then used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 5 to translate the minimal magnetic fre- quency shifts into minimal detectable field intensities Bth, which are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Appendix D: Can detected fields result from dynamo action in previous convective cores?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The stars in which we detected strong core magnetic fields in this study have masses ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='11 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='56 M⊙ (see Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' The lowest-mass stars of the sample have a radiative core during the bulk of their main-sequence evo- lution, but even these stars possessed a small convective core at the beginning of the main sequence because of the burning of 3He and 12C outside of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We tried to determine to what extent the detected fields are com- patible with dynamo-generated fields in the main-sequence convective core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For this purpose, we assumed that a uniform field Br,MS was produced during the main sequence, over a distance corresponding to the maximal extent of the convective core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' After the withdrawal of the convective core, we assumed a conservation of the magnetic flux in each layer, so that the Article number, page 8 of 10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Deheuvels, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Ballot, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Lignières: Strong magnetic fields detected in red giant cores ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 1 for the remaining stars of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Article number, page 9 of 10 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' main Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Red giant branch stars showing strong variations in ∆Π1 among the catalog of Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' KIC Id ∆ν νmax M (a) N ∆Π(meas) 1 ∆Π1 δω0/(2π) ⟨B2 r⟩min (µHz) (µHz) (M⊙) (s) (s) (µHz) kG 6182668 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='12 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='31 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='69 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='09 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='18 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='72 ± 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='01 35 Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (a): Masses from Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Table D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Minimal main-sequence field intensities required to account for current measured fields in the cores of our sample of red giants, assuming conservation of the magnetic flux along the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' KIC Id Br,MS (kG) 6182668 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1 9474201 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='2 6842204 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='7 8560280 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='8 8689270 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='6 3216736 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='9 5180345 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='8 6975038 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='7 3109742 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='3 6614684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='8 7728945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='8 field intensity varies as 1/r(m)2 for a layer at Lagrangian coordinate m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We could then calculate the average field strength ⟨B2 r⟩0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='5 = �´ ro ri K(r)B2r dr �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='5 at each step of the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For each star of the sample, we calculated the intensity of the main-sequence radial field Br,MS that is re- quired to produce the minimal average fields ⟨B2 r⟩0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='5 min that we measured in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' We thus found main-sequence field strengths ranging from about 1 to 26 kG (see Table D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' These values correspond to lower limits on Br,MS, the actual value depending on the geometry of the current radial mag- netic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' For instance, if B2 r has an axisymmetric dipolar configuration (asymmetry parameter a = 2/5), the recov- ered values of Br,MS range from 1 to 40 kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} +page_content=' Article number, page 10 of 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfWvzh/content/2301.01308v1.pdf'} diff --git a/RtA0T4oBgHgl3EQfDv-e/content/tmp_files/2301.02008v1.pdf.txt b/RtA0T4oBgHgl3EQfDv-e/content/tmp_files/2301.02008v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..eea735ca24ceaf2685d45836224caaf3fbc300d5 --- /dev/null +++ b/RtA0T4oBgHgl3EQfDv-e/content/tmp_files/2301.02008v1.pdf.txt @@ -0,0 +1,757 @@ +EXPRESSIVE SPEECH-DRIVEN FACIAL ANIMATION WITH +CONTROLLABLE EMOTIONS +Yutong Chen∗, Junhong Zhao† and Wei-Qiang Zhang‡ +∗Tsinghua University, chenyt19@mails.tsinghua.edu.cn +†Victoria University of Wellington, New Zealand, junhong.jennifer@gmail.com +‡Tsinghua University, wqzhang@tsinghua.edu.cn, . +ABSTRACT +It is in high demand to generate facial animation with high re- +alism, but it remains a challenging task. Existing approaches +of speech-driven facial animation can produce satisfactory +mouth movement and lip synchronization, but show weak- +ness in dramatic emotional expressions and flexibility in emo- +tion control. +This paper presents a novel deep learning- +based approach for expressive facial animation generation +from speech that can exhibit wide-spectrum facial expres- +sions with controllable emotion type and intensity. We pro- +pose an emotion controller module to learn the relationship +between the emotion variations (e.g., types and intensity) and +the corresponding facial expression parameters. It enables +emotion-controllable facial animation, where the target ex- +pression can be continuously adjusted as desired. The qual- +itative and quantitative evaluations show that the animation +generated by our method is rich in facial emotional expres- +siveness while retaining accurate lip movement, outperform- +ing other state-of-the-art methods. +Index Terms— 3D facial animation, 3D avatar, talking +head, emotion, expressive facial animation +1. INTRODUCTION +Facial animation is a growing research topic that has been +widely adopted in many applications, such as education, Vir- +tual Reality (VR), and digital entertainment. +Commercial +products often require high plausibility and expressiveness +in the animated characters to meet users’ needs for immer- +sive engagement. Creating such realistic facial animation is +a great challenge. Many recent works have resorted to deep +learning to automate face animating based on easy-acquire +data. Speech-driven facial animation is one critical compo- +nent in this field and has drawn much attention. It is to em- +ulate life-like facial motion based on the information carried +by a vocal audio track. Our work focuses on speech-driven +facial animation in 3D. Compared with animating 2D im- +ages, modulating a 3D model is directly applicable to most +3D applications like 3D games and visual aftereffects. At- +tempts have been made to explore the dependencies between +audio and 3D face movement, most of which focus on the +lower face and lip movements and their synchronization with +speech [1, 2, 3, 4, 5, 6]. Although they can produce plausible +basic facial motions, they are far from satisfactory in synthe- +sizing facial expressions, especially in presenting emotions. +To realize emotional animation based on only an audio +track is a challenging task. Although the dependency between +sound production and lip movement for the same person is +deterministic, its dependencies with the expressions of differ- +ent emotion categories and intensities are highly ambiguous. +There are many different personalized conveying ways of one +emotion for different users. Such inherent ambiguity makes +neural networks hard to handle emotion variations based on +the audio input, even given long-contextual information. The +works by Karras et al. and Pham et al. [7, 8] tried to extract +emotion features from speech and implicitly embed them into +their neural network to realize emotion synthesis. However, +their solution inevitably suffers from over-smoothed regres- +sion due to the limited training data, and often results in lim- +ited expressiveness. In practice, we found that even given the +best expressive speech, the animations generated by existing +methods still fail to achieve drastic emotion dynamics. +We present a novel approach to realize emotion- +controllable facial animation. Instead of recovering lip move- +ment from audio, our method enriches the emotion expres- +sivity and enables the adjustment of the intensity of emo- +tion effects to satisfy the animators’ needs. We propose an +emotion controller module, which includes an emotion pre- +dictor followed by an emotion augment network, to explicitly +model the relationship between emotion variations and corre- +sponding facial expression parameters. Image-based emotion +recognition was used to generate emotion information as pri- +ors to guide the training process. During inference, the spec- +ified emotion condition will apply to the speech-driven facial +animation to realize emotion enhancement and customization. +By explicitly modelling emotion, our animator-friendly sys- +tem enables emotion control with a given emotion type and +an intensity value. Our method shows promising results in +synthesizing controllable emotional facial animation while re- +taining high-accuracy lip synchronization, outperforming the +arXiv:2301.02008v1 [cs.CV] 5 Jan 2023 + +state-of-the-arts. The implementation code will be publicly +available online. +2. RELATED WORK +Despite much work focusing on facial animation from image +or video [9, 10, 11, 12] or generate speech-driven 2D talk- +ing head [13, 14, 15, 16, 17], we concentrate our efforts on +speech-driven 3D facial animation, mainly targeting to im- +prove the emotional expression synthesis. We review the most +relevant deep learning-based approaches. +Speech-driven 3D models. +Earlier speech-driven 3D fa- +cial animation methods are based on phonetic annotation +and viseme-model blending [1, 18]. +VisemeNet [1] lever- +aged LSTMs for phoneme grouping and facial landmark pre- +diction, and used their results to regress viseme parameters +for lip animation. VOCA [5] tried to encode the identity- +dependent information into animation and synthesized vari- +ous speaking styles. MeshTalk [6] proposed a method to dis- +entangle audio-correlated and audio-uncorrelated information +to generate more plausible dynamics on the upper face while +attaining accurate lip motion. +FaceFormer [2] proposed a +transformer-based autoregressive model to encode long-term +audio context and synthesize improved lip motions. Taylor et +al. [3] proposed a sliding-window regression method to pre- +dict the active appearance model (AAM) parameters of a ref- +erence lower face based on phoneme labels. Liu et al. [19] +considered the influence of geometry representation and uti- +lize them to produce generalized speaker-independent facial +animation. Although considerable progress has been achieved +in the field, most of these prior works focused on lip mo- +tion accuracy without considering expressiveness, which lim- +its the realism of their results. +Conditional emotion synthesis. Some recent facial anima- +tion methods [15, 7, 8, 20] tackled some issues in emotion +synthesis to make more vivid facial animations. +Pham et +al. used LSTMs in [8] (improved in [21] with CNN-RNN) +to model the long-contextual relationship between acoustic +features and facial expressions to realize emotion awareness +in the generated animation. The method proposed by Kar- +ras et al. [7] extracts a latent emotion representation from +the audio without identifying emotion categories. +In both +methods, emotions are implicitly represented and thus lack +meaningful guidance to emotion intensity control. +Ji et +al. [15] decomposed speech into emotion and content com- +ponents to generate emotion-controlled 2D talking heads. +Chun et al. [20] introduced an emotion-guided method where +emotion-expressive blendshapes are enhanced by emotion +recognition guidance and then fused with mouth-expressive +blendshapes. Nevertheless, the technique needs manual effort +to prepare each emotion template, and the generated expres- +sions often lack upper-face dynamics. +3. METHOD +3.1. Network Architecture Design +Figure 1. illustrates our pipeline. Our goal is to enrich emo- +tion expressivity in speech-driven facial animation and enable +users’ control over emotion variations. We first design a neu- +ral network to estimate the facial movement represented by +FLAME parameters [22] (See supplementary) from the input +audio, using both local and long-contextual information. The +problem can be formulated as follows: +(⃗φt, ⃗θt) = F(S(⃗xt±∆t) → ⃗ψt; W(⃗xt±∆t) → ⃗ωt) +(1) +Given the audio segment xt at time t and its neighboring +frames, the local content features (content vector ⃗ωt) and +global style features (style vector ⃗ψt) learned by wav2vec2.0 +W(·) and a transformer encoder S(·) are first extracted sep- +arately. +Then they are concatenated together to predict +FLAME parameters. The mapping between the audio and the +FLAME parameters is learned by an Auido2FLAME mod- +ule F(·), which is a multi-layer CNN. The predicted FLAME +parameters, including expression parameters ⃗φt and pose pa- +rameters ⃗θt, combined with the shape parameters of the given +identity, are converted to 3D mesh as the output. The lip syn- +chronization will be primarily focused on and preciseness in +mouth motion will be ensured at this phase. +We introduce an emotion control module that includes a +bi-LSTM-based emotion predictor followed by an embedding +layer to generate emotion-related latent features (emotion fea- +ture vectors), and a CNN-based emotion-augment network +to enhance the expressivity of FLAME parameters based on +emotion features. The emotion augmentation process can be +represented as: +E(⃗ψt±∆t, ⃗ωt±∆t, γu,t) : (⃗φt, ⃗θt) → (⃗φ′ +t, ⃗θ′ +t) +(2) +Where E(·) denotes the emotion control module. With the +audio features and emotion conditions γu,t customized by the +user, it maps (⃗φt, ⃗θt) predicted by the Audio2FLAME model +to the emotion-enhanced facial parameters (φ′ +t, θ′ +t). We in- +corporate the emotion-augment network in a residual man- +ner in our pipeline, which allows dedicated optimization on +emotion-related expressions while retaining content-related +expressions. Using this way, the user can explicitly design +the emotion intensities and categories at the frame level and +regulate the animation output. +3.2. Emotion Control Module +The core challenge to realizing full control of emotion sim- +ulation is to make the model adaptive to not only emotional +state changes, but also emotion strength variations to allow +straightforward intensity adjustment. Our proposed training +and inference pipeline is illustrated in Fig. 2. + +Audio +Emotion Condition +(Animator-Controlled) +FLAME Parameters +Style vector +Content vector +wav2vec 2.0 +Transformer +Audio2FLAME +Pose +Shape +Expression +Emotion Predictor +Emotion-augment network +Emotion Control Module +Enhanced FLAME Parameters +Pose +Shape +Expression +Fig. 1: An overview of our pipeline. +Emotion prediction and control. In the training phase, to +make the network see emotion variations in the input, we +leverage the image-based emotion recognition model to ob- +tain frame-level emotion information as emotion priors to fa- +cilitate the model training. DAN model from Weng et.al. [23] +was used in our experiments, which could be substituted by +other similar models. We assume emotion features decoded +from 2D visual images are more reliable and informative than +that from audio, which is beneficial for the model learning. +However, the emotion classification probabilities from +DAN can not indicate the magnitude of emotions. Instead, +we found that the emotion logits before the final softmax +layer of the emotion recognition network, featuring seven- +dimensional vector for seven emotions including happiness, +anger and etc., are highly in agreement with the perceived +emotion intensity. Therefore, we use them as emotion priors +for model training and hinge them with users’ emotion con- +trol. See supplementary material in detail about our simpli- +fied conceptual proof of the linear relation between emotion +logits and emotion intensities. In our experiment, we found +that the emotion logits are effective in emotion synthesis and +work well with the adjustment of both emotion categories and +intensities. The emotion augment network is resilient to the +prediction error of the emotion priors from the DAN module, +and can produce general emotional expressions from them. +In the inference phase, emotion priors are extracted from +audio through a bi-LSTM network, and altered by the cus- +tomized emotion category and intensity. The bi-LSTM net- +work was trained by maximizing the mutual information be- +tween audio-based and video-based emotion priors, using +video-based emotion priors as a pseudo ground-truth. The +emotion conditions given by the user will be transformed +to a one-hot vector with values ranging from 0 to 1 (γu,t) +and added to the decentralized audio-based emotion priors +(segment-level mean normalization (γa) ), bringing in the fi- +nal emotion priors γt = γu,t + (γa,t − γa). +Emotion augment network. Before inputting into the emo- +tion augment network, the emotion priors will be transformed +into emotion feature vectors through an embedding layer to +drive 3D facial expression augmentation. +The embedding +Audio +Enhanced FLAME parameters +Audio +Video +DAN +Train +Emotion-augment network +Inference +Embedding layer +Emotion Condition +(Animator-Controlled) +Emotion +Predictor +Raw FLAME parameters +Emotion priors +Fig. 2: Training/inference pipeline of our proposed emotion control module. +layer is a learnable 2D matrix that will be optimized dur- +ing training. The 7-dimensional priors vector will be trans- +formed to a size of 128 emotion feature vectors by multiply- +ing with the learned embedding matrix, and then input into +the emotion augment network built with CNN blocks to real- +ize emotion-guided facial expression enhancement. To attain +a proper balance between lip synchronization and emotion ex- +pressivity, we added the emotion augment network on a resid- +ual basis (see Eq. 3). The raw facial parameters extracted +from Audio2FLAME are expected to have a preferable lip +synchronization. The gap between the raw facial parameters +and ground truth ( ⃗ +φgt,t; ⃗ +θgt,t) is mainly caused by their differ- +ent emotional expressions, making the augment network A(·) +more effective to learn on. +A(⃗φt; ⃗θt) + (⃗φt; ⃗θt) ∼= ( ⃗ +φgt,t; ⃗ +θgt,t) +(3) +3.3. Loss Design +We train the model using the loss function: +L = w1Lvx + w2Llm +(4) +where Lvx is vertex position loss, Llm is mouth shape loss, +and w is the weight . +Vertex position loss: A 3D mesh will be transformed from +the predicted FLAME parameters, and the L1 difference be- +tween the converted vertices and ground truth vertices will be + +Happiness +Fear +0.50 +0.75 +1.0 +Fig. 3: Results of emotion animation from the same speech with various emotion classes and customized intensities. The texture is derived from the ground +truth video for visualization. Left: animation frames of happiness; Right: animation frames of fear. From top to bottom, the intensities are 1.0, 0.75, and 0.5. +calculated as Lvx. A vertex mask is applied to the mesh to +cover the front face area and exclude ears and eyes. +Mouth shape loss: To ensure lip synchronization, we select +the positions of the top (vt), bottom (vb), leftmost (vl), and +rightmost (vr) vertex and calculate the height of the mouth, +V = |vt − vb|, and the width of the mouth, H = |vl − vr|, +as the shape description. The L1 distance of the height and +width between ground truth and the predicted mouth shape is +used as the mouth shape loss. +Llm = d1 ∗ ||Hp − Hg||1 + d2 ∗ ||Vp − Vg||1. +(5) +We set d1 = 1/0.0476 and d2 = 1/0.017 in our experiment. +3.4. Implementation details +We train the model end to end with the Adam optimizer, and +the learning rate decays from 1e−4 to 1e−5 for every ten +epochs. A low-pass filtering step is applied to the output an- +imation sequence to restrain high-frequency noise. We also +refine the predicted FLAME parameters to ensure the bilat- +eral symmetry of the animated face. Please refer to our sup- +plementary for more implementation details. +Fig. 4: Results of EMOCA [24] face geometry reconstruction. +4. EXPERIMENTS +Data setups: +Manually-labelled 3D datasets with a rich +emotion diversity are limited. +Our model is trained us- +ing a mixture of 3D and 2D datasets that contain emotion- +rich speeches, videos, or 3D model pairs, including VO- +CASET [5](3D) and CREMAD[25](2D). For the 2D CRE- +MAD dataset, EMOCA [24] method was used to reconstruct +3D facial models from 2D images (see Figure 4 the results) +and extract FLAME parameters as the ground truth. +4.1. Emotion animation control +Conditioning on users’ control to realize 3D emotional facial +animation is one of the key contributions of our method. Fig- +ure 3 shows controllable facial animation results on happi- +ness and fear emotions with intensities of 0.5, 0.75, and 1.0, +respectively. All the examples are driven by the same audio +input. The results demonstrate that driven by the same audio +input, our model can generate diverse emotional facial ani- +mation effects that reveal users’ emotion customization. The +generated facial expressions were influenced by both users’ +alteration and audio input. +Specifically, with the same audio input, the mouth shows +a lift in fear rather than a drop in happy emotion. A con- +tinuous emotion magnitude change can also lead to satis- +fied expression variations. +For example, in the fear emo- +tion animation, the mouth opens wider, and the mouth cor- +ners move downwards with emotion intensity increasing from +0.5 to 1.0. Similar to happiness, in which the mouth corners +and cheeks raise harder for the intensity of 1.0 while drop- +ping to neutrally closed for the intensity of 0.5. With our +emotion-controllable approach, users can edit the animation +in any keyframe by specifying desired intensity values with- +out preparing any other dependencies, which is more straight- + +AN +Ty +y +YAFig. 5: Qualitative state-of-the-art comparisons in angry animation. +The +sentence ”Dogs are sitting by the door” does not come up in our training +dataset. Different animation frames were selected from the sentence’s begin- +ning (top), intermediate (middle), and end (bottom) phases. See the submit- +ted video for more visualizations. +forward and efficient than traditional methods like [20]. +4.2. Comparisons +We compare our method with the state-of-the-art speech- +driven facial animation methods to show how our method +performs in both emotion synthesis and lip synchroniza- +tion. We chose the works by Pham et al. [21] and Chun et +al. [20] for the emotion synthesis comparison, and VOCA [5] +and FaceFormer [2] for the lip synchronization comparision +(each denoted as [Pham17], [Chun21], [VOCA19] and [Face- +Former22] respectively). +Emotion synthesis. Figure 5 shows an example of facial an- +imation with angry emotion. We observed that [VOCA19] +and [FaceFormer22] show accurate lip movement but neu- +tral expressions. +While our method can put on an extra +layer of emotion variations without sacrificing lip synchro- +nization and comprehensibility of the spoken content. In our +results, eyebrows drop, cheek contraction appears, and the +degree of mouth opening/closing varies according to mood +peaks and troughs. [Pham17] can bring in some emotional +expressions, but the lip movement is not as accurate as ours, +and the dynamics in the upper face are pretty limited (nearly +still). [Chun21] provided emotional expressions with precise +lip synchronization, but since the emotion template was given +manually and applied uniformly to all frames, the generated +animation lacks emotion dynamics alongside the speaking +process, especially in the upper face region (see the begin- +ning and end frames). In contrast, ours have more emotional +swings along the animation that look natural as real humans +expression. In addition, note that our method is capable of +transferring emotions from one identity to another, benefit- +ing from the FLAME method’s disentanglement of shape and +expression, which [Chun21] and [Pham17] are not. +Lip synchronization. It’s essential for speech-driven anima- +tion to have accurate lip movements in conveying content in- +formation. For a fair comparison in terms of lip synchroniza- +tion, we used the FLAME parameters from Audio2FLAME +to compare with other prior works quantitatively. The test- +ing data keeps the same as what was used by FaceFormer [2], +which includes two subjects’ data from VOCASET, each con- +taining 20 sentence samples. +To avoid introducing additional alignment errors among +different meshes output by different methods, we perform a +linear 3D transformation before metrics calculations to make +them comparable. After alignment, we uniformly selected 24 +key-point vertices around the lip and calculated their distance +with ground truth as the measurement of lip synchronization. +The lip movement errors of a key-point vertex k in frame j of +sequence i are calculated by: +Di,j,k = +� +(xk− ˆxk)+(yk− ˆyk)+(zk− ˆzk) +(6) +The overall mean and maximal distance are calculated as two +metrics. +Table 1 shows that our method achieved the best results +compared with all the other methods, demonstrating that our +proposed pipeline is capable of generating decent lip synchro- +nization for emotion enhancement. The RandInit model cre- +ated from randomized initialization without training steps has +the lowest accuracy, as expected. It serves as a reference to +show the improvement brought by different training methods. +Table 1: Comparisons on lip movement error (mm). +RandInit +[VOCA19] +[FaceFormer22] +Ours +Mean↓ +2.31 +1.94 +1.97 +1.92 +Max↓ +3.87 +3.41 +3.33 +3.24 +4.3. Ablation Study +We ablate the loss items, network components, and training +datasets to see their contributions. Table 2 summarizes the +effects of our proposed model learned without vertex posi- +tion loss, without mouth shape loss, and without style vec- +tor extracted from the transformer encoder. We observed that +both vertex position loss and mouth shape loss contribute to +lip movement accuracy. Removing any of them caused an +increase in error metrics. Style vectors also facilitate the net- +work to capture long-contextual features from the audio input +and improve lip synchronization. + +VOCA19] +[Pham17] +[Chun21] +Ours +Ours(w/o ECM) +[Faceformer22]Table 2: Ablation study measured by lip movement error. +Mean(mm)↓ +Maximum(mm)↓ +Ours +1.92 +3.24 +w/o Lvx loss +2.01 +3.39 +w/o Llm loss +2.02 +3.34 +w/o style vector +2.03 +3.38 +VOCASET-only +1.97 +3.30 +w/ LRS2 +1.99 +3.34 +We also tried different training datasets to investigate the +benefits of proper training data. +Only using the 3D VO- +CASET dataset cuts down lip movement accuracy. We also +investigated the LRS2 [26] dataset. 3000 portrait video clips +from BBC television that contain various subjects, back- +ground noise, and environments are selected for training. Al- +though the result shows no improvement in lip movement +accuracy on the experimental dataset, we observe that it im- +proves our model’s generalization on in-the-wild testing. +We also compared the network structure with and with- +out the emotion control module qualitatively. +Figure 5 +shows the results based on the FLAME parameters from +Auido2FLAME (denoted as Ours (w/o ECM)) and the emo- +tion control module (denoted as Ours). +We can see that +the emotion control module can improve the expressivity of +the animation and realize more drastic emotional expression, +compared with the neutral animation generated without it. +5. CONCLUSION AND FUTURE WORK +We presented a novel deep learning-based approach to gener- +ate controllable speech-driven emotional facial animation. An +emotion controller module is proposed to enrich emotion ex- +pressivity and enables animator customization on emotion in- +tensity and classes. An image-based emotion recognition was +used to generate emotion priors to facilitate explicit emotion +learning. Future works could consider pushing the boundary +of extreme emotion generation with accurate lip synchroniza- +tion and improving the animation generation by upgrading the +temporal performance of video-based emotion recognition. +6. 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Graph., vol. 36, no. 6, pp. 194–1, 2017. +[23] Zhengyao Wen, Wenzhong Lin, Tao Wang, and Ge Xu, +“Distract +your attention: Multi-head cross attention network for facial expres- +sion recognition,” arXiv preprint arXiv:2109.07270, 2021. + +[24] Radek Danˇeˇcek, Michael J Black, and Timo Bolkart, “Emoca: Emotion +driven monocular face capture and animation,” in CVPR, 2022, pp. +20311–20322. +[25] Houwei Cao, David G Cooper, Michael K Keutmann, Ruben C Gur, +Ani Nenkova, and Ragini Verma, “Crema-d: Crowd-sourced emotional +multimodal actors dataset,” IEEE transactions on affective computing, +vol. 5, no. 4, pp. 377–390, 2014. +[26] Triantafyllos Afouras, Joon Son Chung, Andrew Senior, Oriol Vinyals, +and Andrew Zisserman, “Deep audio-visual speech recognition,” IEEE +transactions on pattern analysis and machine intelligence, 2018. +7. SUPPLEMENTARY MATERIALS: EXPRESSIVE +SPEECH-DRIVEN FACIAL ANIMATION WITH +CONTROLLABLE EMOTIONS +7.1. FLAME +FLAME is a statistical 3D head model that is differentiable +and disentangles expression and shape. The model includes +identity shape parameters ⃗β, pose parameters ⃗θ, and facial ex- +pression parameters ⃗φ. With all the parameters set, FLAME +generates a mesh with a set of vertices (nv = 5023) and faces +(nv = 9976). In FLAME, the shape and expression are dis- +entangled by learning the difference between the neutral and +expressive faces of the same person. Some state-of-the-art +face reconstruction methods, including DECA and the follow- +up EMOCA, used FLAME as the face model and recovered +high-detailed and expressive 3D facial expressions from sin- +gle monocular images. Our work also uses FLAME as the 3D +facial model. +7.2. Emotion Logits as emotion priors in training +In this work, we use emotional logits generated from the +layer before softmax in a facial recognition network as the +emotion priors for model training. Here we give a simpli- +fied conceptual proof that the emotion logits have a linear +relation with the difference in perceptual emotion strength +of various emotional states. Given a binary (emotion state +A and B) classification as an example, we define ak and +bk as the strength of emotion A and B of sample Sk and +ηa,b +k += ak − bk is their difference. The possibility of sam- +ple Sk to be emotion A is P(Sk = A|ηa,b +k ). We assume that +all classes are equally distributed and continuously differen- +tiable. Then, the prior possibility for each class is the same, +i.e., P(Sk = A) = P(Sk = B). Therefore, according to +Bayes theorem, we get +P(Sk =A|ηa,b +k )= +P(ηa,b +k |Sk = A) +P(ηa,b +k |Sk = A)+P(ηa,b +k |Sk = B) +. (7) +We define the joint distribution of (ak, bk) given Sk = A +is faˆb(η), and the joint distribution of (ak, bk) given Sk = B +is fˆab(η). And we assume they both follow a normal distribu- +tion N(µ, σ2), then +P(ηa,b +k |Sk = A) = faˆb(ηk) = +� +C1 +faˆb(ak, bk)ds +(8) +where C1 is ak − bk = ηk. With symmetric distribution as- +sumption, we got that: +P(ηa,b +k |Sk =B)= +� +C1 +fˆab(ak, bk)ds= +� +C2 +faˆb(ak, bk)ds +(9) +where C2 is bk − ak = ηk, which can be converted to ak − +bk = −ηk. Thus: +P(ηa,b +k |Sk = B) = faˆb(−ηk). +(10) +Considering the softmax projection based on the emo- +tion logits Zak and Zbk in the emotion recognition network +is P(Sk = A|ηa,b +k ) = +exp(Zak) +� +j exp(Zbk), the deviation of emotion +logits: +∆z,k = ln( P(Sk =A|ηa,b +k ) +P(Sk =B|ηa,b +k ) +) = ln(P(ηa,b +k |Sk = A) +P(ηa,b +k |Sk = B) +). (11) +According to formula 8, 9, and 10, we derive that: +∆z,k =ln( faˆb(ηk) +faˆb(−ηk))= (ηk+µ)2−(ηk−µ)2 +2σ2 += 2µ +σ2 ηk +(12) +which shows the deviation of emotion logits ∆z,k has linear +relation with the deviation of perceptual emotion strength ηk. +The deducing process can be spread to multi-class classifica- +tion tasks similarly. +7.3. More implementation details +The audio is converted to 16000Hz and is clipped into seg- +ments using a sliding window with 100ms windows-length +and 33ms stride. Each audio segment and its left and right +neighbors (∆t = 1) are used as the audio input of the net- +work. The video is downsampled to 30 fps and applied with +a portrait cropping to adapt them for 3D facial model recon- +struction. Wav2vec2 and EMOCA use the model provided by +the original paper to extract audio features and reconstruct 3D +facial models. Both of them are kept frozen during training. +For global style vector extraction, we use a transformer +encoder that contains four transformer encoder blocks com- +posed of a self-attention layer followed by an addition and +normalization layer. The 64-dimension vector output from the +last token of the final attention layer is used as the style vector. +As a three-layer CNN, Audio2FLAME has a (256, 128, 1, 1, +ReLU), (128, 128, 1, 3, ReLU), and (128, 56, 1, 1) setup for +input and output channels, kernel size of 1-dimension convo- +lution, stride, and activation functions of each layer. + +The emotion predictor comprises one bi-directional +LSTM layer with a 128 hidden size. +The predicted emo- +tion priors will be re-scaled with max-min normalization, us- +ing minimum and maximum values obtained from statistics +of emotion logits extracted from the training video dataset. +Emotion-augment network has the same structure as Au- +dio2FLAME, except a layer normalization is applied before +the first convolutional layer. With the concatenation of 128- +dimensional emotion feature vectors and 56-dimensional raw +FLAME parameters from Audio2FLAME as the input (186- +dimensions in total), it outputs 56-dimensional enhanced +FLAME parameters. +In the training phase, the seed in PyTorch and NumPy +randomization is set to 1000. More details can be found in +our released code. + diff --git a/RtA0T4oBgHgl3EQfDv-e/content/tmp_files/load_file.txt b/RtA0T4oBgHgl3EQfDv-e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b9f7599ea2dfcc87bb1cc909ac9a9e2ea1b88ce --- /dev/null +++ b/RtA0T4oBgHgl3EQfDv-e/content/tmp_files/load_file.txt @@ -0,0 +1,377 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf,len=376 +page_content='EXPRESSIVE SPEECH-DRIVEN FACIAL ANIMATION WITH CONTROLLABLE EMOTIONS Yutong Chen∗, Junhong Zhao† and Wei-Qiang Zhang‡ ∗Tsinghua University, chenyt19@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='cn †Victoria University of Wellington, New Zealand, junhong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='jennifer@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='com ‡Tsinghua University, wqzhang@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='cn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' ABSTRACT It is in high demand to generate facial animation with high re- alism, but it remains a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Existing approaches of speech-driven facial animation can produce satisfactory mouth movement and lip synchronization, but show weak- ness in dramatic emotional expressions and flexibility in emo- tion control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' This paper presents a novel deep learning- based approach for expressive facial animation generation from speech that can exhibit wide-spectrum facial expres- sions with controllable emotion type and intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We pro- pose an emotion controller module to learn the relationship between the emotion variations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=', types and intensity) and the corresponding facial expression parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' It enables emotion-controllable facial animation, where the target ex- pression can be continuously adjusted as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The qual- itative and quantitative evaluations show that the animation generated by our method is rich in facial emotional expres- siveness while retaining accurate lip movement, outperform- ing other state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Index Terms— 3D facial animation, 3D avatar, talking head, emotion, expressive facial animation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' INTRODUCTION Facial animation is a growing research topic that has been widely adopted in many applications, such as education, Vir- tual Reality (VR), and digital entertainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Commercial products often require high plausibility and expressiveness in the animated characters to meet users’ needs for immer- sive engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Creating such realistic facial animation is a great challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Many recent works have resorted to deep learning to automate face animating based on easy-acquire data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Speech-driven facial animation is one critical compo- nent in this field and has drawn much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' It is to em- ulate life-like facial motion based on the information carried by a vocal audio track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Our work focuses on speech-driven facial animation in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Compared with animating 2D im- ages, modulating a 3D model is directly applicable to most 3D applications like 3D games and visual aftereffects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' At- tempts have been made to explore the dependencies between audio and 3D face movement, most of which focus on the lower face and lip movements and their synchronization with speech [1, 2, 3, 4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Although they can produce plausible basic facial motions, they are far from satisfactory in synthe- sizing facial expressions, especially in presenting emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' To realize emotional animation based on only an audio track is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Although the dependency between sound production and lip movement for the same person is deterministic, its dependencies with the expressions of differ- ent emotion categories and intensities are highly ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' There are many different personalized conveying ways of one emotion for different users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Such inherent ambiguity makes neural networks hard to handle emotion variations based on the audio input, even given long-contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The works by Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' and Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' [7, 8] tried to extract emotion features from speech and implicitly embed them into their neural network to realize emotion synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' However, their solution inevitably suffers from over-smoothed regres- sion due to the limited training data, and often results in lim- ited expressiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' In practice, we found that even given the best expressive speech, the animations generated by existing methods still fail to achieve drastic emotion dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We present a novel approach to realize emotion- controllable facial animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Instead of recovering lip move- ment from audio, our method enriches the emotion expres- sivity and enables the adjustment of the intensity of emo- tion effects to satisfy the animators’ needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We propose an emotion controller module, which includes an emotion pre- dictor followed by an emotion augment network, to explicitly model the relationship between emotion variations and corre- sponding facial expression parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Image-based emotion recognition was used to generate emotion information as pri- ors to guide the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' During inference, the spec- ified emotion condition will apply to the speech-driven facial animation to realize emotion enhancement and customization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' By explicitly modelling emotion, our animator-friendly sys- tem enables emotion control with a given emotion type and an intensity value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Our method shows promising results in synthesizing controllable emotional facial animation while re- taining high-accuracy lip synchronization, outperforming the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='02008v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='CV] 5 Jan 2023 state-of-the-arts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The implementation code will be publicly available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' RELATED WORK Despite much work focusing on facial animation from image or video [9, 10, 11, 12] or generate speech-driven 2D talk- ing head [13, 14, 15, 16, 17], we concentrate our efforts on speech-driven 3D facial animation, mainly targeting to im- prove the emotional expression synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We review the most relevant deep learning-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Speech-driven 3D models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Earlier speech-driven 3D fa- cial animation methods are based on phonetic annotation and viseme-model blending [1, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' VisemeNet [1] lever- aged LSTMs for phoneme grouping and facial landmark pre- diction, and used their results to regress viseme parameters for lip animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' VOCA [5] tried to encode the identity- dependent information into animation and synthesized vari- ous speaking styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' MeshTalk [6] proposed a method to dis- entangle audio-correlated and audio-uncorrelated information to generate more plausible dynamics on the upper face while attaining accurate lip motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' FaceFormer [2] proposed a transformer-based autoregressive model to encode long-term audio context and synthesize improved lip motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' [3] proposed a sliding-window regression method to pre- dict the active appearance model (AAM) parameters of a ref- erence lower face based on phoneme labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' [19] considered the influence of geometry representation and uti- lize them to produce generalized speaker-independent facial animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Although considerable progress has been achieved in the field, most of these prior works focused on lip mo- tion accuracy without considering expressiveness, which lim- its the realism of their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Conditional emotion synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Some recent facial anima- tion methods [15, 7, 8, 20] tackled some issues in emotion synthesis to make more vivid facial animations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' used LSTMs in [8] (improved in [21] with CNN-RNN) to model the long-contextual relationship between acoustic features and facial expressions to realize emotion awareness in the generated animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The method proposed by Kar- ras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' [7] extracts a latent emotion representation from the audio without identifying emotion categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' In both methods, emotions are implicitly represented and thus lack meaningful guidance to emotion intensity control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' [15] decomposed speech into emotion and content com- ponents to generate emotion-controlled 2D talking heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Chun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' [20] introduced an emotion-guided method where emotion-expressive blendshapes are enhanced by emotion recognition guidance and then fused with mouth-expressive blendshapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Nevertheless, the technique needs manual effort to prepare each emotion template, and the generated expres- sions often lack upper-face dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' METHOD 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Network Architecture Design Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' illustrates our pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Our goal is to enrich emo- tion expressivity in speech-driven facial animation and enable users’ control over emotion variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We first design a neu- ral network to estimate the facial movement represented by FLAME parameters [22] (See supplementary) from the input audio, using both local and long-contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The problem can be formulated as follows: (⃗φt, ⃗θt) = F(S(⃗xt±∆t) → ⃗ψt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' W(⃗xt±∆t) → ⃗ωt) (1) Given the audio segment xt at time t and its neighboring frames, the local content features (content vector ⃗ωt) and global style features (style vector ⃗ψt) learned by wav2vec2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='0 W(·) and a transformer encoder S(·) are first extracted sep- arately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Then they are concatenated together to predict FLAME parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The mapping between the audio and the FLAME parameters is learned by an Auido2FLAME mod- ule F(·), which is a multi-layer CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The predicted FLAME parameters, including expression parameters ⃗φt and pose pa- rameters ⃗θt, combined with the shape parameters of the given identity, are converted to 3D mesh as the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The lip syn- chronization will be primarily focused on and preciseness in mouth motion will be ensured at this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We introduce an emotion control module that includes a bi-LSTM-based emotion predictor followed by an embedding layer to generate emotion-related latent features (emotion fea- ture vectors), and a CNN-based emotion-augment network to enhance the expressivity of FLAME parameters based on emotion features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The emotion augmentation process can be represented as: E(⃗ψt±∆t, ⃗ωt±∆t, γu,t) : (⃗φt, ⃗θt) → (⃗φ′ t, ⃗θ′ t) (2) Where E(·) denotes the emotion control module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' With the audio features and emotion conditions γu,t customized by the user, it maps (⃗φt, ⃗θt) predicted by the Audio2FLAME model to the emotion-enhanced facial parameters (φ′ t, θ′ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We in- corporate the emotion-augment network in a residual man- ner in our pipeline, which allows dedicated optimization on emotion-related expressions while retaining content-related expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Using this way, the user can explicitly design the emotion intensities and categories at the frame level and regulate the animation output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Emotion Control Module The core challenge to realizing full control of emotion sim- ulation is to make the model adaptive to not only emotional state changes, but also emotion strength variations to allow straightforward intensity adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Our proposed training and inference pipeline is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Audio Emotion Condition (Animator-Controlled) FLAME Parameters Style vector Content vector wav2vec 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='0 Transformer Audio2FLAME Pose Shape Expression Emotion Predictor Emotion-augment network Emotion Control Module Enhanced FLAME Parameters Pose Shape Expression Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 1: An overview of our pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Emotion prediction and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' In the training phase, to make the network see emotion variations in the input, we leverage the image-based emotion recognition model to ob- tain frame-level emotion information as emotion priors to fa- cilitate the model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' DAN model from Weng et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' [23] was used in our experiments, which could be substituted by other similar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We assume emotion features decoded from 2D visual images are more reliable and informative than that from audio, which is beneficial for the model learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' However, the emotion classification probabilities from DAN can not indicate the magnitude of emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Instead, we found that the emotion logits before the final softmax layer of the emotion recognition network, featuring seven- dimensional vector for seven emotions including happiness, anger and etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=', are highly in agreement with the perceived emotion intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Therefore, we use them as emotion priors for model training and hinge them with users’ emotion con- trol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' See supplementary material in detail about our simpli- fied conceptual proof of the linear relation between emotion logits and emotion intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' In our experiment, we found that the emotion logits are effective in emotion synthesis and work well with the adjustment of both emotion categories and intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The emotion augment network is resilient to the prediction error of the emotion priors from the DAN module, and can produce general emotional expressions from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' In the inference phase, emotion priors are extracted from audio through a bi-LSTM network, and altered by the cus- tomized emotion category and intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The bi-LSTM net- work was trained by maximizing the mutual information be- tween audio-based and video-based emotion priors, using video-based emotion priors as a pseudo ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The emotion conditions given by the user will be transformed to a one-hot vector with values ranging from 0 to 1 (γu,t) and added to the decentralized audio-based emotion priors (segment-level mean normalization (γa) ), bringing in the fi- nal emotion priors γt = γu,t + (γa,t − γa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Emotion augment network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Before inputting into the emo- tion augment network, the emotion priors will be transformed into emotion feature vectors through an embedding layer to drive 3D facial expression augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The embedding Audio Enhanced FLAME parameters Audio Video DAN Train Emotion-augment network Inference Embedding layer Emotion Condition (Animator-Controlled) Emotion Predictor Raw FLAME parameters Emotion priors Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 2: Training/inference pipeline of our proposed emotion control module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' layer is a learnable 2D matrix that will be optimized dur- ing training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The 7-dimensional priors vector will be trans- formed to a size of 128 emotion feature vectors by multiply- ing with the learned embedding matrix, and then input into the emotion augment network built with CNN blocks to real- ize emotion-guided facial expression enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' To attain a proper balance between lip synchronization and emotion ex- pressivity, we added the emotion augment network on a resid- ual basis (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The raw facial parameters extracted from Audio2FLAME are expected to have a preferable lip synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The gap between the raw facial parameters and ground truth ( ⃗ φgt,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' ⃗ θgt,t) is mainly caused by their differ- ent emotional expressions, making the augment network A(·) more effective to learn on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' A(⃗φt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' ⃗θt) + (⃗φt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' ⃗θt) ∼= ( ⃗ φgt,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' ⃗ θgt,t) (3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Loss Design We train the model using the loss function: L = w1Lvx + w2Llm (4) where Lvx is vertex position loss, Llm is mouth shape loss, and w is the weight .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Vertex position loss: A 3D mesh will be transformed from the predicted FLAME parameters, and the L1 difference be- tween the converted vertices and ground truth vertices will be Happiness Fear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 3: Results of emotion animation from the same speech with various emotion classes and customized intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The texture is derived from the ground truth video for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Left: animation frames of happiness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Right: animation frames of fear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' From top to bottom, the intensities are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='75, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' calculated as Lvx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' A vertex mask is applied to the mesh to cover the front face area and exclude ears and eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Mouth shape loss: To ensure lip synchronization, we select the positions of the top (vt), bottom (vb), leftmost (vl), and rightmost (vr) vertex and calculate the height of the mouth, V = |vt − vb|, and the width of the mouth, H = |vl − vr|, as the shape description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The L1 distance of the height and width between ground truth and the predicted mouth shape is used as the mouth shape loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Llm = d1 ∗ ||Hp − Hg||1 + d2 ∗ ||Vp − Vg||1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' (5) We set d1 = 1/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='0476 and d2 = 1/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='017 in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Implementation details We train the model end to end with the Adam optimizer, and the learning rate decays from 1e−4 to 1e−5 for every ten epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' A low-pass filtering step is applied to the output an- imation sequence to restrain high-frequency noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We also refine the predicted FLAME parameters to ensure the bilat- eral symmetry of the animated face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Please refer to our sup- plementary for more implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 4: Results of EMOCA [24] face geometry reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' EXPERIMENTS Data setups: Manually-labelled 3D datasets with a rich emotion diversity are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Our model is trained us- ing a mixture of 3D and 2D datasets that contain emotion- rich speeches, videos, or 3D model pairs, including VO- CASET [5](3D) and CREMAD[25](2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' For the 2D CRE- MAD dataset, EMOCA [24] method was used to reconstruct 3D facial models from 2D images (see Figure 4 the results) and extract FLAME parameters as the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Emotion animation control Conditioning on users’ control to realize 3D emotional facial animation is one of the key contributions of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Fig- ure 3 shows controllable facial animation results on happi- ness and fear emotions with intensities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='75, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' All the examples are driven by the same audio input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The results demonstrate that driven by the same audio input, our model can generate diverse emotional facial ani- mation effects that reveal users’ emotion customization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The generated facial expressions were influenced by both users’ alteration and audio input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Specifically, with the same audio input, the mouth shows a lift in fear rather than a drop in happy emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' A con- tinuous emotion magnitude change can also lead to satis- fied expression variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' For example, in the fear emo- tion animation, the mouth opens wider, and the mouth cor- ners move downwards with emotion intensity increasing from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='5 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Similar to happiness, in which the mouth corners and cheeks raise harder for the intensity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='0 while drop- ping to neutrally closed for the intensity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' With our emotion-controllable approach, users can edit the animation in any keyframe by specifying desired intensity values with- out preparing any other dependencies, which is more straight- AN Ty y YAFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 5: Qualitative state-of-the-art comparisons in angry animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The sentence ”Dogs are sitting by the door” does not come up in our training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Different animation frames were selected from the sentence’s begin- ning (top), intermediate (middle), and end (bottom) phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' See the submit- ted video for more visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' forward and efficient than traditional methods like [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Comparisons We compare our method with the state-of-the-art speech- driven facial animation methods to show how our method performs in both emotion synthesis and lip synchroniza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We chose the works by Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' [21] and Chun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' [20] for the emotion synthesis comparison, and VOCA [5] and FaceFormer [2] for the lip synchronization comparision (each denoted as [Pham17], [Chun21], [VOCA19] and [Face- Former22] respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Emotion synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Figure 5 shows an example of facial an- imation with angry emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We observed that [VOCA19] and [FaceFormer22] show accurate lip movement but neu- tral expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' While our method can put on an extra layer of emotion variations without sacrificing lip synchro- nization and comprehensibility of the spoken content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' In our results, eyebrows drop, cheek contraction appears, and the degree of mouth opening/closing varies according to mood peaks and troughs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' [Pham17] can bring in some emotional expressions, but the lip movement is not as accurate as ours, and the dynamics in the upper face are pretty limited (nearly still).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' [Chun21] provided emotional expressions with precise lip synchronization, but since the emotion template was given manually and applied uniformly to all frames, the generated animation lacks emotion dynamics alongside the speaking process, especially in the upper face region (see the begin- ning and end frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' In contrast, ours have more emotional swings along the animation that look natural as real humans expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' In addition, note that our method is capable of transferring emotions from one identity to another, benefit- ing from the FLAME method’s disentanglement of shape and expression, which [Chun21] and [Pham17] are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Lip synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' It’s essential for speech-driven anima- tion to have accurate lip movements in conveying content in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' For a fair comparison in terms of lip synchroniza- tion, we used the FLAME parameters from Audio2FLAME to compare with other prior works quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The test- ing data keeps the same as what was used by FaceFormer [2], which includes two subjects’ data from VOCASET, each con- taining 20 sentence samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' To avoid introducing additional alignment errors among different meshes output by different methods, we perform a linear 3D transformation before metrics calculations to make them comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' After alignment, we uniformly selected 24 key-point vertices around the lip and calculated their distance with ground truth as the measurement of lip synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The lip movement errors of a key-point vertex k in frame j of sequence i are calculated by: Di,j,k = � (xk− ˆxk)+(yk− ˆyk)+(zk− ˆzk) (6) The overall mean and maximal distance are calculated as two metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Table 1 shows that our method achieved the best results compared with all the other methods, demonstrating that our proposed pipeline is capable of generating decent lip synchro- nization for emotion enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The RandInit model cre- ated from randomized initialization without training steps has the lowest accuracy, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' It serves as a reference to show the improvement brought by different training methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Table 1: Comparisons on lip movement error (mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' RandInit [VOCA19] [FaceFormer22] Ours Mean↓ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='92 Max↓ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='41 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='33 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Ablation Study We ablate the loss items, network components, and training datasets to see their contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Table 2 summarizes the effects of our proposed model learned without vertex posi- tion loss, without mouth shape loss, and without style vec- tor extracted from the transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We observed that both vertex position loss and mouth shape loss contribute to lip movement accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Removing any of them caused an increase in error metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Style vectors also facilitate the net- work to capture long-contextual features from the audio input and improve lip synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' VOCA19] [Pham17] [Chun21] Ours Ours(w/o ECM) [Faceformer22]Table 2: Ablation study measured by lip movement error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Mean(mm)↓ Maximum(mm)↓ Ours 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='92 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='24 w/o Lvx loss 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='39 w/o Llm loss 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='34 w/o style vector 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='38 VOCASET-only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='97 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='30 w/ LRS2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='99 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='34 We also tried different training datasets to investigate the benefits of proper training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Only using the 3D VO- CASET dataset cuts down lip movement accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We also investigated the LRS2 [26] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 3000 portrait video clips from BBC television that contain various subjects, back- ground noise, and environments are selected for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Al- though the result shows no improvement in lip movement accuracy on the experimental dataset, we observe that it im- proves our model’s generalization on in-the-wild testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We also compared the network structure with and with- out the emotion control module qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Figure 5 shows the results based on the FLAME parameters from Auido2FLAME (denoted as Ours (w/o ECM)) and the emo- tion control module (denoted as Ours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We can see that the emotion control module can improve the expressivity of the animation and realize more drastic emotional expression, compared with the neutral animation generated without it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK We presented a novel deep learning-based approach to gener- ate controllable speech-driven emotional facial animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' An emotion controller module is proposed to enrich emotion ex- pressivity and enables animator customization on emotion in- tensity and classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' An image-based emotion recognition was used to generate emotion priors to facilitate explicit emotion learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Future works could consider pushing the boundary of extreme emotion generation with accurate lip synchroniza- tion and improving the animation generation by upgrading the temporal performance of video-based emotion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' REFERENCES [1] Yang Zhou, Zhan Xu, Chris Landreth, Evangelos Kalogerakis, Subhransu Maji, and Karan Singh, “Visemenet: Audio-driven 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Vinyals, and Andrew Zisserman, “Deep audio-visual speech recognition,” IEEE transactions on pattern analysis and machine intelligence, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' SUPPLEMENTARY MATERIALS: EXPRESSIVE SPEECH-DRIVEN FACIAL ANIMATION WITH CONTROLLABLE EMOTIONS 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' FLAME FLAME is a statistical 3D head model that is differentiable and disentangles expression and shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The model includes identity shape parameters ⃗β, pose parameters ⃗θ, and facial ex- pression parameters ⃗φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' With all the parameters set, FLAME generates a mesh with a set of vertices (nv = 5023) and faces (nv = 9976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' In FLAME, the shape and expression are dis- entangled by learning the difference between the neutral and expressive faces of the same person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Some state-of-the-art face reconstruction methods, including DECA and the follow- up EMOCA, used FLAME as the face model and recovered high-detailed and expressive 3D facial expressions from sin- gle monocular images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Our work also uses FLAME as the 3D facial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Emotion Logits as emotion priors in training In this work, we use emotional logits generated from the layer before softmax in a facial recognition network as the emotion priors for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Here we give a simpli- fied conceptual proof that the emotion logits have a linear relation with the difference in perceptual emotion strength of various emotional states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Given a binary (emotion state A and B) classification as an example, we define ak and bk as the strength of emotion A and B of sample Sk and ηa,b k = ak − bk is their difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The possibility of sam- ple Sk to be emotion A is P(Sk = A|ηa,b k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' We assume that all classes are equally distributed and continuously differen- tiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Then, the prior possibility for each class is the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=', P(Sk = A) = P(Sk = B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Therefore, according to Bayes theorem, we get P(Sk =A|ηa,b k )= P(ηa,b k |Sk = A) P(ηa,b k |Sk = A)+P(ηa,b k |Sk = B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' (7) We define the joint distribution of (ak, bk) given Sk = A is faˆb(η), and the joint distribution of (ak, bk) given Sk = B is fˆab(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' And we assume they both follow a normal distribu- tion N(µ, σ2), then P(ηa,b k |Sk = A) = faˆb(ηk) = � C1 faˆb(ak, bk)ds (8) where C1 is ak − bk = ηk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' With symmetric distribution as- sumption, we got that: P(ηa,b k |Sk =B)= � C1 fˆab(ak, bk)ds= � C2 faˆb(ak, bk)ds (9) where C2 is bk − ak = ηk, which can be converted to ak − bk = −ηk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Thus: P(ηa,b k |Sk = B) = faˆb(−ηk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' (10) Considering the softmax projection based on the emo- tion logits Zak and Zbk in the emotion recognition network is P(Sk = A|ηa,b k ) = exp(Zak) � j exp(Zbk), the deviation of emotion logits: ∆z,k = ln( P(Sk =A|ηa,b k ) P(Sk =B|ηa,b k ) ) = ln(P(ηa,b k |Sk = A) P(ηa,b k |Sk = B) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' (11) According to formula 8, 9, and 10, we derive that: ∆z,k =ln( faˆb(ηk) faˆb(−ηk))= (ηk+µ)2−(ηk−µ)2 2σ2 = 2µ σ2 ηk (12) which shows the deviation of emotion logits ∆z,k has linear relation with the deviation of perceptual emotion strength ηk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The deducing process can be spread to multi-class classifica- tion tasks similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' More implementation details The audio is converted to 16000Hz and is clipped into seg- ments using a sliding window with 100ms windows-length and 33ms stride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Each audio segment and its left and right neighbors (∆t = 1) are used as the audio input of the net- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The video is downsampled to 30 fps and applied with a portrait cropping to adapt them for 3D facial model recon- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Wav2vec2 and EMOCA use the model provided by the original paper to extract audio features and reconstruct 3D facial models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Both of them are kept frozen during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' For global style vector extraction, we use a transformer encoder that contains four transformer encoder blocks com- posed of a self-attention layer followed by an addition and normalization layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The 64-dimension vector output from the last token of the final attention layer is used as the style vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' As a three-layer CNN, Audio2FLAME has a (256, 128, 1, 1, ReLU), (128, 128, 1, 3, ReLU), and (128, 56, 1, 1) setup for input and output channels, kernel size of 1-dimension convo- lution, stride, and activation functions of each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The emotion predictor comprises one bi-directional LSTM layer with a 128 hidden size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' The predicted emo- tion priors will be re-scaled with max-min normalization, us- ing minimum and maximum values obtained from statistics of emotion logits extracted from the training video dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' Emotion-augment network has the same structure as Au- dio2FLAME, except a layer normalization is applied before the first convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' With the concatenation of 128- dimensional emotion feature vectors and 56-dimensional raw FLAME parameters from Audio2FLAME as the input (186- dimensions in total), it outputs 56-dimensional enhanced FLAME parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' In the training phase, the seed in PyTorch and NumPy randomization is set to 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} +page_content=' More details can be found in our released code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtA0T4oBgHgl3EQfDv-e/content/2301.02008v1.pdf'} diff --git a/T9FLT4oBgHgl3EQfQi86/content/tmp_files/2301.12033v1.pdf.txt b/T9FLT4oBgHgl3EQfQi86/content/tmp_files/2301.12033v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b737e51b728eb5b4aa158c664fb7d70e92067dd2 --- /dev/null +++ b/T9FLT4oBgHgl3EQfQi86/content/tmp_files/2301.12033v1.pdf.txt @@ -0,0 +1,3211 @@ +Norm-based Generalization Bounds +for Compositionally Sparse Neural Networks +Tomer Galanti 1 Mengjia Xu 1 2 Liane Galanti 3 Tomaso Poggio 1 +Abstract +In this paper, we investigate the Rademacher com- +plexity of deep sparse neural networks, where +each neuron receives a small number of inputs. +We prove generalization bounds for multilayered +sparse ReLU neural networks, including convo- +lutional neural networks. These bounds differ +from previous ones, as they consider the norms +of the convolutional filters instead of the norms +of the associated Toeplitz matrices, independently +of weight sharing between neurons. +As we show theoretically, these bounds may be +orders of magnitude better than standard norm- +based generalization bounds and empirically, they +are almost non-vacuous in estimating general- +ization in various simple classification problems. +Taken together, these results suggest that compo- +sitional sparsity of the underlying target function +is critical to the success of deep neural networks. +1. Introduction +Over the last decade, deep learning with large neural net- +works has greatly advanced the solution of a wide range of +tasks including image classification (He et al., 2016; Doso- +vitskiy et al., 2021; Zhai et al., 2021), language process- +ing (Vaswani et al., 2017; Devlin et al., 2019; Brown et al., +2020), interacting with open-ended environments (Silver +et al., 2016; Arulkumaran et al., 2019), and code synthe- +sis (Chen et al., 2021). Despite traditional theories (Vapnik, +1998), recent findings (Zhang et al., 2017; Belkin, 2021) +show that deep neural networks can generalize well even +when their size far exceeds the number of training samples. +To address this question, recent efforts in deep learning the- +ory study the generalization performance of deep networks +by analyzing the complexity of the learned function. +*Equal contribution +1Massachusetts Institute of Technol- +ogy +2Brown University +3Tel-Aviv University. +Correspon- +dence to: Tomer Galanti , Tomaso Poggio +. +Preprint +Recent work has suggested generalization guarantees for +deep neural networks based on various norms of their weight +matrices (Neyshabur et al., 2015; Golowich et al., 2017; +Bartlett & Mendelson, 2001; Harvey et al., 2017; Bartlett +et al., 2017; Neyshabur et al., 2018; Cao & Gu, 2019; +Daniely & Granot, 2019; Wei & Ma, 2019; Allen-Zhu et al., +2019; Li et al., 2018). Many efforts have been made to +improve the applicability of these bounds to realistic scales. +Some studies have focused on developing norm-based gen- +eralization bounds for complex network architectures, such +as residual networks (He et al., 2019). Other studies investi- +gated ways to reduce the dependence of the bounds on the +product of spectral norms (Wei & Ma, 2019; Nagarajan & +Kolter, 2019), or to use compression bounds based on PAC- +Bayes theory (Zhou et al., 2019; Lotfi et al., 2022), or on +the optimization procedure used to train the networks (Cao +& Gu, 2019; Arora et al., 2019; Richards & Kuzborskij, +2021). However, most of these studies have focused on +fully-connected networks which empirically have lower per- +formance compared to other architectures. In particular, +these studies cannot directly explain the success of current +successful architectures (LeCun et al., 1998; Vaswani et al., +2017; Dosovitskiy et al., 2020). +To fully understand the success of deep learning, it is neces- +sary to analyze a wider scope of architectures beyond fully- +connected networks. An interesting recent direction (Ledent +et al., 2021; Long & Sedghi, 2020) suggests better general- +ization bounds for neural networks with shared parameters, +such as convolutional neural networks. In fact, (Ledent +et al., 2021) was the first to show that convolutional layers +contribute to generalization bounds with a norm component +smaller than the norm of the associated linear transformation. +However, many questions remain unanswered, including (a) +Why certain compositionally sparse architectures, such as +convolutional networks, perform better than fully-connected +architectures? (b) Is weight sharing necessary for the suc- +cess of convolutional neural networks? In this paper, we +contribute to an understanding of both of these questions. +1.1. Related Work +Approximation guarantees for multilayer sparse net- +works. +While fully-connected networks, including shal- +arXiv:2301.12033v1 [cs.LG] 28 Jan 2023 + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +low networks, are universal approximators (Cybenko, 1989; +Hornik, 1991) of continuous functions, they are largely lim- +ited in theory and in practice. Classic results (Mhaskar, +1996; Maiorov & Pinkus, 1999; Maiorov et al., 1999; +Maiorov, 1999; Hanin & Sellke, 2017) show that, in the +worst-case, approximating r-continuously differentiable +target functions (with bounded derivatives) using fully- +connected networks requires Θ(ϵ−d/r) parameters, where d +is the input dimension and ϵ is the approximation rate. The +exponential dependence on d is also known as the “curse of +dimensionality”. +A recent line of work (Mhaskar et al., 2017; Poggio et al., +2020; Poggio, 2022) shows that the curse of dimensionality +can be avoided by deep, sparse networks, when the target +function is itself compositionally sparse. Furthermore, it +has been conjectured that efficiently computable functions, +that is functions that are computable by a Turing machine +in polynomial time, are compositionally sparse. This sug- +gests, in turns, that, for practical functions, deep and sparse +networks can avoid the curse of dimensionality. +These results, however, lack any implication about general- +ization; in particular, they do not show that overparametrized +sparse networks have good generalization properties. +Norm-based generalization bounds. +A recent thread in +the literature (Neyshabur et al., 2015; Golowich et al., 2017; +Bartlett & Mendelson, 2001; Harvey et al., 2017; Bartlett +et al., 2017; Neyshabur et al., 2018; Cao & Gu, 2019; +Daniely & Granot, 2019; Wei & Ma, 2019) has introduced +norm-based generalization bounds for neural networks. In +particular, let S = {(xi, yi)}m +i=1 be a training dataset of m +independently drawn samples from a probability measure +P defined on the sample space X × Y, where X ⊂ Rd and +Y = {±1}. A fully-connected network is defined as +fw(x) = W Lσ(W L−1σ(. . . σ(W 2σ(W 1x)) . . . )), (1) +where L is the depth of the network, W l ∈ Rdl+1×dl +and σ(x) is the element-wise ReLU activation function +max(0, x). +A common approach for estimating the +gap between the train and test errors of a neural net- +work is to use the Rademacher complexity of the net- +work. For example, in (Neyshabur et al., 2015), an up- +per bound on the Rademacher complexity is introduced +based on the norms of the weight matrices of the net- +work of order O( 2L +√m +�L +l=1 ∥W l∥F ). +Later, (Golowich +et al., 2017) showed that the exponential dependence +on the depth can be avoided by using the contraction +lemma and obtained a bound that scales with O( +√ +L). +In (Bartlett et al., 2017), a Rademacher complexity bound +based on covering numbers was introduced, which scales +as ˜O +� +�L +l=1 ∥W l∥2 +√m +· +��L +l=1 +∥(W l−M l)⊤∥2/3 +2,1 +∥W l∥2/3 +2 +�3/2� +, where +M l ∈ Rdl+1×dl are fixed reference matrices and ∥·∥2 is the +spectral norm. +While these results provide solid upper bounds on the test +error of deep neural networks, they only take into account +very limited information about the architectural choices of +the network. In particular, when applied to convolutional +networks, the matrices W l represent the linear operation per- +formed by a convolutional layer whose filters are wl. How- +ever, since W l applies wl to several patches (dl patches), +we have ∥W l∥F = √dl∥wl∥F . As a result, the bound +scales with O( +��L +l=1 dl), that grows exponentially with L. +This means that the bound is not suitable for convolutional +networks with many layers as it would be very loose in +practice. In this work, we establish generalization bounds +that are customized for convolutional networks and scale +with �L +l=1 ∥wl∥F instead of �L +l=1 ∥W l∥F . +In (Jiang et al., 2020) they conducted a large-scale ex- +periment evaluating multiple norm-based generalization +bounds, including those of (Bartlett et al., 2017; Golowich +et al., 2017). They argued that these bounds are highly +non-vacuous and negatively correlated with the test error. +However, in all of these experiments, they trained the neu- +ral networks with the cross-entropy loss which implicitly +maximizes the network’s weight norms once the network +perfectly fits the training data. This can explain the observed +negative correlation between the bounds and the error. +In this work, we empirically show that our bounds provide +relatively tight estimations of the generalization gap for +convolutional networks trained with weight normalization +and weight decay using the MSE loss. +Generalization bounds for convolutional networks. +Several recent papers have introduced generalization bounds +for convolutional networks that take into account the unique +structure of these networks. In (Li et al., 2018), a gener- +alization bound for neural networks with weight sharing +was introduced. However, this bound only holds under the +assumption that the weight matrices are orthonormal, which +is not realistic in practice. Other papers introduce general- +ization bounds based on parameter counting for convolu- +tional networks that improve classic guarantees for fully- +connected networks but are typically still vacuous by several +orders of magnitude. In (Long & Sedghi, 2020), norm-based +generalization bounds for convolutional networks were in- +troduced by addressing their weight-sharing. However, this +bound scales roughly as the square root of the number of +parameters. In (Du et al., 2018), size-free bounds for con- +volutional networks in terms of the number of trainable +parameters for two-layer networks were proved. In (Ledent +et al., 2021), the generalization bounds in (Bartlett et al., +2017) were extended for convolutional networks where the +linear transformation W l at each layer is replaced with the +trainable parameters. While this paper provides generaliza- + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +tion bounds in which each convolutional filter contributes +only once to the bound, it does not hold when different fil- +ters are used for different patches, even if their norms are +the same. In short, their analysis treats different patches +as “datapoints” in an augmented problem where only one +linear function is applied at each layer. If several choices of +linear functions (different weights for different patches) are +allowed, the capacity of the function class would increase. +While all of these papers provide generalization bounds for +convolutional networks, they all rely on the number of train- +able parameters or depend on weight sharing. None of these +works, in particular, address the question of whether weight +sharing is necessary for convolutional neural networks to +generalize well. +1.2. Contributions +In this work, we study the generalization performance of +multilayered sparse neural networks (Mhaskar et al., 2017), +such as convolutional neural networks. Sparse, deep neural +networks are networks of neurons represented as a Directed +Acyclic Graph (DAG), where each neuron is a function +of a small set of other neurons. We derive norm-based +generalization bounds for these networks. Unlike previous +bounds (Long & Sedghi, 2020; Ledent et al., 2021), our +bounds do not rely on weight sharing and provide favor- +able guarantees for sparse neural networks that do not use +weight sharing. These results suggest that it is possible to +obtain good generalization performance with sparse neural +networks without relying on weight sharing. +Finally, we conduct multiple experiments to evaluate our +bounds for convolutional neural networks trained on simple +classification problems. We show that our bound is relatively +tight, even in the overparameterized regime. +2. Problem Setup +We consider the problem of training a model for a standard +classification problem. Formally, the target task is defined +by a distribution P over samples (x, y) ∈ X × Y, where +X ⊂ Rc0d0 is the instance space (e.g., images), and Y ⊂ +Rk is a label space containing the k-dimensional one-hot +encodings of the integers 1, . . . , k. +We consider a hypothesis class F ⊂ {f ′ : X → Rk} +(e.g., a neural network architecture), where each function +fw ∈ F is specified by a vector of parameters w ∈ RN +(i.e., trainable parameters). A function fw ∈ F assigns a +prediction to an input point x ∈ X, and its performance on +the distribution P is measured by the expected error +errP (fw) := E(x,y)∼P [I[sign(fw(x)) ̸= y]] , +(2) +where I : {True, False} → {0, 1} be the indicator function +(i.e., I[True] = 1 and vice versa). +Since we do not have direct access to the full population +distribution P, the goal is to learn a predictor, fw, from +some training dataset S = {(xi, yi)}m +i=1 of independent and +identically distributed (i.i.d.) samples drawn from P along +with regularization to control the complexity of the learned +model. In addition, since I is a non-continuous function, we +typically use a surrogate loss function ℓ : Rk × Y → [0, ∞) +is a non-negative, differentiable, loss function (e.g., MSE or +cross-entropy losses). +2.1. Rademacher Complexities +In this paper, we examine the generalization abilities of +overparameterized neural networks by investigating their +Rademacher complexity. This quantity can be used to upper +bound the worst-case generalization gap (i.e., the distance +between train and test errors) of functions from a certain +class. It is defined as the expected performance of the class +when averaged over all possible labelings of the data, where +the labels are chosen independently and uniformly at ran- +dom from the set {±1}. In other words, it is the average +performance of the function class on random data. For more +information, see (Mohri et al., 2018a; Shalev-Shwartz & +Ben-David, 2014; Bartlett & Mendelson, 2002). +Definition 2.1 (Rademacher Complexity). Let F be a set +of real-valued functions fw : X → R defined over a set X. +Given a fixed sample X ∈ X m, the empirical Rademacher +complexity of H is defined as follows: +RX(H) := +1 +mEξ +� +sup +fw∈F +��� +m +� +i=1 +ξifw(xi) +��� +� +. +The expectation is taken over ξ = (ξi, . . . , ξm), where, +ξi ∈ {±1} are i.i.d. and uniformly distributed samples. +In contrast to the Vapnik–Chervonenkis (VC) dimension, +the Rademacher complexity has the added advantage that it +is data-dependent and can be measured from finite samples. +The Rademacher complexity can be used to upper bound +the generalization gap of a certain class of functions (Mohri +et al., 2018a). In particular, we can easily upper bound +the test classification error errP (fw) using the Rademacher +complexity for models fw that perfectly fit the training sam- +ples, i.e., fw(xi) = yi = ±1. +Lemma 2.2. Let P be a distribution over Rc0d0 × {±1} +and F ⊂ {f ′ : X → {±1}}. Let S = {(xi, yi)}m +i=1 be a +dataset of i.i.d. samples selected from P and X = {xi}m +i=1. +Then, with probability at least 1 − δ over the selection of S, +for any fw ∈ F that perfectly fits the data (i.e., fw(xi) = +yi), we have +errP (fw) ≤ 2RX(F) + 3 +� +log(2/δ) +2m +. +(3) + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +Proof. We apply a standard Rademacher complexity-based +generalization bound with the ramp loss function. The ramp +loss function is defined as follows: +ℓramp(y, y′) := +� +� +� +� +� +1, +if +yy′ ≤ 0 +1 − yy′, +if +0 ≤ yy′ ≤ 1 +0, +if +yy′ ≥ 1 +. +By Theorem 3.3 in (Mohri et al., 2018b), with prob- +ability at least 1 − δ, for any function fw +∈ +F, +E(x,y)∼P [ℓramp(fw(x), y)] is bounded by +1 +m +m +� +i=1 +ℓramp(fw(xi), yi) + 2RX(F) + 3 +� +log(2/δ) +2m +. +We note that for any function fw for which fw(xi) = yi = +±1, we have ℓramp(fw(xi), yi) = 0. In addition, for any +function fw and pair (x, y), we have ℓramp(fw(x), y) ≥ +I[sign(fw(x)) ̸= y], hence, E(x,y)∼P [ℓramp(fw(x), y)] ≥ +errP (fw). Therefore, we conclude that with probability at +least 1 − δ, for any function fw ∈ F that perfectly fits the +training data, we have the desired inequality. +The above lemma provides an upper bound on the test er- +ror of a trained model fw that perfectly fits the training +data. The bound is decomposed into two parts; one is the +Rademacher complexity and the second scales as O(1/√m) +which is small when m is large. In section 3 we derive +norm-based bounds on the Rademacher complexity of com- +positionally sparse neural networks. +2.2. Architectures +A neural network architecture can be formally defined using +a Directed Acyclic Graph (DAG) G = (V, E). The class +of neural networks associated with this architecture is de- +noted as FG. The set of neurons in the network is given by +V = �L +l=0{zl +1, . . . , zl +dl}, which is organized into L layers. +An edge (zl +i, zl−1 +j +) ∈ E indicates a connection between a +neuron in layer l − 1 and a neuron in layer l. The full set of +neurons at the layer lth is denoted by vl := (zl +j)dl +j=1. +A neural network function fw : Rc0d0 → Rk takes “flat- +tened” images x as input, where c0 is the number of input +channels and d0 is the image dimension represented as a +vector. Each neuron zl +i : Rc0d0 → Rcl computes a vec- +tor of size cl (the number of channels in layer l). The +set of predecessor neurons of zl +i, denoted by pred(l, i), +is the set of j ∈ [dl−1] such that (zl +i, zl−1 +j +) ∈ E, and +vl +i := (zl +j)j∈pred(l,i) denotes the set of predecessor neurons +of zl +i. The neural network zL +j0(x) := zL +1 (x) := fw(x) is +recursively defined as follows: +zl +i(x) := wl +iσ(vl−1 +i +(x)), +where wl +i ∈ Rcl×(cl−1·|pred(l−1,i)|) is a weight matrix, x = +(z0 +i (x))d0 +i=1 and each z0 +i (x) is a vector of dimension c0 rep- +resenting a “pixel” in the image x. +The degree of sparsity of a neural network can be measured +using the degree, which is defined as the maximum number +of predecessors for each neuron. +deg(G) := +max +l∈[L],j∈[dl] |pred(l, j)|. +A compositionally sparse neural network is a neural network +architecture G for which the degree deg(G) = O(1) (with +respect to maxi=0,...,L(di) and L). These considerations +extend easily to networks that contain sparse layers as well +as fully-connected layers. +Convolutional neural networks. +A special type of com- +positionally sparse neural networks is convolutional neural +networks. In such networks, each neuron acts upon a set +of nearby neurons from the previous layer, using a kernel +shared across the neurons of the same layer. +To formally analyze convolutional networks, we consider +a broader set of neural network architectures that includes +sparse networks with shared weights. Specifically, for an +architecture G with |pred(l, j)| = kl for all j ∈ [dl], we +define the set of neural networks Fsh +G to consist of all neural +networks fw ∈ Fsh +G that satisfy the weight sharing prop- +erty wl := wl +j1 = wl +j2 for all j1, j2 ∈ [dl] and l ∈ [L]. +Convolutional neural networks are essentially sparse neural +networks with shared weights and locality (each neuron is a +function of a set of nearby neurons of its preceding layer). +Norms of neural networks. +As mentioned earlier, pre- +vious papers (Golowich et al., 2017; Neyshabur et al., +2018; Arora et al., 2018; Neyshabur et al., 2017; Bartlett +et al., 2017) have proposed different generalization bounds +based on different norms measuring the complexity of fully- +connected networks. One approach that was suggested +by (Golowich et al., 2017) is to use the product of the norms +of the weight matrices given by ˜ρ(w) := �L +l=1 ∥W l∥F . +In this work, we derive generalization bounds based on the +product of the maximal norms of the kernel matrices across +layers, defined as: +ρ(w) := ∥wL +1 ∥2 · +L−1 +� +l=1 +max +j∈[dl] ∥wl +j∥F , +(4) +where ∥ · ∥F and ∥ · ∥2 are the Frobenius and the spectral +norms. Specifically, for a convolutional neural network, we +have a simplified form of ρ(w) = ∥wL∥2 · �L−1 +l=1 ∥wl∥F , +due to the weight sharing property. +We observe that this quantity is significantly smaller than the +quantity ˜ρ(w) = �L +l=1 +��dl +j=1 ∥wl +j∥2 +F used by (Golowich +et al., 2017). For instance, when weight sharing is applied, +we can see that ˜ρ(w) = ρ(w) · +��L +l=1 dl. +Classes of interest. +In the next section, we study the +Rademacher complexity of classes of compositionally + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +sparse neural networks that are bounded in norm. We focus +on two classes: FG,ρ := {fw ∈ FG | ρ(w) ≤ ρ} and +Fsh +G,ρ := {fw ∈ Fsh +G | ρ(w) ≤ ρ}, where G is a composi- +tionally sparse neural network architecture and ρ is a bound +on the norm of the network parameters. +3. Theoretical Results +In this section, we introduce our main theoretical results. +The following theorem provides an upper bound on the +Rademacher complexity of the class FG,ρ of neural net- +works of architecture G of norm ≤ ρ. +Proposition 3.1. Let G be a neural network architecture +of depth L and let ρ > 0. Let X = {xi}m +i=1 be a set of +samples. Then, +RX(FG,ρ) ≤ +ρ +m · +� +1 + +� +2L log(2deg(G)) +� +· +� +� +� +� max +j1,...,jL +L +� +l=1 +|pred(l, jL−l)| · +m +� +i=1 +∥z0 +jL(xi)∥2, +where the maximum is taken over j1, . . . , jL, such that, +jL−l+1 ∈ pred(l, jL−l) for all l ∈ [L]. +The proof for this theorem is provided in Appendix B and +builds upon the proof of Theorem 1 in (Golowich et al., +2017). A summary of the proof is presented in section 3.1. +As we show next, by combining Lemma 2.2 and Proposi- +tion 3.1 we can obtain an upper bound on the test error of +compositionally sparse neural networks fw that perfectly fit +the training data (i.e., for all i ∈ [m] : fw(xi) = yi). +Theorem 3.2. Let P be a distribution over Rc0d0 × {±1}. +Let S = {(xi, yi)}m +i=1 be a dataset of i.i.d. samples selected +from P. Then, with probability at least 1 − δ over the +selection of S, for any fw ∈ FG that perfectly fits the data +(for all i ∈ [m] : fw(xi) = yi), we have +errP (fw) ≤ (ρ(w) + 1) +m +� +1 + +� +2L log(2deg(G)) +� +· +� +� +� +� max +j1,...,jL +L +� +l=1 +|pred(l, jL−l)| +m +� +i=1 +∥z0 +jL(xi)∥2 ++ 3 +� +log(2(ρ(w) + 2)2/δ) +2m +, +where the maximum is taken over j1, . . . , jL, such that, +jL−l+1 ∈ pred(l, jL−l) for all l ∈ [L]. +The theorem above provides a generalization bound +for neural networks of a given architecture G. +To +understand this bound, +we first analyze the term +maxj1,...,jL +�L +l=1 |pred(l, jL−l)| · �m +i=1 ∥z0 +jL(xi)∥2. +We +consider a setting where d0 = 2L, cl = 1 and each neuron +takes two neurons as input, kl := |pred(l, j)| = 2 for all l ∈ +[L] and j ∈ [dl]. In particular, �L +l=1 kl = 2L and z0 +j (xi) = +xij is the jth pixel of xi. By assuming that the norms of the +pixels are β-balanced, i.e., ∀i ∈ [m] : maxj∈[d0] ∥xij∥2 ≤ +β Avgj∈[d0][∥xij∥2] (for some constant β > 0), we obtain +that �L +l=1 kl · maxj +�m +i=1 ∥z0 +j (xi)∥2 ≤ β �m +i=1 ∥xi∥2. In +addition, we note that the second term in the bound is typi- +cally smaller than the first term as it scales with +� +log(ρ(w)) +instead of ρ(w) and has no dependence on the size of the +network. Therefore, our bound can be simplified to +O +� +ρ(w) +√m +� +Lβ log(deg(G)) Avgm +i=1[∥xi∥2] +� +. +(5) +Similar to the bound in (Golowich et al., 2017), our bound +scales with O( +√ +L), where L is the depth of the network. +Convolutional neural networks. +As previously stated in +section 2, convolutional neural networks utilize weight shar- +ing neurons in each layer, with each neuron in the lth layer +having an input dimension of kl. The norm of the network +is calculated as ρ(w) = �L +l=1 ∥wl∥F , and the degree of the +network is determined by the maximum input dimension +across all layers, deg(G) = maxl∈[L] kl. This results in a +simplified version of the theorem. +Corollary 3.3 (Rademacher Complexity of ConvNets). Let +G be a neural network architecture of depth L and let ρ > 0. +Let X = {xi}m +i=1 be a set of samples. Then, +RS(F sh +G,ρ) ≤ +ρ +m +� +1 + +� +2L log(2 deg(G)) +� +· +� +� +� +� +L +� +l=1 +kl · max +j∈[d0] +m +� +i=1 +∥z0 +j (xi)∥2, +where kl denotes the kernel size in the l’th layer. +Comparison with the bound of (Golowich et al., 2017). +The result in Corollary 3.3 is a refined version of the analysis +in (Golowich et al., 2017) for the specific case of convo- +lutional neural networks. Theorem 1 in (Golowich et al., +2017) can of course be applied to convolutional networks by +treating their convolutional layers as fully-connected layers. +However, this approach yields a substantially worse bound +compared to the one proposed in Corollary 3.3. +Consider a convolutional neural network architecture G. +The lth convolutional layer takes the concatenation of +(σ(zl +1), . . . , σ(zl +dl)) as input and returns (zl+1 +1 +, . . . , zl+1 +dl+1) +as its output. +Each zl+1 +j +is computed as follows +zl+1 +j += wl+1σ(vl +j(x)). +Therefore, the matrix W l+1 +associated with the convolutional layer contains dl+1 +copies of wl+1 and its Frobenius norm is therefore +� +dl+1 · ∥wl+1∥F . In particular, by applying Theorem 1 +in (Golowich et al., 2017), we obtain a bound that scales as +O +� +ρ +m +� +L �L +l=1 dl · �m +i=1 ∥xi∥2 +� +. In particular, if each +convolutional layer has kl = 2 with no overlaps and +d0 = 2L, then, dl = 2L−l and the bound therefore scales +as O +� +ρ +√m +� +L20.5L(L−1) · Avgm +i=1[∥xi∥2] +� +. On the other + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +hand, as we discussed earlier, if the norms of the pixels of +each sample x are β-balanced (for some constant β > 0), +our bound scales as O +� +ρ +√m +� +L Avgm +i=1[∥xi∥2] +� +which is +smaller by a factor of 20.25L(L−1) than the previous bound. +Comparison with the bound of (Long & Sedghi, 2020). +A recent paper (Long & Sedghi, 2020) introduced general- +ization bounds for convolutional networks based on param- +eter counting. This bound roughly scales like +O +�� +N(�L +l=1 ∥wl∥2+log(1/γ))+log(1/δ) +m +� +, +where γ is a margin (typically smaller than 1), and N is +the number of trainable parameters (taking weight sharing +into account by counting each parameter of convolutional +filters only once). While these bounds provide improved +generalization guarantees when reusing parameters, it scales +as Ω( +� +N/m) which is very large in practice. For example, +the standard ResNet-50 architecture has approximately N = +23M trainable parameters while the MNIST dataset has only +m = 50000 training samples. +Comparison with the bound of (Ledent et al., 2021). +A +recent paper (Ledent et al., 2021) introduces a generaliza- +tion bound for convolutional networks that is similar to the +analysis presented in (Bartlett et al., 2017). Specifically, the +bounds in Theorem 17 of (Ledent et al., 2021) roughly scale +as +O +� +� +� +L +� +l=1 +∥W l∥2 +√m +�L−1 +� +l=1 +k +α +2 +l +∥(wl−ul)⊤∥α +2,1 +∥wl∥α +2 ++ +∥wL∥α +2 +max +i +∥wL +i,:∥α +2 +� 1 +α +Iα +� +� +� , +where kl is the kernel size of the lth layer and W l is the +matrix corresponding to the linear operator associated with +the lth convolutional layer, wi,: is the ith row of a matrix +w, α is either 2 or 2/3, Iα = L if α = 2 and Iα = 1 +otherwise and ul are predefined “reference” matrices of the +same dimensions as wl. +In general, our bounds and the ones in (Ledent et al., 2021) +cannot be directly compared, with each being better in +different cases. However, our bound has a significantly +better explicit dependence on the depth L than the bound +in (Ledent et al., 2021). To see this, consider the simple case +where each convolutional layer operates on non-overlapping +patches and we choose ul = 0 for all l ∈ [L − 1] (which +is a standard choice of reference matrices). We notice that +∥W l∥2 = ∥wl∥2 and that for any matrix A ∈ Rn×m, the fol- +lowing inequalities hold: rank(A) ≥ ∥A⊤∥2,1 +∥A∥2 +≥ ∥A∥F +∥A∥2 ≥ 1 +and rank(A) ≥ +∥A∥2 +maxi ∥Ai,:∥2 ≥ 1. Therefore, the bound +in (Ledent et al., 2021) is at least +�L +l=1 ∥wl∥2 +√m +· L3/2, which +scales at least as Ω(L3/2) with respect to L, while our bound +scales as O( +√ +L) (when ρ(w) is independent of L), mean- +ing that the dependence on the depth is significantly better +than that of the bound in (Ledent et al., 2021). +3.1. Proof Sketch +We propose an extension to a well-established method for +bounding the Rademacher complexity of norm-bounded +deep neural networks. This approach, originally developed +by (Neyshabur et al., 2015) and later improved by (Golowich +et al., 2017), utilizes a “peeling” argument, where the com- +plexity bound for a depth L network is reduced to a complex- +ity bound for a depth L − 1 network and applied repeatedly. +Specifically, the lth step bounds the complexity bound for +depth l by using the product of the complexity bound for +depth l − 1 and the norm of the lth layer. By the end of +this process, we obtain a bound that depends on the term +Eξg(| �m +i=1 ξixi|) (g(x) = x in (Neyshabur et al., 2015) +and g = exp in (Golowich et al., 2017)), which can be fur- +ther bounded using maxx∈X ∥x∥2. The final bound scales +with ˜ρ(w) · maxx∈X ∥x∥. Our extension aims to further +improve the tightness of these bounds by incorporating ad- +ditional information about the network’s degrees of sparsity. +To bound RX(FG,ρ) using ρ(w) · maxx∈X ∥x∥, we notice +that each neuron operates on a small subset of the neurons +from the previous layer. Therefore, we can bound the contri- +bution of a certain constituent function zl +j(x) = wl +jvl−1 +j +(x) +in the network using the norm ∥wl +j∥F and the complexity +of vl−1 +j +(x) instead of the full layer vl−1(x). +To explain this process, we provide a proof sketch of Propo- +sition 3.1 for convolutional networks G = (V, E) with non- +overlapping patches. For simplicity, we assume that d0 = +2L, cl = 1, and the strides and kernel sizes at each layer +are k = 2. In particular, the network fw can be represented +as a binary tree, where the output neuron is computed as +fw(x) = zL +j0(x) = wL · σ(zL−1 +1 +(x), zL−1 +2 +(x)), zL−1 +1 +(x) = +wL−1 · σ(zL−2 +1 +(x), zL−2 +2 +(x)) and zL−1 +2 +(x) = wL−1 · +σ(zL−2 +3 +(x), zL−2 +4 +(x)) and so on. +Similar to (Golowich +et al., 2017), we first bound the Rademacher complexity +using Jensen’s inequality, +mRX(FG,ρ) = 1 +λ log exp +� +λEξ sup +fw +m +� +i=1 +ξifw(xi) +� +≤ 1 +λ log +� +Eξ sup +fw +exp +� +λ +m +� +i=1 +ξifw(xi) +�� +, (6) +where λ > 0 is an arbitrary parameter. As a next step, +we rewrite the Rademacher complexity in the following +manner: +Eξ sup +fw +exp +����� +m +� +i=1 +ξi · fw(xi) +����� += Eξ sup +fw +exp +� +� +� +� +����� +m +� +i=1 +ξi · wL · σ(zL−1 +1 +(xi), zL−1 +2 +(xi)) +����� +2 +≤ Eξ sup +fw +exp +� +� +� +�∥wL∥2 +2 · +2 +� +j=1 +����� +m +� +i=1 +ξi · σ(zL−1 +j +(xi)) +����� +2 +. (7) + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +We notice that each zL−1 +j +(x) is itself a depth L − 1 +binary-tree neural network. +Therefore, intuitively we +would like to apply the same argument L − 1 more +times. +However, in contrast to the above, the net- +works σ(zL−1 +1 +(x)) = σ(wL−1(zL−2 +1 +(x), zL−2 +2 +(x))) and +σ(zL−1 +2 +(x)) = σ(wL−1(zL−2 +3 +(x), zL−2 +4 +(x))) end with +a ReLU activation. +To address this issue, (Neyshabur +et al., 2015; Golowich et al., 2017) proposed a “peeling +process” based on Equation 4.20 in (Ledoux & Talagrand, +1991) that can be used to bound terms of the form +Eξ +sup +f ′∈F′,W : ∥W ∥F ≤R +exp[ +� +α ∥�m +i=1 ξi · σ(Wf ′(xi))∥2]. +However, this bound is not directly applicable when there +is a sum inside the square root, as in equation 7 which +includes a sum over j ∈ {1, 2}. Therefore, a modified +peeling lemma is required to deal with this case. +Lemma 3.4 (Peeling Lemma). Let σ be a 1-Lipschitz, +positive-homogeneous activation function which is applied +element-wise (such as the ReLU). Then for any class of +vector-valued functions F ⊂ {f = (f1, . . . , fq) | ∀j ∈ +[q] : fj : Rd → Rp}, and any convex and monotonically +increasing function g : R → [0, ∞), +Eξ +sup +f∈F +Wj: ∥Wj∥≤R +g +� +� +� +� +� +� +q +� +j=1 +����� +m +� +i=1 +ξi · σ(Wjfj(xi)) +����� +2� +� +≤ 2Eξ +sup +j∈[q], f∈F +g +� +√qR +����� +m +� +i=1 +ξi · fj(xi) +����� +� +. +By applying this lemma L − 1 times with g = exp and f +representing the neurons preceding a certain neuron at a +certain layer, we obtain the following inequality +Eξ sup +fw +exp +����� +m +� +i=1 +ξi · fw(xi) +����� +≤ 2LEξ sup +j,w +exp +� +� +� +�∥wL∥2 +2 +L−1 +� +l=1 +∥wl∥2 +F · 2L +����� +m +� +i=1 +ξixij +����� +2 +≤ 2L +d +� +j=1 +Eξ exp +� +λ2L/2ρ · +����� +m +� +i=1 +ξixij +����� +� +≤ 4L sup +j +exp +� +� λ22Lρ2·�m +i=1 x2 +ij +2 ++ λ2L/2ρ · +� +� +� +� +m +� +i=1 +x2 +ij +� +� , +where the last inequality follows from standard concentra- +tion bounds. Finally, by equation 6 and properly adjusting +λ, we can finally bound RX(FG,ρ) as O( +√ +Lρ +√m ). +4. Experiments +In section 3 we showed that the Rademacher complexity +of compositionally sparse networks is bounded by O( ρ(w) +√m ) +(when maxi∈[m] ∥xi∥ and L are constants). In this section, +103 +104 +105 +m +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +103 +(w) +m +Train error +Test error +Generalization gap +103 +104 +105 +m +65 +70 +75 +80 +85 +90 +95 +100 +(w) +Figure 1. Comparing our bound with the train and test errors +and the generalization gap of a 5-layers network. (top) We +report ρ(w) +√m , the train error errS(fw), the test error errP (fw) and +the generalization gap |errP (fw) − errS(fw)| when varying the +number of training samples (in logarithmic scales). (bottom) We +display the value of ρ(w) when varying the number of training +samples. We used the following hyperparameters: ρl = 0.1, +λ = 1e−3. More detailed results can be found in Table 1. +we conduct an empirical evaluation of the performance, the +term ρ(w) +√m and ρ(w) for neural networks trained with a vary- +ing number of training samples. Further, in Appendix A, we +provide additional experiments that illustrate the evolution +of these quantities throughout the training process. +Network architecture. +We use two types of deep neural +network architectures. Both consist of four hidden convo- +lutional layers, which use 3 × 3 convolutions, stride 2, and +padding 0, and have output channel numbers of 32, 64, 128, +and 128, respectively. The final fully connected layer maps +the 3200-dimensional output of the final convolutional layer +to 2 outputs, with ReLU activation applied to all layers ex- +cept the last one. The second architecture is identical, but +it replaces the last linear fully connected layer with two +fully connected layers that project the 3200-dimensional + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +103 +104 +105 +m +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +103 +(w) +m +Train error +Test error +Generalization gap +103 +104 +105 +m +250 +260 +270 +280 +290 +300 +310 +320 +330 +(w) +Figure 2. Comparing our bound with the generalization gap +and the test error of a 6-layers network. See Figure 1 for details. +More detailed numerical results can be found in Table 2. We used +the following hyperparameters: ρl = 0.01, λ = 5e−4. +output of the final convolutional layer to a 128-dimensional +vector before mapping it to 2 outputs. The total number of +parameters in the first model is 246886 and in the second is +650343 parameters. +Optimization process. +In Theorem 3.2, a general- +ization bound is proposed that scales with ρ(w) += +∥wL∥2 · �L−1 +l=1 ∥wl∥F . +To control ρ(w), we regularize +�L +l=1 ∥wl∥F ≥ ρ(w) by applying weight normalization +to all trainable layers, except for the last one, which is left +un-normalized. Specifically, we fix the norm of the weights +wl in each layer by decomposing them into a direction and +magnitude, such that wl = ρlvl (where ∥vl∥F = 1). To +initialize wl, we use the default PyTorch initialization and +normalize vl to have a norm of 1. We only update vl us- +ing the method described in (Salimans & Kingma, 2016) +while keeping ρ1, . . . , ρL−1 constant. This way we can reg- +ularize �L +l=1 ∥wl∥F , by applying weight decay of rate λ +exclusively to the weights of the top layer. +Each model was trained using MSE-loss minimization be- +tween the logits of the network and the one-hot encodings of +the training labels. To train the model we used the Stochastic +Gradient Descent (SGD) optimizer with an initial learning +rate µ = 0.03, momentum of 0.9, batch size 128, and a +cosine learning rate scheduler (Loshchilov & Hutter, 2016). +Varying the number of samples. +In this experiment we +trained the same model for binary classification between +the first two classes of the CIFAR-5m dataset (Nakkiran +et al., 2020) with a varying number of training samples. +This dataset contains 6 million synthetic CIFAR-10-like +images (including the CIFAR10 dataset). It was generated +by sampling the DDPM generative model of (Ho et al., +2020), which was trained on the CIFAR-10 training set. For +each number of samples m, we chose m random training +samples from the dataset and trained the model on these +samples for 5000 epochs over 5 different runs. +In Figures 1-2, we report the values of ρ(w), ρ(w) +√m , the train +and test errors, and the generalization gap for each model +as a function of m. Each quantity is averaged over the last +100 training epochs (i.e., epochs 4900-5000). Since we +do not have access to the complete population distribution +P, we estimated the test error by using 1000 test samples +per class. As seen in the figures, ρ(w) +√m provides a relatively +tight estimation of the generalization gap even though the +network is overparameterized. For example, when m is +greater than 10000, the quantity ρ(w) +√m is smaller than 1 for +the 5-layer model. Additionally, it is observed that ρ(w) is +bounded as a function of m, even though it could potentially +increase with the size of the training dataset. Therefore, +ρ(w) +√m appears to decrease at a rate of O(1/√m). +5. Conclusions +We studied the question of why certain deep learning archi- +tectures, such as CNNs and Transformers, perform better +than others on real-world datasets. To tackle this question, +we derived Rademacher complexity generalization bounds +for sparse neural networks, which are orders of magnitude +better than a naive application of standard norm-based gen- +eralization bounds for fully-connected networks. In contrast +to previous papers (Long & Sedghi, 2020; Ledent et al., +2021), our results do not rely on parameter sharing between +filters, suggesting that the sparsity of the neural networks +is the critical component to their success. This sheds new +light on the central question of why certain architectures +perform so well and suggests that sparsity may be a key +factor in their success. Even though our bounds are not prac- +tical in general, our experiments show that they are quite +tight for simple classification problems, unlike other bounds +based on parameter counting, suggesting that the underlying +theory is sound and does not need a basic reformulation. + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +Acknowledgments +We thank Akshay Rangamani, Eran Malach and Antoine +Ledent for illuminating discussions during the preparation of +this manuscript. This material is based upon work supported +by the Center for Minds, Brains and Machines (CBMM), +funded by NSF STC award CCF-1231216. +References +Allen-Zhu, Z., Li, Y., and Liang, Y. Learning and gener- +alization in overparameterized neural networks, going +beyond two layers. In Advances in Neural Information +Processing Systems. Curran Associates, Inc., 2019. +Arora, S., Ge, R., Neyshabur, B., and Zhang, Y. 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Scaling +vision transformers, 2021. +Zhang, C., Bengio, S., Hardt, M., Recht, B., and Vinyals, +O. Understanding deep learning requires rethinking gen- +eralization. In International Conference on Learning +Representations, 2017. URL https://openreview. +net/forum?id=Sy8gdB9xx. +Zhou, W., Veitch, V., Austern, M., Adams, R. P., and Or- +banz, P. Non-vacuous generalization bounds at the ima- +genet scale: a PAC-bayesian compression approach. In +International Conference on Learning Representations, +2019. URL https://openreview.net/forum? +id=BJgqqsAct7. + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +A. Additional Experiments +A.1. Additional Details for the Experiments in Figures 1-2 +In Figures 1-2 we provided multiple plots demonstrating the behaviors of various quantities (e.g., ρ(w), the train and test +errors) when varying the number of training samples m. For completeness, in Tables 1-2 we explicitly report the values of +each quantity reported in Figures 1-2. +m +ρ(w) +Train error +Test error +Train loss +Test loss +Generalization gap +ρ(w) +√m +500 +99.499 +0.007 +0.188 +0.032 +0.161 +0.182 +4.450 +1000 +82.050 +0.013 +0.087 +0.038 +0.085 +0.074 +2.595 +2000 +71.701 +0.021 +0.057 +0.042 +0.062 +0.036 +1.603 +3000 +72.128 +0.028 +0.053 +0.045 +0.058 +0.025 +1.317 +4000 +69.004 +0.030 +0.052 +0.047 +0.056 +0.022 +1.091 +5000 +68.359 +0.031 +0.05 +0.048 +0.056 +0.019 +0.967 +7500 +69.241 +0.033 +0.048 +0.048 +0.052 +0.016 +0.800 +9000 +69.172 +0.034 +0.047 +0.048 +0.052 +0.013 +0.729 +10000 +68.003 +0.035 +0.048 +0.049 +0.052 +0.013 +0.68 +20000 +64.326 +0.029 +0.032 +0.044 +0.046 +0.003 +0.455 +40000 +64.598 +0.029 +0.031 +0.044 +0.045 +0.003 +0.323 +80000 +64.904 +0.028 +0.029 +0.043 +0.044 +0.004 +0.229 +100000 +65.418 +0.028 +0.030 +0.043 +0.043 +0.001 +0.207 +150000 +64.530 +0.029 +0.029 +0.044 +0.044 +0.001 +0.167 +182394 +65.040 +0.029 +0.026 +0.044 +0.043 +0.003 +0.152 +Table 1. We report the averaged values of the norm ρ(w), the train and test errors, the training and test losses, the generalization gap, and +ρ(w) +√m for the experiment in Figure 1. +m +ρ(w) +Train error +Test error +Train loss +Test loss +Generalization gap +ρ(w) +√m +500 +277.829 +0.000 +0.210 +0.005 +0.177 +0.210 +12.425 +1000 +263.867 +0.000 +0.085 +0.008 +0.068 +0.085 +8.344 +2000 +287.343 +0.002 +0.065 +0.012 +0.052 +0.062 +6.425 +3000 +297.993 +0.007 +0.054 +0.015 +0.045 +0.048 +5.441 +4000 +312.316 +0.011 +0.044 +0.017 +0.037 +0.033 +4.938 +5000 +298.258 +0.014 +0.042 +0.019 +0.036 +0.028 +4.218 +7500 +293.125 +0.015 +0.035 +0.021 +0.032 +0.021 +3.385 +9000 +298.155 +0.018 +0.034 +0.022 +0.031 +0.016 +3.143 +10000 +298.442 +0.017 +0.032 +0.022 +0.029 +0.014 +2.984 +20000 +272.198 +0.016 +0.026 +0.018 +0.024 +0.010 +1.925 +40000 +265.294 +0.018 +0.020 +0.019 +0.021 +0.003 +1.326 +80000 +261.114 +0.018 +0.020 +0.020 +0.020 +0.004 +0.923 +100000 +262.034 +0.018 +0.021 +0.019 +0.021 +0.004 +0.829 +150000 +258.874 +0.019 +0.019 +0.020 +0.020 +0.003 +0.668 +182394 +259.392 +0.018 +0.019 +0.020 +0.020 +0.002 +0.607 +Table 2. We report the averaged values of the norm ρ(w), the train and test errors, the training and test losses, the generalization gap, and +ρ(w) +√m for the experiment in Figure 2. +A.2. Evaluating Networks During Training +In section 4, we examined the behavior of ρ(w), the train and test errors, and our bound while varying the number of training +samples. In this section, we conduct supplementary experiments that compare these quantities throughout the training +process. Additionally, to add diversity to the study, we utilize a slightly different training method in these experiments. + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +0 +100 +200 +300 +400 +500 +epoch +2 +15 +2 +13 +2 +11 +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +0 +100 +200 +300 +400 +500 +epoch +2 +15 +2 +13 +2 +11 +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +0 +100 +200 +300 +400 +500 +epoch +2 +15 +2 +13 +2 +11 +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +0 +100 +200 +300 +400 +500 +epoch +2 +15 +2 +13 +2 +11 +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +0 +100 +200 +300 +400 +500 +epoch +2 +15 +2 +13 +2 +11 +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +0 +100 +200 +300 +400 +500 +epoch +2 +15 +2 +13 +2 +11 +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +L = 6 +L = 7 +L = 8 +Figure 3. Varying the number of layers. We report ρ(w) +√m , the train error errS(fw), the test error errP (fw) and the generalization gap +|errP (fw) − errS(fw)| of CONV-L-1000 trained on MNIST with a varying number of layers. We trained the models with batch size 64 +and learning rate µ = 1. For the top plots we used λ = 2e−3 and for the bottom ones we used λ = 3e−3. +Network architecture. +In this experiment, we employed a simple convolutional network architecture denoted by CONV- +L-H. The network consists of a stack of two 2 × 2 convolutional layers with a stride of 2 and zero padding, utilizing ReLU +activations. This is followed by L − 2 stacks of 3 × 3 convolutional layers with H channels, a stride of 1, and padding of 1, +also followed by ReLU activations. The final layer is a fully-connected layer. No biases are used in any of the layers. +Optimization process. +In the current experiment we trained each model with a standard weight normalization (Salimans +& Kingma, 2016) for each parametric layer. Each model was trained using MSE-loss minimization between the logits of the +network and the one-hot encodings of the training labels. To train the model we used the Stochastic Gradient Descent (SGD) +optimizer with an initial learning rate µ that is decayed by a factor of 0.1 at epochs 60, 100, 300, momentum of 0.9 and +weight decay with rate λ. +In Figure 3, we present the results of our experimentation where we trained models of varying depths on the MNIST +dataset (LeCun & Cortes, 2010). One of the key observations from our experiment is that as we increase the depth of the +model, the term ρ(w) +√m empirically generally decreases, even though the overall number of training parameters grows with the +number of layers. This is in correlation with the fact that the generalization gap is lower for deeper networks. suggests that +deeper models have a better generalization ability despite having more parameters. Furthermore, we also observed that in all +cases, the term ρ(w) +√m is quite small, reflecting the tightness of our bound. +In Figures 4 and 5, we present the results of an experiment where we varied the number of channels H in models trained +on MNIST and Fashion MNIST (respectively). As can be seen, the bound remains largely unchanged when increasing +H despite the network’s size scaling as Θ(H2). We also observed that, after the network achieves good performance, the +bound is highly correlated with the generalization gap. Specifically, for MNIST, the generalization gap and the bound are +relatively stable, while for Fashion-MNIST, the bound seems to grow at the same rate as the generalization gap. Since the +results are presented in log-scales, this suggests that the generalization gap is empirically proportional to our bound. +B. Proofs +Lemma B.1 (Peeling Lemma). Let σ be a 1-Lipschitz, positive-homogeneous activation function which is applied element- +wise (such as the ReLU). Then for any class of vector-valued functions F ⊂ {f = (f1, . . . , fq) | ∀j ∈ [q] : fj : Rd → Rp}, + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +0 +100 +200 +300 +400 +500 +epoch +2 +15 +2 +13 +2 +11 +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +0 +100 +200 +300 +400 +500 +epoch +2 +15 +2 +13 +2 +11 +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +0 +100 +200 +300 +400 +500 +epoch +2 +15 +2 +13 +2 +11 +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +H = 50 +H = 500 +H = 1500 +Figure 4. Varying the number of channels. We report ρ(w) +√m , the train error errS(fw), the test error errP (fw) and the generalization gap +|errP (fw) − errS(fw)| of CONV-6-H trained on MNIST with a varying number of channels. We trained the models with batch size 64, +µ = 1, and λ = 3e−3. +and any convex and monotonically increasing function g : R → [0, ∞), we have +Eξ +sup +f∈F +Wj: ∥Wj∥≤R +g +� +� +� +� +� +� +q +� +j=1 +����� +m +� +i=1 +ξi · σ(Wjfj(xi)) +����� +2� +� ≤ 2Eξ +sup +j∈[q], f∈F +g +� +√qR +����� +m +� +i=1 +ξi · fj(xi) +����� +� +. +Proof. Let W ∈ Rh×p be a matrix and let w1, . . . , wh be the rows of the matrix W. Define a function Qj(w) := +��m +i=1 ξi · σ( w⊤ +r +∥wr∥fj(xi)) +�2 +for some fixed functions fj. We notice that +q +� +j=1 +����� +m +� +i=1 +ξi · σ(Wjfj(xi)) +����� +2 += +q +� +j=1 +h +� +r=1 +∥wjr∥2 +� m +� +i=1 +ξi · σ( +w⊤ +jr +∥wjr∥fj(xi)) +�2 += +q +� +j=1 +h +� +r=1 +∥wjr∥2 · Qj( wjr +∥wjr∥). +For any wj1, . . . , wjh, we have +h +� +r=1 +∥wjr∥2 · Qj( wjr +∥wjr∥) ≤ R · max +r +Qj( wjr +∥wjr∥), +(8) +which is obtained for ˆwj1, . . . , ˆwjh, where ˆwji = 0 for all i ̸= r∗ and ˆwjr∗ of norm R for some r∗ ∈ [h]. Together with the +fact that g is a monotonically increasing function, we obtain +Eξ +sup +f∈F +Wj: ∥Wj∥≤R +g +� +� +� +� +� +� +q +� +j=1 +����� +m +� +i=1 +ξi · σ(Wjfj(xi)) +����� +2� +� ≤ Eξ +sup +f∈F +w1...,wq: ∥wj∥=R +g +� +� +� +� +� +� +q +� +j=1 +�� +m +� +i=1 +ξi · σ(w⊤ +j fj(xi)) +��2 +� +� +≤ Eξ +sup +j∈[q], f∈F +w1...,wq: ∥wj∥=R +g +� +� +� +� +� +�q · +�� +m +� +i=1 +ξi · σ(w⊤ +j fj(xi)) +��2 +� +� += Eξ +sup +j∈[q], f∈F +w: ∥w∥=R +g +� +√q · +�� +m +� +i=1 +ξi · σ(w⊤fj(xi)) +�� +� +. + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +0 +100 +200 +300 +400 +500 +epoch +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +0 +100 +200 +300 +400 +500 +epoch +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +0 +100 +200 +300 +400 +500 +epoch +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +0 +100 +200 +300 +400 +500 +epoch +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +0 +100 +200 +300 +400 +500 +epoch +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +0 +100 +200 +300 +400 +500 +epoch +2 +9 +2 +7 +2 +5 +2 +3 +2 +1 +(w) +m +Train error +Test error +Generalization gap +H = 200 +H = 500 +H = 2000 +Figure 5. Varying the number of channels. We report ρ(w) +√m , the train error errS(fw), the test error errP (fw) and the generalization gap +|errP (fw) − errS(fw)| of CONV-8-H trained on Fashion-MNIST with a varying number of channels. We trained the models with batch +size 128 and learning rate µ = 1. For the top plots we used λ = 1e−3, and for the bottom ones we used λ = 2e−3. +Since g(|z|) ≤ g(z) + g(−z), +Eξ +sup +j∈[q], f∈F +w: ∥w∥=R +g +� +√q · +�� +m +� +i=1 +ξi · σ(w⊤fj(xi)) +�� +� +≤ Eξ +sup +j∈[q], f∈F +w: ∥w∥=R +g +� +√q · +m +� +i=1 +ξi · σ(w⊤fj(xi)) +� ++ Eξ +sup +j∈[q], f∈F +w: ∥w∥=R +g +� +−√q · +m +� +i=1 +ξi · σ(w⊤fj(xi)) +� += 2Eξ +sup +j∈[q], f∈F +w: ∥w∥=R +g +� +√q · +m +� +i=1 +ξi · σ(w⊤fj(xi)) +� +, +where the last equality follows from the symmetry in the distribution of the ξi random variables. By Equation 4.20 +in (Ledoux & Talagrand, 1991), the right-hand side can be upper bounded as follows +2Eξ +sup +j∈[q], f∈F +w: ∥w∥=R +g +� +√q · +m +� +i=1 +ξi · σ(w⊤fj(xi)) +� +≤ 2Eξ +sup +j∈[q], f∈F +w: ∥w∥=R +g +� +√q · +m +� +i=1 +ξi · w⊤fj(xi) +� +≤ 2Eξ +sup +j∈[q], f∈F +w: ∥w∥=R +g +� +√q · ∥w∥ +����� +m +� +i=1 +ξi · fj(xi) +����� +� +≤ 2Eξ +sup +j∈[q], f∈F +g +� +√qR +����� +m +� +i=1 +ξi · fj(xi) +����� +� +. +Proposition B.2. Let G be a neural network architecture of depth L and let ρ > 0. Let X = {xi}m +i=1 be a set of samples. +Then, +RX(FG,ρ) ≤ +ρ +m · +� +1 + +� +2L log(2deg(G)) +� +· +� +� +� +� max +j1,...,jL +L +� +l=1 +|pred(l, jL−l)| · +m +� +i=1 +∥z0 +jL(xi)∥2, + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +where the maximum is taken over j1, . . . , jL, such that, jL−l+1 ∈ pred(l, jL−l) for all l ∈ [L]. +Proof. Due to the homogeneity of the ReLU function, each function fw ∈ FG,ρ can be rewritten as f ˆ +w, where ˆwL +1 := ρ +wL +1 +∥wL +1 ∥2 +and for all l < L and jl ∈ [dl], ˆwl +jl := +wl +jl +maxj∈[dl] ∥wl +j∥F . In particular, we have FG,ρ ⊂ ˆFG,ρ := {fw | ∥wL +1 ∥2 ≤ ρ and ∀i < +L, jl ∈ [dl] : ∥wl +jl∥F ≤ 1} since the ReLU function is homogeneous. For simplicity, we denote by f ˜ +w an arbitrary member +of ˜FG,ρ and ˆwl +jl the weights of the jlth neuron of the lth layer. In addition, we denote vl +j1(xi) = (zl +j2(xi))j2∈pred(L,j1) and +zl +j(xi) = σ( ˆwl +jvl−1 +j +(xi)) and we denote j0 = 1. +We apply Jensen’s inequality, +mR := mRX( ˆFG,ρ) = Eξ +� +sup +ˆ +w +m +� +i=1 +ξif ˆ +w (xi) +� +≤ +1 +λ log Eξ sup +ˆ +w +exp +� +λ +m +� +i=1 +ξif ˆ +w (xi) +� +, +where the supremum is taken over the weights ˆwl +jl (l ∈ [L], jl ∈ [dl]) that are described above. Since ∥ ˆwL +j0∥2 ≤ ρ, we have +mR ≤ +1 +λ log Eξ sup +ˆ +w +exp +� +λ +m +� +i=1 +ξif ˆ +w (xi) +� += +1 +λ log Eξ sup +ˆ +w +exp +� +λ +����� +m +� +i=1 +ξi · ˆwL +j0 · vL−1 +j0 +(xi) +����� +� +≤ +1 +λ log +� +� +�Eξ sup +ˆ +w +exp +� +� +�λρ · +� +� +� +� +����� +m +� +i=1 +ξi · vL−1 +j0 +(xi) +����� +2 +� +� +� +� +� +� . +Next, we use Lemma 3.4, +mR ≤ +1 +λ log +� +�Eξ sup +ˆ +w +exp +� +�λρ · +� +� +� +� +� +j1∈pred(L,j0) +����� +m +� +i=1 +ξi · zL−1 +j1 +(xi) +����� +2� +� +� +� += +1 +λ log +� +�Eξ sup +ˆ +w +exp +� +�λρ · +� +� +� +� +� +j1∈pred(L,j0) +����� +m +� +i=1 +ξi · ˆwL−1 +j1 +· σ(vL−2 +j1 +(xi)) +����� +2� +� +� +� +≤ +1 +λ log +� +� +�2Eξ sup +j1, ˆ +w +exp +� +� +�λρ · +� +� +� +�|pred(L, j0)| · +����� +m +� +i=1 +ξi · vL−2 +j1 +(xi) +����� +2 +� +� +� +� +� +� += +1 +λ log +� +�2Eξ sup +j1, ˆ +w +exp +� +�λρ · +� +� +� +�|pred(L, j0)| +� +j2∈pred(L−1,j1) +· +����� +m +� +i=1 +ξi · zL−2 +j2 +(xi) +����� +2� +� +� +� , +where the supremum is taken over the parameters of f ˆ +w and j1 ∈ pred(L, j0). By applying this process recursively L times, +we obtain the following inequality, +mR = Eξ +� +sup +ˆ +w +m +� +i=1 +ξif ˆ +w (xi) +� +≤ +1 +λ log +� +�2LEξ +sup +j1,...,jL +exp +� +�λρ · +� +� +� +� +L +� +l=1 +|pred(l, jL−l)| · +����� +m +� +i=1 +ξi · z0 +jL(xi) +����� +� +� +� +� , +(9) + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +where the supremum is taken over j1, . . . , jL, such that, jl+1 ∈ pred(l, jL−l). We notice that +Eξ +sup +j1,...,jL +exp +� +�λρ · +� +� +� +� +L +� +l=1 +|pred(l, jL−l)| · +����� +m +� +i=1 +ξi · z0 +jL(xi) +����� +� +� +≤ +� +j1,...,jL +Eξ exp +� +�λρ · +� +� +� +� +L +� +l=1 +|pred(l, jL−l)| · +����� +m +� +i=1 +ξi · z0 +jL(xi) +����� +� +� +≤ deg(G)L · max +j1,...,jL Eξ exp +� +�λρ · +� +� +� +� +L +� +l=1 +|pred(l, jL−l)| · +����� +m +� +i=1 +ξi · z0 +jL(xi) +����� +� +� . +(10) +Following the proof of Theorem 1 in (Golowich et al., 2017), by applying Jensen’s inequality and Theorem 6.2 in (Boucheron +et al., 2013) we obtain that for any α > 0, +Eξ exp +� +α +����� +m +� +i=1 +ξi · z0 +jL(xi) +����� +� +≤ exp +� +�α2 �m +i=1 ∥z0 +jL(xi)∥2 +2 ++ α +� +� +� +� +m +� +i=1 +∥z0 +jL(xi)∥2 +� +� . +(11) +Hence, by combining equations 9-11 with α = λρ · +��L +l=1 |pred(l, jL−l)|, we obtain that +mR = Eξ +� +sup +f∈F +m +� +i=1 +ξif (xi) +� +≤ 1 +λ log +� +�(2deg(G))L · max +j1,...,jL Eξ exp +� +�λρ · +� +� +� +� +L +� +l=1 +|pred(l, jL−l)| · +����� +m +� +i=1 +ξi · z0 +jL(xi) +����� +� +� +� +� += 1 +λ max +j1,...,jL log +� +�(2deg(G))L · Eξ exp +� +�λρ · +� +� +� +� +L +� +l=1 +|pred(l, jL−l)| · +����� +m +� +i=1 +ξi · z0 +jL(xi) +����� +� +� +� +� +≤ log(2deg(G))L +λ ++ +λρ2 max +j1,...,jL +�L +l=1 |pred(l, jL−l)| · �m +i=1 ∥z0 +jL(xi)∥2 +2 ++ ρ +� +� +� +� max +j1,...,jL +L +� +l=1 +|pred(l, jL−l)| · +m +� +i=1 +∥z0 +jL(xi)∥2 +The choice λ = +� +2 log(2deg(G))L +ρ2 maxj1,...,jL +�L +l=1 |pred(l,jL−l)|·�m +i=1 ∥z0 +jL(xi)∥2 , yields the desired inequality. +Theorem B.3. Let P be a distribution over Rc0d0 × {±1}. Let S = {(xi, yi)}m +i=1 be a dataset of i.i.d. samples selected +from P. Then, with probability at least 1 − δ over the selection of S, for any fw ∈ FG,ρ that perfectly fits the data (for all +i ∈ [m] : fw(xi) = yi), we have +errP (fw) ≤ (ρ(w) + 1) +m +� +1 + +� +2L log(2deg(G)) +� +· +� +� +� +� max +j1,...,jL +L +� +l=1 +|pred(l, jL−l)| +m +� +i=1 +∥z0 +jL(xi)∥2 + 3 +� +log(2(ρ(w) + 2)2/δ) +2m +where the maximum is taken over j1, . . . , jL, such that, jL−l+1 ∈ pred(l, jL−l) for all l ∈ [L]. +Proof. Let t ∈ N ∪ {0} and Gt = FG,ρ. By Lemma 2.2, with probability at least 1 − +δ +t(t+1), for any function fw ∈ Gt that +perfectly fits the training data, we have +errP (fw) ≤ 2RX(Gt) + 3 +� +log(2(t + 1)2/δ) +2m +. +(12) + +Norm-based Generalization Bounds for Compositionally Sparse Neural Networks +By Proposition 3.1, we have +RX(Gt) ≤ t · +� +1 + +� +2L log(2deg(G)) +� +· +� +� +� +� max +j1,...,jL +L +� +l=1 +|pred(l, jL−l)| · +m +� +i=1 +∥z0 +jL(xi)∥2 +(13) +because of the union bound over all t ∈ N, equation 3 holds uniformly for all t ∈ N and fw ∈ Gt with probability at least +1 − δ. For each fw with norm ρ(w) we then apply the bound with t = ⌈ρ(w)⌉ since fw ∈ Gt, and obtain, +errP (fw) ≤ +t +� +1 + +� +2L log(2deg(G)) +� � +max +j1,...,jL +�L +l=1 |pred(l, jL−l)| �m +i=1 ∥z0 +jL(xi)∥2 +m ++ 3 +� +log(2(t + 1)2/δ) +2m +≤ +(ρ(w) + 1) +� +1 + +� +2L log(2deg(G)) +� � +max +j1,...,jL +�L +l=1 |pred(l, jL−l)| �m +i=1 ∥z0 +jL(xi)∥2 +m ++ 3 +� +log(2(ρ(w) + 2)2/δ) +2m +, +which proves the desired bound. + diff --git a/T9FLT4oBgHgl3EQfQi86/content/tmp_files/load_file.txt b/T9FLT4oBgHgl3EQfQi86/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..00dff8fc44136d550d249163d4dc444d74ec5bec --- /dev/null +++ b/T9FLT4oBgHgl3EQfQi86/content/tmp_files/load_file.txt @@ -0,0 +1,1861 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf,len=1860 +page_content='Norm-based Generalization Bounds for Compositionally Sparse Neural Networks Tomer Galanti 1 Mengjia Xu 1 2 Liane Galanti 3 Tomaso Poggio 1 Abstract In this paper, we investigate the Rademacher com- plexity of deep sparse neural networks, where each neuron receives a small number of inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' We prove generalization bounds for multilayered sparse ReLU neural networks, including convo- lutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' These bounds differ from previous ones, as they consider the norms of the convolutional filters instead of the norms of the associated Toeplitz matrices, independently of weight sharing between neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' As we show theoretically, these bounds may be orders of magnitude better than standard norm- based generalization bounds and empirically, they are almost non-vacuous in estimating general- ization in various simple classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' Taken together, these results suggest that compo- sitional sparsity of the underlying target function is critical to the success of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' Introduction Over the last decade, deep learning with large neural net- works has greatly advanced the solution of a wide range of tasks including image classification (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' Doso- vitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=', 2021), language process- ing (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=', 2020), interacting with open-ended environments (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' Arulkumaran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=', 2019), and code synthe- sis (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' Despite traditional theories (Vapnik, 1998), recent findings (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' Belkin, 2021) show that deep neural networks can generalize well even when their size far exceeds the number of training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' To address this question, recent efforts in deep learning the- ory study the generalization performance of deep networks by analyzing the complexity of the learned function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' Equal contribution 1Massachusetts Institute of Technol- ogy 2Brown University 3Tel-Aviv University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9FLT4oBgHgl3EQfQi86/content/2301.12033v1.pdf'} +page_content=' Correspon- dence to: Tomer Galanti b, we use the asymmetric probabilistic +method to derive sufficient conditions for the existence of an Eulerian extension +of G with m edges. +The paper is organized as follows: In Section 2, we state and prove our main +result regarding Eulerian extensions with edge constraints. +2 +Edge Constrained Eulerian Extensions +Let G = (V, E) be a graph with vertex set V and edge set E. The vertices u +and v are said to be adjacent in G if the edge (u, v) with endvertices u and v is +present in E. We define dG(v) to be the degree of vertex v, i.e., the number of +vertices adjacent to v in G. +A sequence of vertices W := (u1, u2, . . . , ut) is said to be a walk if ui is +adjacent to ui+1 for each 1 ≤ i ≤ t − 1. If in addition the vertex ut is also +adjacent to u1, then W is said to be a circuit. We say that W is an Eulerian +⋆ Corresponding Author + +2 +G. Ganesan +circuit if each edge of the graph G occurs exactly once in W. The graph G is +said to be an Eulerian graph if G contains an Eulerian circuit. +Let G be any graph. We say that a graph H is an Eulerian extension of G +if G and H share the same vertex set, G is a subgraph of H and H is Eulerian. +Definition 1. For an integer m ≥ 1, we say that a graph G is m-Eulerian +extendable if there exists an Eulerian extension H of G containing exactly m +edges. +We have the following result regarding m-Eulerian extendability. Through- +out, constants do not depend on n. +Theorem 1. For every pair of constants 0 < α, β < 1 satisfying β + 40α2 < 1 +2 +strictly, there exists a constant N = N(α, β) ≥ 1 such that the following holds +for all n ≥ N: Let m be any integer satisfying +2n ≤ m ≤ α · n +3 +2 +(2.1) +and let G ⊂ Kn be any connected graph containing n vertices, b edges and a +maximum vertex degree ∆. If +∆ ≤ β · n and b ≤ m − n, +(2.2) +then G is m-Eulerian extendable. +To see the necessity of the bound b ≤ m − n, we use the fact that a graph H +is Eulerian if and only if H is connected and each vertex of H has even degree +(Theorem 1.2.26, pp. 27, West (2001)). Therefore to obtain an Eulerian extension +of G, we only need to convert all odd degree vertices into even degree vertices. +Suppose that n is even and all the vertices in G have an odd degree. Because +the degree of each vertex is at most n +2 − 1 (see (2.2)) the sum of neighbourhood +sizes of 2i − 1 and 2i is at most n − 2. Therefore for each 1 ≤ i ≤ n +2 , there exists +a vertex wi neither adjacent to 2i − 1 nor adjacent to 2i in the graph G. Adding +the n edges {(wi, 2i − 1), (wi, 2i)}1≤i≤ n +2 gives us an Eulerian extension of G. +In our proof of Theorem 1 below, we use the asymmetric probabilistic method +for higher values of m to obtain walks of predetermined lengths between pairs +of odd degree vertices and thereby construct the desired extension, +Proof of Theorem 1 +As before, we use the fact that a graph H is Eulerian if and only if H is connected +and each vertex of H has even degree. We assume that the vertex set of G is V := +{0, 1, 2, . . ., n − 1} and also let T be the set of all odd degree vertices in G so +that the number of odd degree vertices #T , is even. If there are vertices u, v ∈ T +that are not adjacent to each other in G, then we mark the edge (u, v) and also +the endvertices u and v. We then pick two new non-adjacent vertices x and y +in T \ {u, v} and repeat the procedure. We continue this process until we reach +one of the following two scenarios: Either the number of marked edges is m−b in + +Edge Constrained Eulerian Extensions +3 +which case, we simply add the marked edges to G and get the desired Eulerian +extension H. Or, we are left with a set of marked edges of cardinality, say l and +a clique C := {u1, . . . , u2z} ⊂ T containing 2z unmarked vertices. +Let G0 be the graph obtained by adding all the l ≤ n +2 marked edges to G. +If ∆0 and b0 denote the maximum vertex degree and the number of edges in G0, +respectively, then +∆0 ≤ ∆ + 1 and b0 = b + l ≤ b + n. +(2.3) +We now pair the vertices in C as {u2i−1, u2i}1≤i≤z assuming that z ≥ 1 (If +not, we simply remove a marked edge e from G0 and label the endvertices +of e as u1 and u2). We use the probabilistic method to obtain z edge-disjoint +walks {Wi}1≤i≤z containing no edge of G0 such that each walk Wi has w edges +and u2i−1 and u2i as endvertices, where w satisfies +b0 + z · w = m. +(2.4) +Adding the walks {Wi}1≤i≤z to G0 would then give us the desired m-Eulerian +extension. +In (2.4) we have assumed for simplicity that w = m−b0 +z +is an integer. If not, we +write m−b0 = z·w+r where 0 ≤ r ≤ w−1 and construct the z−1 walks Wi, 1 ≤ +i ≤ z − 1 each of length w edges and the last walk Wz of length w + r ≤ 2w. +Again adding these walks to G0 would give us the desired Eulerian extension +with m edges. For future use we remark that the length w of each walk added +in the above process is bounded above by +w = m − b0 +z +≤ m ≤ α · n +3 +2 . +(2.5) +We begin with the pair of vertices u1 and u2. Let {Xi}1≤i≤w be independent +and identically distributed (i.i.d.) random variables uniformly distributed in the +set {0, 1, . . ., n − 1}. Letting S := (u1, X1, . . . , Xw, u2), we would like to convert +the sequence S into a walk W1 with endvertices u1 and u2 and containing no +edge of G0. The construction of W1 is split into two parts: In the first part, we +collect the preliminary relevant properties of S and in the second part, we obtain +the walk W1. +Preliminary definitions and estimates: An entry in S is defined to be a vertex +and we define (u1, X1), (Xw, u2) and {(Xi, Xi+1)}1≤i≤w−1 to be the edges of S. +The neighbour set of a vertex v in S the set of vertices u such that either (v, u) +or (u, v) appears as an edge of S. The neighbour set of v in the multigraph G0∪S +is the union of the neighbour set of v in the graph G0 and the neighbour set of v +in S. The degree of a vertex v in G0 ∪ S is defined to be the sum of the degree +of v in G0 and the degree of v in S. +The three main ingredients used in the construction of the walk W1 are: +(1) The degree of a vertex in the multigraph G0 ∪ S, +(2) the number of “bad” vertices in G0 ∪ S and +(3) the number of “bad” edges in S. +Below, we define and estimate each of the three quantities in that order. + +4 +G. Ganesan +We first estimate the degree of each vertex in the multigraph G0 ∪ S. For +any 0 ≤ v ≤ n − 1 and any 1 ≤ i ≤ w, let Ii = 11(Xi = v) be the indicator +function of the event that Xi = v. We have P(Ii = 1) = 1 +n and so if Dv = �w +i=1 Ii +denotes the number of times the entry v appears in the sequence (X1, . . . , Xw), +then EDv = w +n and so by the standard deviation estimate (A.1) in Appendix, +we have +P +� +Dv ≥ 2w +n +� +≤ 2 exp +� +− w +16n +� +. +(2.6) +If w +n ≥ 100 log n then we get from (2.6) that P(Dv ≥ 2w +n ) ≤ +1 +n2 . Else, we use +Chernoff bound directly to get that +P (Dv ≥ 100 log n) ≤ 1 +n2 . +(2.7) +Therefore setting an := max +� 2w +n , 100 log n +� +, we get that +P (Dv ≥ an) ≤ 1 +n2 . +(2.8) +If the event +Edeg := +� +0≤v≤n−1 +{Dv ≤ an} +(2.9) +occurs, then in G0 ∪ S each vertex has degree at most ∆0 + 1 + an, with the +extra term 1 to account for the fact that vertices X1 and Xw are also adjacent +to u1 and u2, respectively. By the union bound and (2.6) we therefore have +P(Edeg) ≥ 1 − 1 +n. +(2.10) +The next step is to estimate the number of “bad” vertices in G0∪S. Let X0 := +u1, Xw+1 := u2 and for 0 ≤ i ≤ w − 1, say that vertex Xi is bad if Xi = Xi+1 +or Xi = Xi+2. For simplicity define Xw to be bad always. If Ji is the indicator +function of the event that vertex Xi is bad, then for 0 ≤ i ≤ w − 1, we have that +1 +n ≤ P(Ji = 1) ≤ 2 +n. +(2.11) +The term Nv,bad := �w−1 +i=0 Ji + 1 denotes the total number of bad vertices in the +sequence S. To estimate Nv,bad we split Nv,bad −1 = J(A)+J(B)+J(C), where +J(A) = J1 + J4 + . . . , J(B) = J2 + J5 + . . . and J(C) = J3 + J6 + . . . +so that each J(u), u ∈ {A, B, C} is a sum of i.i.d. random variables. +The term J(A) contains at least w +3 − 1 and at most w +3 random variables. As +in the proof of (2.8), we use (2.11) and the standard deviation estimate (A.1) in +Appendix to obtain that +P +� +J(A) ≥ an +3 +� +≤ 1 +n2 + +Edge Constrained Eulerian Extensions +5 +for all n large. A similar estimate holds for J(B) and J(C) and so combining +these estimates and using the union bound, we get that +P (Ev,bad) ≥ 1 − 3 +n2 +(2.12) +where Ev,bad := {Nv,bad ≤ an + 1} denotes the event that the number of bad +vertices in S is at most an + 1. +The final estimate involves counting the number of bad edges in the se- +quence S. For 0 ≤ i ≤ w say that (Xi, Xi+1) is a bad edge if one of the following +two conditions hold: +(d1) Either {Xi, Xi+1} is an edge of G0 or +(d2) There exists i + 2 ≤ j ≤ w such that {Xi, Xi+1} = {Xj, Xj+1}. +To estimate the probability of occurrence of (d1), let e be an edge of G0 with +endvertices u and v. We have that +P ({Xi, Xi+1} = {u, v}) ≤ 2 +n2 . +Similarly for any i + 2 ≤ j ≤ w, the possibility (d2) also occurs with probability +at most +2 +n2 . Therefore if Li is the indicator function of the event that (Xi, Xi+1) +is a bad edge, we have that +P(Li = 1) ≤ +b0 +� +l=1 +2 +n2 + +w +� +j=i+2 +2 +n2 ≤ 2(b0 + w) +n2 +. +(2.13) +If Ne,bad := �w +i=0 Li denotes the total number of bad edges in S, then +from (2.13) and the fact that L0 ≤ 1 we have +ENe,bad ≤ 1 + 2(b0 + w)w +n2 +=: cn. +(2.14) +Letting Ee,bad := {Ne,bad ≤ K · cn} denote the event that the number of bad +edges in S is at most K · cn, for some large integer constant K ≥ 1 to be +determined later, we get from Markov inequality that +P (Ee,bad) ≥ 1 − 1 +K , +(2.15) +If Evalid denotes the event that the first and last edges (X0, X1) and (Xw, Xw+1) +are valid edges not in G0, then using the fact that the degree of any vertex in G0 +is at most n +2 (see (2.3) and (2.2) in the statement of the Theorem) we get that +P(Evalid) ≥ +�1 +2 − 1 +n +�2 +. +(2.16) +Defining the joint event +Ejoint := Evalid ∩ Edeg ∩ Ev,bad ∩ Ee,bad + +6 +G. Ganesan +and using +P +� +A +� +l� +i=1 +Bi +� +≥ P(A) − P +� l� +i=1 +Bc +i +� +≥ P(A) − +l +� +i=1 +P (Bc +i ) . +(2.17) +with A = Evalid we get from (2.10), (2.12), (2.15) and (2.16) that +P(Ejoint) ≥ +�1 +2 − 1 +n +�2 +− 1 +K − 1 +n − 3 +n2 ≥ 1 +21 +(2.18) +for all n large, provided the constant K = 5, which we fix henceforth. This com- +pletes the preliminary estimates used in the construction of the walk W1. +Construction of the walk W1: Assuming that the event Ejoint occurs, we now +convert S0 := S into a walk W1. We begin by “correcting” all bad vertices. +Let Xi1, Xi2, . . . , Xit, i1 < i2 < . . . < it be the set of all bad vertices. Thus for +example either Xi1 = Xi1+1 or Xi1 = Xi1+2. Because the event Edeg occurs, we +get from the discussion following (2.9) that the degree of each vertex in G0 ∪ S0 +is at most ∆0 + an + 1. From (2.3) and the first condition in (2.2) we get that +∆0 ≤ ∆ + 1 ≤ n +3 + 1 +(2.19) +and from the definition of an prior to (2.8) and the upper bound w ≤ n +3 +2 in (2.5), +we get that +an = max +� +100 log n, 2w +n +� +≤ 100 log n + 2w +n ≤ 100 log n + 2√n ≤ 3√n (2.20) +for all n large. Consequently, using β < 1 +2 strictly (see statement of Theorem 1) +, +∆0 + an + 1 ≤ β · n + 1 + 3√n ≤ n +2 − 5 +(2.21) +for all n large. From (2.21), we therefore get that there exists a vertex v1 that +is not a neighbour of Xi1 in G0 ∪ S. Similarly, the total number of neighbours +of v1 and Xi1+3 in G0 ∪ S is at most +2∆0 + 2an + 2 ≤ 2β · n + 2 + 6√n < n − 10 +(2.22) +for all n large and so there exists a vertex v2 ̸= Xi1 that is not a neighbour of v1 +and also not a neighbour of Xi1+3 in G0 ∪ S. +We now set X(1) +i1+1 = v1 and X(1) +i1+2 = v2 and X(1) +j += Xj for j ̸= i1 + 1, i1 + 2 +and call the resulting sequence as S1 := (X(1) +1 , . . . , X(1) +w ). By construction the +degree of each vertex in the multigraph G0 ∪ S1 is at most ∆ + an + 1 + 2 and + +Edge Constrained Eulerian Extensions +7 +there are at most t − 1 bad vertices in S1. We now pick the bad vertex with the +least index in S1 and repeat the above procedure with S1 to get a sequence S2 +containing at most t − 2 bad vertices. +After k ≤ t iterations of the above procedure, the degree of each vertex in +the multigraph G0 ∪ Sk would be at most +∆0 + an + 1 + 2k ≤ ∆0 + an + 1 + 2t ≤ ∆0 + 3an + 3 +(2.23) +because the event Ejoint ⊆ Ev,bad occurs and so t ≤ an + 1. Again using (2.19) +and (2.20) and arguing as in (2.22), we get that the sum of the degrees of any +two vertices in G0 ∪St is at most n−10 for all n large. Thus the above procedure +indeed proceeds for t iterations and by construction, the sequence St obtained +at the end, has no bad vertices. +We now perform an analogous procedure for correcting all bad edges in St. For +example if (Xl, Xl+1) is a bad edge in St, then following an analogous argument +as before we pick a vertex Yl+1 that is neither adjacent to Xl nor adjacent to Xl+2 +in the sequence St. We replace Xl+1 with Yl+1 to get a new sequence St+1. In the +union G0 ∪St+1 the degree of each vertex is at most ∆0 +3an +3+2 (see (2.23)) +and the number of bad edges is at most r−1. At the end of r ≤ K ·cn iterations, +we obtain a multigraph G0 ∪ St+r, where the degree of each vertex is at most +∆0 + 3an + 3 + 2r ≤ ∆0 + 3an + 3 + 2Kcn, +since the event Ee,bad occurs and therefore Ne,bad ≤ K·cn (see discussion preced- +ing (2.15)). Substituting the expression for cn from (2.14) and using the second +estimate for an in (2.20), we get that ∆0 + 3an + 3 + 2Kcn is at most +∆0 + 300 log n + 6w +n + 3 + 2K + 2K(b0 + w)w +n2 +≤ +∆0 + 301 log n + 6w +n + 2K(b0 + w)w +n2 +(2.24) +for all n large. Recalling that u1 and u2 are the endvertices of the starting se- +quence S0, we get that the final sequence St+r contains no bad edge and is +therefore the desired walk W1 with endvertices u1 and u2. This completes the +construction of the walk W1. +Rest of the walks: We now repeat the above procedure to construct the rest of +the walks. We set G1 := G0 ∪ W1 and argue as above to obtain a walk W2 +with w edges present in G1 and containing u3 and u4 as endvertices. Adding +the walk W2 to G1 we get a new graph G2. In effect, to the graph G1 contain- +ing b0+w edges, we have added w edges and by an argument analogous to (2.24), +we have increased the degree of a vertex by at most +301 log n + 6w +n + 2K(b0 + 2w)w +n2 +, +in obtaining the graph G2. We recall that (see first paragraph of the proof) there +are z such walks to be created of which z − 1 have length w and the final walk + +8 +G. Ganesan +has length at most 2w. Therefore after z iterations, we get a graph Gz with m +edges and whose maximum vertex degree ∆z is at most +∆z ≤ ∆0 + +� +301 log n + 6w +n +� +· (z − 1) + 301 log n + 12w +n ++ 2K +z−1 +� +k=1 +(b0 + k · w)w +n2 ++ 2K (b0 + (z − 1) · w)2w +n2 +. +(2.25) +By construction Gz is an Eulerian graph. +To verify the obtainability of Gz, we estimate ∆z as follows. The term z is no +more than the size of a maximum clique in the original graph G (see discussion +prior to (2.3)) and since there are m ≤ n +3 +2 edges in G, the maximum size of a +clique in G is at most n +3 +4 . Therefore +z ≤ n +3 +4 . +(2.26) +Also using (2.4) and (2.2), we get that zw ≤ m ≤ α · n +3 +2 and so +wz +n ≤ α · √n ≤ √n +(2.27) +Finally from (2.3) we have that b0 ≤ b + n and so the second line in (2.25) is at +most +z +� +k=1 +(b + n + k · w)w +n2 +≤ z(b + n + zw)2w +n2 +≤ 2m(n + m) +n2 +≤ 2√n + 2m2 +n2 +≤ √n + 2α2 · n +(2.28) +where the second inequality in (2.28) follows from the estimate b + zw ≤ b0 + +zw = m (see (2.4)), the third and fourth estimates in (2.28) follow from the +bound m ≤ α · n +3 +2 (see (2.1)). +Plugging (2.28), (2.27) and (2.26) into (2.25) we get that +∆z ≤ ∆0 + 301n +3 +4 · log n + √n +�12 +10 + 2K +� ++ 4Kα2 · n +≤ (β + 4Kα2) · n + 1 + 301n +3 +4 · log n + √n +�12 +10 + 2K +� +(2.29) +for all n large, where the second inequality in (2.29) is obtained by using +∆0 ≤ ∆ + 1 ≤ β · n + 1 (see (2.3) and the first condition in (2.2)). From the +statement of Theorem 1 and using K = 5, we have that +β + 4Kα2 = β + 20α2 < 1 +2 + +Edge Constrained Eulerian Extensions +9 +strictly and so the degree of any vertex in Gz is strictly less than n +2 and also, +the sum of degrees of any two vertices in Gz is at most +(2β + 40α2) · n + 3 + 602n +3 +4 · log n + 12√n +10 +< n − 10 +for all n large. Thus the graph Gz can be obtained by the above probabilistic +method as in the discussion following (2.21). +Appendix +Throughout we use the following deviation estimate. Let Zi, 1 ≤ i ≤ t be inde- +pendent Bernoulli random variables satisfying +P(Zi = 1) = pi = 1 − P(Zi = 0). +If Wt = �t +i=1 Zi and µt = EWt, then for any 0 < ǫ < 1 +2 we have that +P (|Wt − µt| ≥ ǫµt) ≤ 2 exp +� +−ǫ2 +4 µt +� +. +(A.1) +For a proof of (A.1), we refer to Corollary A.1.14, pp. 312, Alon and Spencer +(2008). +Acknowledgement +I thank Professors V. Raman, C. R. Subramanian and the referees for crucial +comments that led to an improvement of the paper. I also thank IMSc for my +fellowships. +References +1. N. Alon and J. Spencer. (2008). The Probabilisitic Method. Wiley Interscience. +2. F. T. Boesch, C. Suffel and R. Tindell. (1977). The Spanning Subgraph of Eulerian +Graphs, Journal of Graph Theory, 1, pp. 79–84. +3. F. Dorn, H. Moser, R. Niedermeier and M. Weller. (2013). Efficient Algorithms +for Eulerian Extension and Rural Postman Problem. SIAM Journal on Discrete +Mathematics, 27, pp. 75–94. +4. F. V. Fomin and P. A. Golovach. (2012). Parameterized Complexity of Connected +Even/Odd Subgraph Problems. Proceedings 29th STACS, 14, pp. 432–440. +5. W. H¨ohn, T. Jacobs, and N. Megow. (2012). On Eulerian Extensions and Their +Application to No-Wait Flowshop Scheduling. Journal of Scheduling, 15, pp. 295– +309. +6. L. Lesniak and O. R. Oellermann. (1986). An Eulerian Exposition. Journal of +Graph Theory, 10 (1986), pp. 277–297. +7. D. B. West. (2001). Introduction to Graph Theory. Prentice Hall. + diff --git a/UtE5T4oBgHgl3EQfBA5d/content/tmp_files/load_file.txt b/UtE5T4oBgHgl3EQfBA5d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..50ba67eb2cae57a14e16ff79f8371c8f848d0fc7 --- /dev/null +++ b/UtE5T4oBgHgl3EQfBA5d/content/tmp_files/load_file.txt @@ -0,0 +1,328 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf,len=327 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='05383v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='CO] 13 Jan 2023 Edge Constrained Eulerian Extensions Ghurumuruhan Ganesan ⋆ IISER Bhopal, gganesan82@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='com Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' In this paper we study Eulerian extensions with edge con- straints and use the probabilistic method to establish sufficient condi- tions for a given connected graph to be a subgraph of a Eulerian graph containing m edges, for a given number m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Keywords: Eulerian Extensions, Edge Constraint, Probabilistic Method 1 Introduction In the Eulerian extension problem, a given graph is to be converted into an Eulerian graph by addition of as few edges as possible and such problems have applications in routing and scheduling (Dorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Boesch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (1977) studied conditions under which a graph G can be extended to an Eulerian graph and later Lesniak and Oellermann (1986) presented a detailed survey on sub- graphs and supergraphs of Eulerian graphs and multigraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' For applications of Eulerian extensions to scheduling and parametric aspects, we refer to H¨ohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2012) and Fomin and Golovach (2012), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' In this paper, we construct Eulerian extension of graphs with a predetermined number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Specifically, given a graph G with maximum degree ∆ and b number of edges and given an integer m > b, we use the asymmetric probabilistic method to derive sufficient conditions for the existence of an Eulerian extension of G with m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' The paper is organized as follows: In Section 2, we state and prove our main result regarding Eulerian extensions with edge constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' 2 Edge Constrained Eulerian Extensions Let G = (V, E) be a graph with vertex set V and edge set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' The vertices u and v are said to be adjacent in G if the edge (u, v) with endvertices u and v is present in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We define dG(v) to be the degree of vertex v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=', the number of vertices adjacent to v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' A sequence of vertices W := (u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' , ut) is said to be a walk if ui is adjacent to ui+1 for each 1 ≤ i ≤ t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' If in addition the vertex ut is also adjacent to u1, then W is said to be a circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We say that W is an Eulerian ⋆ Corresponding Author 2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Ganesan circuit if each edge of the graph G occurs exactly once in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' The graph G is said to be an Eulerian graph if G contains an Eulerian circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Let G be any graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We say that a graph H is an Eulerian extension of G if G and H share the same vertex set, G is a subgraph of H and H is Eulerian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' For an integer m ≥ 1, we say that a graph G is m-Eulerian extendable if there exists an Eulerian extension H of G containing exactly m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We have the following result regarding m-Eulerian extendability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Through- out, constants do not depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' For every pair of constants 0 < α, β < 1 satisfying β + 40α2 < 1 2 strictly, there exists a constant N = N(α, β) ≥ 1 such that the following holds for all n ≥ N: Let m be any integer satisfying 2n ≤ m ≤ α · n 3 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='1) and let G ⊂ Kn be any connected graph containing n vertices, b edges and a maximum vertex degree ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' If ∆ ≤ β · n and b ≤ m − n, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='2) then G is m-Eulerian extendable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' To see the necessity of the bound b ≤ m − n, we use the fact that a graph H is Eulerian if and only if H is connected and each vertex of H has even degree (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='26, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' 27, West (2001)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Therefore to obtain an Eulerian extension of G, we only need to convert all odd degree vertices into even degree vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Suppose that n is even and all the vertices in G have an odd degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Because the degree of each vertex is at most n 2 − 1 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='2)) the sum of neighbourhood sizes of 2i − 1 and 2i is at most n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Therefore for each 1 ≤ i ≤ n 2 , there exists a vertex wi neither adjacent to 2i − 1 nor adjacent to 2i in the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Adding the n edges {(wi, 2i − 1), (wi, 2i)}1≤i≤ n 2 gives us an Eulerian extension of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' In our proof of Theorem 1 below, we use the asymmetric probabilistic method for higher values of m to obtain walks of predetermined lengths between pairs of odd degree vertices and thereby construct the desired extension, Proof of Theorem 1 As before, we use the fact that a graph H is Eulerian if and only if H is connected and each vertex of H has even degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We assume that the vertex set of G is V := {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=', n − 1} and also let T be the set of all odd degree vertices in G so that the number of odd degree vertices #T , is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' If there are vertices u, v ∈ T that are not adjacent to each other in G, then we mark the edge (u, v) and also the endvertices u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We then pick two new non-adjacent vertices x and y in T \\ {u, v} and repeat the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We continue this process until we reach one of the following two scenarios: Either the number of marked edges is m−b in Edge Constrained Eulerian Extensions 3 which case, we simply add the marked edges to G and get the desired Eulerian extension H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Or, we are left with a set of marked edges of cardinality, say l and a clique C := {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' , u2z} ⊂ T containing 2z unmarked vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Let G0 be the graph obtained by adding all the l ≤ n 2 marked edges to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' If ∆0 and b0 denote the maximum vertex degree and the number of edges in G0, respectively, then ∆0 ≤ ∆ + 1 and b0 = b + l ≤ b + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='3) We now pair the vertices in C as {u2i−1, u2i}1≤i≤z assuming that z ≥ 1 (If not, we simply remove a marked edge e from G0 and label the endvertices of e as u1 and u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We use the probabilistic method to obtain z edge-disjoint walks {Wi}1≤i≤z containing no edge of G0 such that each walk Wi has w edges and u2i−1 and u2i as endvertices, where w satisfies b0 + z · w = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='4) Adding the walks {Wi}1≤i≤z to G0 would then give us the desired m-Eulerian extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='4) we have assumed for simplicity that w = m−b0 z is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' If not, we write m−b0 = z·w+r where 0 ≤ r ≤ w−1 and construct the z−1 walks Wi, 1 ≤ i ≤ z − 1 each of length w edges and the last walk Wz of length w + r ≤ 2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Again adding these walks to G0 would give us the desired Eulerian extension with m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' For future use we remark that the length w of each walk added in the above process is bounded above by w = m − b0 z ≤ m ≤ α · n 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='5) We begin with the pair of vertices u1 and u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Let {Xi}1≤i≤w be independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=') random variables uniformly distributed in the set {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=', n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Letting S := (u1, X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' , Xw, u2), we would like to convert the sequence S into a walk W1 with endvertices u1 and u2 and containing no edge of G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' The construction of W1 is split into two parts: In the first part, we collect the preliminary relevant properties of S and in the second part, we obtain the walk W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Preliminary definitions and estimates: An entry in S is defined to be a vertex and we define (u1, X1), (Xw, u2) and {(Xi, Xi+1)}1≤i≤w−1 to be the edges of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' The neighbour set of a vertex v in S the set of vertices u such that either (v, u) or (u, v) appears as an edge of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' The neighbour set of v in the multigraph G0∪S is the union of the neighbour set of v in the graph G0 and the neighbour set of v in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' The degree of a vertex v in G0 ∪ S is defined to be the sum of the degree of v in G0 and the degree of v in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' The three main ingredients used in the construction of the walk W1 are: (1) The degree of a vertex in the multigraph G0 ∪ S, (2) the number of “bad” vertices in G0 ∪ S and (3) the number of “bad” edges in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Below, we define and estimate each of the three quantities in that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' 4 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Ganesan We first estimate the degree of each vertex in the multigraph G0 ∪ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' For any 0 ≤ v ≤ n − 1 and any 1 ≤ i ≤ w, let Ii = 11(Xi = v) be the indicator function of the event that Xi = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We have P(Ii = 1) = 1 n and so if Dv = �w i=1 Ii denotes the number of times the entry v appears in the sequence (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' , Xw), then EDv = w n and so by the standard deviation estimate (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='1) in Appendix, we have P � Dv ≥ 2w n � ≤ 2 exp � − w 16n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='6) If w n ≥ 100 log n then we get from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='6) that P(Dv ≥ 2w n ) ≤ 1 n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Else, we use Chernoff bound directly to get that P (Dv ≥ 100 log n) ≤ 1 n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='7) Therefore setting an := max � 2w n , 100 log n � , we get that P (Dv ≥ an) ≤ 1 n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='8) If the event Edeg := � 0≤v≤n−1 {Dv ≤ an} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='9) occurs, then in G0 ∪ S each vertex has degree at most ∆0 + 1 + an, with the extra term 1 to account for the fact that vertices X1 and Xw are also adjacent to u1 and u2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' By the union bound and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='6) we therefore have P(Edeg) ≥ 1 − 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='10) The next step is to estimate the number of “bad” vertices in G0∪S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Let X0 := u1, Xw+1 := u2 and for 0 ≤ i ≤ w − 1, say that vertex Xi is bad if Xi = Xi+1 or Xi = Xi+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' For simplicity define Xw to be bad always.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' If Ji is the indicator function of the event that vertex Xi is bad, then for 0 ≤ i ≤ w − 1, we have that 1 n ≤ P(Ji = 1) ≤ 2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='11) The term Nv,bad := �w−1 i=0 Ji + 1 denotes the total number of bad vertices in the sequence S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' To estimate Nv,bad we split Nv,bad −1 = J(A)+J(B)+J(C), where J(A) = J1 + J4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' , J(B) = J2 + J5 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' and J(C) = J3 + J6 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' so that each J(u), u ∈ {A, B, C} is a sum of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' The term J(A) contains at least w 3 − 1 and at most w 3 random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' As in the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='8), we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='11) and the standard deviation estimate (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='1) in Appendix to obtain that P � J(A) ≥ an 3 � ≤ 1 n2 Edge Constrained Eulerian Extensions 5 for all n large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' A similar estimate holds for J(B) and J(C) and so combining these estimates and using the union bound, we get that P (Ev,bad) ≥ 1 − 3 n2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='12) where Ev,bad := {Nv,bad ≤ an + 1} denotes the event that the number of bad vertices in S is at most an + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' The final estimate involves counting the number of bad edges in the se- quence S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' For 0 ≤ i ≤ w say that (Xi, Xi+1) is a bad edge if one of the following two conditions hold: (d1) Either {Xi, Xi+1} is an edge of G0 or (d2) There exists i + 2 ≤ j ≤ w such that {Xi, Xi+1} = {Xj, Xj+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' To estimate the probability of occurrence of (d1), let e be an edge of G0 with endvertices u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We have that P ({Xi, Xi+1} = {u, v}) ≤ 2 n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Similarly for any i + 2 ≤ j ≤ w, the possibility (d2) also occurs with probability at most 2 n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Therefore if Li is the indicator function of the event that (Xi, Xi+1) is a bad edge, we have that P(Li = 1) ≤ b0 � l=1 2 n2 + w � j=i+2 2 n2 ≤ 2(b0 + w) n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='13) If Ne,bad := �w i=0 Li denotes the total number of bad edges in S, then from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='13) and the fact that L0 ≤ 1 we have ENe,bad ≤ 1 + 2(b0 + w)w n2 =: cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='14) Letting Ee,bad := {Ne,bad ≤ K · cn} denote the event that the number of bad edges in S is at most K · cn, for some large integer constant K ≥ 1 to be determined later, we get from Markov inequality that P (Ee,bad) ≥ 1 − 1 K , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='15) If Evalid denotes the event that the first and last edges (X0, X1) and (Xw, Xw+1) are valid edges not in G0, then using the fact that the degree of any vertex in G0 is at most n 2 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='2) in the statement of the Theorem) we get that P(Evalid) ≥ �1 2 − 1 n �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='16) Defining the joint event Ejoint := Evalid ∩ Edeg ∩ Ev,bad ∩ Ee,bad 6 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Ganesan and using P � A � l� i=1 Bi � ≥ P(A) − P � l� i=1 Bc i � ≥ P(A) − l � i=1 P (Bc i ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='17) with A = Evalid we get from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='10), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='12), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='16) that P(Ejoint) ≥ �1 2 − 1 n �2 − 1 K − 1 n − 3 n2 ≥ 1 21 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='18) for all n large, provided the constant K = 5, which we fix henceforth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' This com- pletes the preliminary estimates used in the construction of the walk W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Construction of the walk W1: Assuming that the event Ejoint occurs, we now convert S0 := S into a walk W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We begin by “correcting” all bad vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Let Xi1, Xi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' , Xit, i1 < i2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' < it be the set of all bad vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Thus for example either Xi1 = Xi1+1 or Xi1 = Xi1+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Because the event Edeg occurs, we get from the discussion following (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='9) that the degree of each vertex in G0 ∪ S0 is at most ∆0 + an + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='3) and the first condition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='2) we get that ∆0 ≤ ∆ + 1 ≤ n 3 + 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='19) and from the definition of an prior to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='8) and the upper bound w ≤ n 3 2 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='5), we get that an = max � 100 log n, 2w n � ≤ 100 log n + 2w n ≤ 100 log n + 2√n ≤ 3√n (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='20) for all n large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Consequently, using β < 1 2 strictly (see statement of Theorem 1) , ∆0 + an + 1 ≤ β · n + 1 + 3√n ≤ n 2 − 5 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='21) for all n large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='21), we therefore get that there exists a vertex v1 that is not a neighbour of Xi1 in G0 ∪ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Similarly, the total number of neighbours of v1 and Xi1+3 in G0 ∪ S is at most 2∆0 + 2an + 2 ≤ 2β · n + 2 + 6√n < n − 10 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='22) for all n large and so there exists a vertex v2 ̸= Xi1 that is not a neighbour of v1 and also not a neighbour of Xi1+3 in G0 ∪ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We now set X(1) i1+1 = v1 and X(1) i1+2 = v2 and X(1) j = Xj for j ̸= i1 + 1, i1 + 2 and call the resulting sequence as S1 := (X(1) 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' , X(1) w ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' By construction the degree of each vertex in the multigraph G0 ∪ S1 is at most ∆ + an + 1 + 2 and Edge Constrained Eulerian Extensions 7 there are at most t − 1 bad vertices in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We now pick the bad vertex with the least index in S1 and repeat the above procedure with S1 to get a sequence S2 containing at most t − 2 bad vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' After k ≤ t iterations of the above procedure, the degree of each vertex in the multigraph G0 ∪ Sk would be at most ∆0 + an + 1 + 2k ≤ ∆0 + an + 1 + 2t ≤ ∆0 + 3an + 3 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='23) because the event Ejoint ⊆ Ev,bad occurs and so t ≤ an + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Again using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='19) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='20) and arguing as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='22), we get that the sum of the degrees of any two vertices in G0 ∪St is at most n−10 for all n large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Thus the above procedure indeed proceeds for t iterations and by construction, the sequence St obtained at the end, has no bad vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We now perform an analogous procedure for correcting all bad edges in St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' For example if (Xl, Xl+1) is a bad edge in St, then following an analogous argument as before we pick a vertex Yl+1 that is neither adjacent to Xl nor adjacent to Xl+2 in the sequence St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We replace Xl+1 with Yl+1 to get a new sequence St+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' In the union G0 ∪St+1 the degree of each vertex is at most ∆0 +3an +3+2 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='23)) and the number of bad edges is at most r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' At the end of r ≤ K ·cn iterations, we obtain a multigraph G0 ∪ St+r, where the degree of each vertex is at most ∆0 + 3an + 3 + 2r ≤ ∆0 + 3an + 3 + 2Kcn, since the event Ee,bad occurs and therefore Ne,bad ≤ K·cn (see discussion preced- ing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Substituting the expression for cn from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='14) and using the second estimate for an in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='20), we get that ∆0 + 3an + 3 + 2Kcn is at most ∆0 + 300 log n + 6w n + 3 + 2K + 2K(b0 + w)w n2 ≤ ∆0 + 301 log n + 6w n + 2K(b0 + w)w n2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='24) for all n large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Recalling that u1 and u2 are the endvertices of the starting se- quence S0, we get that the final sequence St+r contains no bad edge and is therefore the desired walk W1 with endvertices u1 and u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' This completes the construction of the walk W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Rest of the walks: We now repeat the above procedure to construct the rest of the walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We set G1 := G0 ∪ W1 and argue as above to obtain a walk W2 with w edges present in G1 and containing u3 and u4 as endvertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Adding the walk W2 to G1 we get a new graph G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' In effect, to the graph G1 contain- ing b0+w edges, we have added w edges and by an argument analogous to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='24), we have increased the degree of a vertex by at most 301 log n + 6w n + 2K(b0 + 2w)w n2 , in obtaining the graph G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' We recall that (see first paragraph of the proof) there are z such walks to be created of which z − 1 have length w and the final walk 8 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Ganesan has length at most 2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Therefore after z iterations, we get a graph Gz with m edges and whose maximum vertex degree ∆z is at most ∆z ≤ ∆0 + � 301 log n + 6w n � (z − 1) + 301 log n + 12w n + 2K z−1 � k=1 (b0 + k · w)w n2 + 2K (b0 + (z − 1) · w)2w n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='25) By construction Gz is an Eulerian graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' To verify the obtainability of Gz, we estimate ∆z as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' The term z is no more than the size of a maximum clique in the original graph G (see discussion prior to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='3)) and since there are m ≤ n 3 2 edges in G, the maximum size of a clique in G is at most n 3 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Therefore z ≤ n 3 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='26) Also using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='2), we get that zw ≤ m ≤ α · n 3 2 and so wz n ≤ α · √n ≤ √n (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='27) Finally from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='3) we have that b0 ≤ b + n and so the second line in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='25) is at most z � k=1 (b + n + k · w)w n2 ≤ z(b + n + zw)2w n2 ≤ 2m(n + m) n2 ≤ 2√n + 2m2 n2 ≤ √n + 2α2 · n (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='28) where the second inequality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='28) follows from the estimate b + zw ≤ b0 + zw = m (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='4)), the third and fourth estimates in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='28) follow from the bound m ≤ α · n 3 2 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Plugging (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='28), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='27) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='26) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='25) we get that ∆z ≤ ∆0 + 301n 3 4 · log n + √n �12 10 + 2K � + 4Kα2 · n ≤ (β + 4Kα2) · n + 1 + 301n 3 4 · log n + √n �12 10 + 2K � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='29) for all n large, where the second inequality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='29) is obtained by using ∆0 ≤ ∆ + 1 ≤ β · n + 1 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='3) and the first condition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' From the statement of Theorem 1 and using K = 5, we have that β + 4Kα2 = β + 20α2 < 1 2 Edge Constrained Eulerian Extensions 9 strictly and so the degree of any vertex in Gz is strictly less than n 2 and also, the sum of degrees of any two vertices in Gz is at most (2β + 40α2) · n + 3 + 602n 3 4 · log n + 12√n 10 < n − 10 for all n large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Thus the graph Gz can be obtained by the above probabilistic method as in the discussion following (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Appendix Throughout we use the following deviation estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Let Zi, 1 ≤ i ≤ t be inde- pendent Bernoulli random variables satisfying P(Zi = 1) = pi = 1 − P(Zi = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' If Wt = �t i=1 Zi and µt = EWt, then for any 0 < ǫ < 1 2 we have that P (|Wt − µt| ≥ ǫµt) ≤ 2 exp � −ǫ2 4 µt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='1) For a proof of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='1), we refer to Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content='14, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' 312, Alon and Spencer (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Acknowledgement I thank Professors V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Raman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Subramanian and the referees for crucial comments that led to an improvement of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' I also thank IMSc for my fellowships.' 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(2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Introduction to Graph Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} +page_content=' Prentice Hall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE5T4oBgHgl3EQfBA5d/content/2301.05383v1.pdf'} diff --git a/V9AyT4oBgHgl3EQfV_fO/content/tmp_files/2301.00156v1.pdf.txt b/V9AyT4oBgHgl3EQfV_fO/content/tmp_files/2301.00156v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ad1454cb384d80c45529625128d4b10a01aef90f --- /dev/null +++ b/V9AyT4oBgHgl3EQfV_fO/content/tmp_files/2301.00156v1.pdf.txt @@ -0,0 +1,305 @@ +arXiv:2301.00156v1 [gr-qc] 31 Dec 2022 +Trans-Planckian censorship and +spacetime singularities +Spiros Cotsakis1,2∗, John Miritzis3† +1Institute of Gravitation and Cosmology, RUDN University +ul. Miklukho-Maklaya 6, Moscow 117198, Russia +2Research Laboratory of Geometry, Dynamical Systems +and Cosmology, University of the Aegean, +Karlovassi 83200, Samos, Greece +3Department of Marine Sciences +University of the Aegean +University Hill, Mytilene 81100, Greece +December 2022 +Abstract +We study the effects of trans-planckian curvature censorship (TCC) bounds on +geodesic completeness of spacetime and the associated existence for an infinite +proper time. Using Gronwall’s lemma, TCC bounds can be derived directly, leading +to a result about the absence of blowup solutions. We show that the TCC provides +part of the required criteria for geodesic completeness, and we then provide the +∗skot@aegean.gr +†imyr@aegean.gr +1 + +remaining ones - the norm of the extrinsic curvature being bounded away from +zero. We also discuss the importance of these results for the classical evolution of +Friedmann universes under the assumptions of global and regular hyperbolicity. +1 +Introduction +It is well-known that the Hawking-Penrose theorems provide sufficient conditions for the +existence of singularities in spacetime [1], while completeness theorems associated with +the work of Y. Choquet-Bruhat give sufficient conditions for the possible geodesic com- +pleteness of spacetimes [2]. In the first case we have geometric and causality conditions +leading to geodesic-incompleteness, while in the second case completeness of geodesics +is established under various analytic criteria. In both cases, such conditions may be +realized in effective theories and, as it has been repeatedly emphasized, such theories +may not be consistent with modern unification ideas, cf. e.g., [3]. +In fact, according to the trans-planckian curvature conjecture, initial fluctuations can +never exit the Hubble radius, and in this sense such information can never classicalize +and become ‘visible’ to classical evolution [4, 3]. +This is like having a cosmological +censor that, in an analogous way as in cosmic censorship, hides any trans-planckian +information. Since a central question in studies of the early structure and evolution of +the universe is the possible presence of singularities, it is important to understand how +the trans-planckian curvature conjecture relates to the possible resolution of cosmological +singularities. +The structure of this paper is as follows. In the next Section, we introduce three +different forms of trans-planckian bounds, and then provide sufficient conditions in the +form of integrability assumptions of the Hubble parameter (i.e., extrinsic curvature) +that lead to two of them. In Section 3, we show how trans-planckian bounds lead to the +absence of a blowup in the classical solutions, and discuss why such bounds alone cannot +provide an overall criterion for the possible geodesic completeness of spacetime. We then +show how one can obtain such criteria by introducing a further condition which we call +2 + +the ‘anti-Gronwall assumption’, that together with the trans-planckian bounds may lead +to a total bound on the norm of the Hubble parameter. We further discuss these results +in the last Section. +2 +Trans-planckian bounds +In this Section, we introduce a new way to derive trans-planckian bounds based on the +Gronwall’s lemma. +We start with the ‘Gronwall hypothesis’ which is contained in the following differential +inequality, +˙a(t) +a(t) ≤ H0(t), +(1) +for the two functions a, H0 defined for all t in the interval [ti, tf] and assumed differen- +tiable and nonnegative (weaker assumptions are possible). Using Gronwall’s lemma (see +for instance [5]), we find, +a(tf) +a(ti) ≤ e +� tf +ti H0(s)ds. +(2) +Let us first consider the case that H0 = const. For each finite tf there is a nonzero +constant Hf such that the right-hand-side of (2) is pointwise bounded, namely, +H0(tf − ti) < ln MP +Hf +. +(3) +Then it follows from the conclusion of the Gronwall’s lemma (2) that, +a(tf) +a(ti) lP < H−1 +f , +(4) +with lP = M−1 +P . (In other notation, setting N = H0(tf − ti) for the number of ‘e-folds’, +if we assume eN < MP/Hf as in (3), then (4) follows.) We note that the trans-planckian +bound in the form stated in Ref. [3] does not hold in the interval [ti, ∞) for each finite +ti, because when the upper endpoint tf → ∞, the left hand side of (3) is infinite. +We move on to the second case that is when H0 is not assumed constant. +We +suppose that H0 is an integrable function on [ti, ∞), and replace the left-hand-side of +3 + +the inequality (3) by the expression +� tf +ti H0(s)ds. We then end up with the pointwise +assumption that for each tf we have, +� tf +ti +H0(s)ds < ln MP +Hf +. +(5) +This implies that the statement of the trans-planckian censorship conjecture as formu- +lated in [3] now becomes a trans-planckian censorship theorem provided H0 is integrable: +for any integrable function H0(t) the integral +� tf +ti H0(s)ds is bounded, and we have, +a(t) +a(ti) lP < H∞, +t ∈ [ti, ∞), +(6) +where H∞ is a suitable constant that provides a uniform bound for the left hand side of +(6). Hence, the integrability of H0 provides a sufficient condition for the validity of the +trans-planckian censorship conjecture. +In other words, under the assumption (1), the inequality (3) (and similarly (5)) +implies (4) (or (6)), but not vice-versa. +Sometimes a stronger version of the trans- +planckian censorship conjecture is stated in the form of a double implication, which, +however, assumes more than just the integrability of H0. The following equivalence, +a(tf) +a(ti) lP < H−1 +f +if and only if +H0(tf − ti) < ln MP +Hf +, +(7) +is true (not just as a one-way implication), provided that the equality ˙a/a = H0 is +assumed instead of the differential inequality (1). +Another possible form is to take the Trans-Planckian Censorship Conjecture to mean +the reverse statement, namely that, (4) ⇒ (3) for any integrable H0, namely, that for +any tf and any nonzero Hf, we have [4], +a(tf) +a(ti) lP < H−1 +f +implies +� tf +ti +H0(s)ds < ln MP +Hf +. +(8) +This statement is different in meaning from both Eqns. (4), (6), or (7), and is true +provided again that H0 is an integrable function. +4 + +3 +A breakdown criterion +In this Section we show that a trans-planckian bound together with the additional as- +sumption of the existence of a lower bound for the scale factor are sufficient conditions +for producing singularity-free universes. +First we show that since any of the trans-plankian bounds discussed in the previous +Section provides an upper bound for a, we can obtain a criterion about the possible +absence of blowup solutions for the scale factor a in any interval of the form [ti, tf]. +For an initial time ti, we take the the ‘initial datum’ to be a(ti) = ai, and consider +the maximal interval of existence of solutions a(t) to be I = (T−, T+) where −∞ ≤ T− < +ti < T+ ≤ ∞. Any trans-plankian bound provides a suitable upper bound for a, and +therefore by the Picard existence and uniqueness theorem (cf. e.g., [5], p. 14) we have a +global solution, that is T+ = ∞, that does not go to infinity in a finite time in the future. +A physical interpretation of this result is that singularities of the finite-time blow-up +kind for a(t) are strictly prohibited when (2) holds and H0 is integrable. +However, in general relativity a singularity is defined as geodesic incompleteness +[1]. +The previous discussion does not of course prove geodesic completeness, and so +cannot provide an argument for a resolution of singularities of spacetime under the +above assumptions. The physical problem is to prove existence for an infinite proper +time, and in this respect the work in Ref. [2] becomes relevant. +In [2], a theorem was proved giving sufficient conditions for geodesic completeness in +the following sense. We assume the standard (3 + 1)-splitting of a globally hyperbolic +spacetime where the lapse function, shift vector field and spatial metric are all bounded +(regular hyperbolicity). If we further take the norms of the spatial gradient of the lapse +function as well as that of the extrinsic curvature to be bounded by integrable functions +on the interval [ti, ∞), then it follows that the spacetime is future timelike and null +geodesically complete. +For example, in the case of an FRW universe with scale factor a, the lapse N = 1, +the shift β = 0, and so the gradient of the lapse vanishes, while the norm of the extrinsic +5 + +curvature is given by |K|g +2 = 3(˙a/a)2 = 3H2. Hence this result tell us that in FRW +universes having their scale factor bounded below will be singular only if there is a finite +time t1 ∈ [ti, ∞) such that the Hubble parameter H is not integrable on the corresponding +interval [t1, ∞). +Previously we assumed the Gronwall bound (2) for H, where H0 could also be neg- +ative and we discussed its importance in the formulation of trans-planckian bounds. +That discussion provides only half of the conditions needed for a complete singularity +resolution, however, we may now discuss the other half. +Let us introduce the following ‘anti-Gronwall’ assumption, namely, +H(t) ≥ b > 0, +(9) +with t ∈ [ti, tf], for some constant b, so that 0 < b ≤ ˙a/a. Integrating on [ti, tf] we find +that, +a(tf) ≥ a(ti) eb(tf −ti), +(10) +that is the scale factor a is bounded from below. +This is a way to circumvent the +singularity at a(t) = 0 for some t earlier than ti that is expected from the Raychaudhuri +equation, because the anti-Gronwall condition (9) is the opposite of the usual one, i.e., +negative expansion (or positive convergence) assumed in the singularity theorems (cf. +[1], Thm. 3, p. 271). +The question is then whether the interval I = (T−, T+) where the scalae factor a is +bounded is finite or infinite. From the results above it follows that using the anti-Gronwall +condition (9) (a be bounded below) together with the trans-plankian bound we find that +the norm |H(t)| will be bounded for all time, not just H, so that the interval I can be +infinite (to the left, right, or both). This is so because according to the completeness +theorem of [2] mentioned above, the integrability of |H|, i.e., |H| is bounded by the +integrable function H0 as in (1)) is also a sufficient condition for geodesic completeness +(the others being that spacetime is globally and regularly hyperbolic) to the past, future, +or both. +6 + +We note here that this argument is independent of the the usual assumption on the +Ricci tensor (e.g., RµνXµXν ≥ 0 for non-spacelike vector fields) because the positive +convergence condition is an independent hypothesis, and leads to the absence of past or +future singularities. +For a Friedman universe in particular, geodesic completeness can only fail if there +exists a time ti such that the norm of the Hubble parameter |H| becomes non-integrable +in the interval [ti, ∞). The non-integrability of |H| provides the only necessary condition +for a Friedman universe to be singular. There are different ways for this non-integrability +to arise, and an exhaustive classification of the nature of possible singularities that occur +this way was presented in [6, 7]. +Therefore we are led to conclude that using the completeness theorem, the trans- +planckian bounds, and the anti-Gronwall assumption, there is a way out of the inevitabil- +ity of the singular nature of Friedman universes either in the past or future, by providing +conditions for the norm of the Hubble parameter to be bounded and so be integrable. +This argument also explains why the trans-planckian censorship conjecture favors +scenarios like the ekpyrotic universe where the scale factor is bounded below, or the +emergent universe scenarios [8] where H is not only integrable but in fact it is asymp- +totically vanishing [6], rather than an inflationary universe where there is a singularity +with a finite H, cf. [9, 6]. +Therefore we conclude that future (or past) geodesic completeness and the associ- +ated absence of future (past) singularities is a necessary consequence of trans-planckian +bounds in any scenario in which the universe satisfies the anti-Gronwall assumption. +4 +Examples +As an application of the previous results, we consider here a few representative examples +that illustrate some of the features of the use of trans-planckian bounds in proving +geodesic completeness. +7 + +A a first example, let us consider the emergent universe scenario of [8]. For this +model, the Gronwall hypothesis, namely that the expansion is sub-Hubblian, together +with the trans-planckian bound (4) implies that the initial (Einstein static universe) scale +factor a(ti) is bounded from below, avoiding the usual fine-tuning issues associated with +the emergent scenario. In addition, the anti-Gronwall bound on the Hubble parameter +(9) implies a large classical expanding universe with scale factor given by (10) at late +times. This universe is also future geodesically complete because the Hubble parameter +is not only bounded by asymptotically vanishing, cf. [6]. +In fact it is not difficult to devise universes with an asymptotically vanishing Hubble +parameter, thus signalling future geodesic completeness. As an example, in any flat or +negatively curved FRW model filled with a perfect fluid and scalar field with a positive, +bounded potential, one can show that not only H but also the fluid density are future +asymptotically vanishing, cf. [10], Proposition 2. Hence in any model with logarithmic or +generalized potentials of the form studied in [11, 12] the trans-planckian bound together +with the anti-Gronwall hypothesis imply a singularity-free evolution. +5 +Discussion +In this paper we have discussed the role of trans-planckian bounds in relation to the for- +mation of singularities. We have first shown that such bounds can be naturally deduced +from the Gronwall hypothesis which provides upper bounds to the Hubble parameter. +This leads to a new criterion for the absence of diverging cosmological solutions either +at a finite time or at infinity. +Further we have shown that trans-planckian bounds, when combined with the condi- +tion that the Hubble parameter is bounded away from zero, lead to geodesically complete +universes satisfying the usual causality assumptions. +We therefore conclude that trans-planckian bounds provide a way to singularity-free +universes if the Hubble parameter is integrable. This opens the way to constructing +8 + +singularity-free cosmologies starting from a trans-planckian bound and examining the +integrability of the expansion parameter. This depends on the kind of matter content of +the universe and may lead to selection rules for non-singular cosmologies, from suitable +restrictions on the fluid or other parameters of the matter fields. We shall consider this +problem elsewhere. +Acknowledgments +S.C. is grateful to Robert Brandenberger for valuable comments on an earlier version of +this work. +References +[1] S. W. Hawking and G. F. R. Ellis, The large-scale structure of space-time (CUP, +1973) +[2] Y. Choquet-Bruhat, S. Cotsakis, Global Hyperbolicity and Completeness, J. Geom. +Phys. 43 (2002) 345-350; arXiv:gr-qc/0201057 +[3] R. Brandenberger, Trans-Planckian Censorship Conjecture and Early Universe Cos- +mology, LHEP 2021 (2021) 198; e-Print: 2102.09641 [hep-th] +[4] A. Bedroya, C. Vafa, Trans-Planckian Censorship and the Swampland, JHEP 09 +(2020) 123; e-Print: 1909.11063 [hep-th] +[5] T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis (AMS, 2006) +[6] S. Cotsakis, I. Klaoudatou, Future Singularities of Isotropic Cosmologies, J. Geom. +Phys. 55 (2005) 306-315; arXiv:gr-qc/0409022 +[7] S. Cotsakis, I. Klaoudatou, Cosmological Singularities and Bel-Robinson Energy, J. +Geom. Phys. 57 (2007) 1303-1312; arXiv:gr-qc/0604029 +9 + +[8] G.F.R. Ellis, R. Maartens, The Emergent Universe: inflationary cosmology with no +singularity, Class.Quant.Grav.21:223-232,2004; arXiv:gr-qc/0211082 +[9] A. Borde, A. H. Guth, Alexander Vilenkin, Inflationary spacetimes are not past- +complete, Phys.Rev.Lett. 90 (2003) 151301; arXiv:gr-qc/0110012 +[10] J. Miritzis, Class. Quantum Grav. 20 (2003) 2981 +[11] P. Parsons and J. D. Barrow, Phys. Rev. D51 (1995) 6757 +[12] J. D. Barrow and P. Parsons, Phys. Rev. D52 (1995) 5576 +10 + diff --git a/V9AyT4oBgHgl3EQfV_fO/content/tmp_files/load_file.txt b/V9AyT4oBgHgl3EQfV_fO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..82b5530d81a897772656c1a8bec3b178356419af --- /dev/null +++ b/V9AyT4oBgHgl3EQfV_fO/content/tmp_files/load_file.txt @@ -0,0 +1,170 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf,len=169 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content='00156v1 [gr-qc] 31 Dec 2022 Trans-Planckian censorship and spacetime singularities Spiros Cotsakis1,2∗, John Miritzis3† 1Institute of Gravitation and Cosmology, RUDN University ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Miklukho-Maklaya 6, Moscow 117198, Russia 2Research Laboratory of Geometry, Dynamical Systems and Cosmology, University of the Aegean, Karlovassi 83200, Samos, Greece 3Department of Marine Sciences University of the Aegean University Hill, Mytilene 81100, Greece December 2022 Abstract We study the effects of trans-planckian curvature censorship (TCC) bounds on geodesic completeness of spacetime and the associated existence for an infinite proper time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Using Gronwall’s lemma, TCC bounds can be derived directly, leading to a result about the absence of blowup solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' We show that the TCC provides part of the required criteria for geodesic completeness, and we then provide the ∗skot@aegean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content='gr †imyr@aegean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content='gr 1 remaining ones - the norm of the extrinsic curvature being bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' We also discuss the importance of these results for the classical evolution of Friedmann universes under the assumptions of global and regular hyperbolicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' 1 Introduction It is well-known that the Hawking-Penrose theorems provide sufficient conditions for the existence of singularities in spacetime [1], while completeness theorems associated with the work of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Choquet-Bruhat give sufficient conditions for the possible geodesic com- pleteness of spacetimes [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' In the first case we have geometric and causality conditions leading to geodesic-incompleteness, while in the second case completeness of geodesics is established under various analytic criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' In both cases, such conditions may be realized in effective theories and, as it has been repeatedly emphasized, such theories may not be consistent with modern unification ideas, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=', [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' In fact, according to the trans-planckian curvature conjecture, initial fluctuations can never exit the Hubble radius, and in this sense such information can never classicalize and become ‘visible’ to classical evolution [4, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' This is like having a cosmological censor that, in an analogous way as in cosmic censorship, hides any trans-planckian information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Since a central question in studies of the early structure and evolution of the universe is the possible presence of singularities, it is important to understand how the trans-planckian curvature conjecture relates to the possible resolution of cosmological singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' The structure of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' In the next Section, we introduce three different forms of trans-planckian bounds, and then provide sufficient conditions in the form of integrability assumptions of the Hubble parameter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=', extrinsic curvature) that lead to two of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' In Section 3, we show how trans-planckian bounds lead to the absence of a blowup in the classical solutions, and discuss why such bounds alone cannot provide an overall criterion for the possible geodesic completeness of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' We then show how one can obtain such criteria by introducing a further condition which we call 2 the ‘anti-Gronwall assumption’, that together with the trans-planckian bounds may lead to a total bound on the norm of the Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' We further discuss these results in the last Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' 2 Trans-planckian bounds In this Section, we introduce a new way to derive trans-planckian bounds based on the Gronwall’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' We start with the ‘Gronwall hypothesis’ which is contained in the following differential inequality, ˙a(t) a(t) ≤ H0(t), (1) for the two functions a, H0 defined for all t in the interval [ti, tf] and assumed differen- tiable and nonnegative (weaker assumptions are possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Using Gronwall’s lemma (see for instance [5]), we find, a(tf) a(ti) ≤ e � tf ti H0(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' (2) Let us first consider the case that H0 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' For each finite tf there is a nonzero constant Hf such that the right-hand-side of (2) is pointwise bounded, namely, H0(tf − ti) < ln MP Hf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' (3) Then it follows from the conclusion of the Gronwall’s lemma (2) that, a(tf) a(ti) lP < H−1 f , (4) with lP = M−1 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' (In other notation, setting N = H0(tf − ti) for the number of ‘e-folds’, if we assume eN < MP/Hf as in (3), then (4) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=') We note that the trans-planckian bound in the form stated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' [3] does not hold in the interval [ti, ∞) for each finite ti, because when the upper endpoint tf → ∞, the left hand side of (3) is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' We move on to the second case that is when H0 is not assumed constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' We suppose that H0 is an integrable function on [ti, ∞), and replace the left-hand-side of 3 the inequality (3) by the expression � tf ti H0(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' We then end up with the pointwise assumption that for each tf we have, � tf ti H0(s)ds < ln MP Hf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' (5) This implies that the statement of the trans-planckian censorship conjecture as formu- lated in [3] now becomes a trans-planckian censorship theorem provided H0 is integrable: for any integrable function H0(t) the integral � tf ti H0(s)ds is bounded, and we have, a(t) a(ti) lP < H∞, t ∈ [ti, ∞), (6) where H∞ is a suitable constant that provides a uniform bound for the left hand side of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Hence, the integrability of H0 provides a sufficient condition for the validity of the trans-planckian censorship conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' In other words, under the assumption (1), the inequality (3) (and similarly (5)) implies (4) (or (6)), but not vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Sometimes a stronger version of the trans- planckian censorship conjecture is stated in the form of a double implication, which, however, assumes more than just the integrability of H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' The following equivalence, a(tf) a(ti) lP < H−1 f if and only if H0(tf − ti) < ln MP Hf , (7) is true (not just as a one-way implication), provided that the equality ˙a/a = H0 is assumed instead of the differential inequality (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Another possible form is to take the Trans-Planckian Censorship Conjecture to mean the reverse statement, namely that, (4) ⇒ (3) for any integrable H0, namely, that for any tf and any nonzero Hf, we have [4], a(tf) a(ti) lP < H−1 f implies � tf ti H0(s)ds < ln MP Hf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' (8) This statement is different in meaning from both Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' (4), (6), or (7), and is true provided again that H0 is an integrable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' 4 3 A breakdown criterion In this Section we show that a trans-planckian bound together with the additional as- sumption of the existence of a lower bound for the scale factor are sufficient conditions for producing singularity-free universes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' First we show that since any of the trans-plankian bounds discussed in the previous Section provides an upper bound for a, we can obtain a criterion about the possible absence of blowup solutions for the scale factor a in any interval of the form [ti, tf].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' For an initial time ti, we take the the ‘initial datum’ to be a(ti) = ai, and consider the maximal interval of existence of solutions a(t) to be I = (T−, T+) where −∞ ≤ T− < ti < T+ ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Any trans-plankian bound provides a suitable upper bound for a, and therefore by the Picard existence and uniqueness theorem (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=', [5], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' 14) we have a global solution, that is T+ = ∞, that does not go to infinity in a finite time in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' A physical interpretation of this result is that singularities of the finite-time blow-up kind for a(t) are strictly prohibited when (2) holds and H0 is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' However, in general relativity a singularity is defined as geodesic incompleteness [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' The previous discussion does not of course prove geodesic completeness, and so cannot provide an argument for a resolution of singularities of spacetime under the above assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' The physical problem is to prove existence for an infinite proper time, and in this respect the work in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' [2] becomes relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' In [2], a theorem was proved giving sufficient conditions for geodesic completeness in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' We assume the standard (3 + 1)-splitting of a globally hyperbolic spacetime where the lapse function, shift vector field and spatial metric are all bounded (regular hyperbolicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' If we further take the norms of the spatial gradient of the lapse function as well as that of the extrinsic curvature to be bounded by integrable functions on the interval [ti, ∞), then it follows that the spacetime is future timelike and null geodesically complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' For example, in the case of an FRW universe with scale factor a, the lapse N = 1, the shift β = 0, and so the gradient of the lapse vanishes, while the norm of the extrinsic 5 curvature is given by |K|g 2 = 3(˙a/a)2 = 3H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Hence this result tell us that in FRW universes having their scale factor bounded below will be singular only if there is a finite time t1 ∈ [ti, ∞) such that the Hubble parameter H is not integrable on the corresponding interval [t1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Previously we assumed the Gronwall bound (2) for H, where H0 could also be neg- ative and we discussed its importance in the formulation of trans-planckian bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' That discussion provides only half of the conditions needed for a complete singularity resolution, however, we may now discuss the other half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Let us introduce the following ‘anti-Gronwall’ assumption, namely, H(t) ≥ b > 0, (9) with t ∈ [ti, tf], for some constant b, so that 0 < b ≤ ˙a/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Integrating on [ti, tf] we find that, a(tf) ≥ a(ti) eb(tf −ti), (10) that is the scale factor a is bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' This is a way to circumvent the singularity at a(t) = 0 for some t earlier than ti that is expected from the Raychaudhuri equation, because the anti-Gronwall condition (9) is the opposite of the usual one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=', negative expansion (or positive convergence) assumed in the singularity theorems (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' [1], Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' 271).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' The question is then whether the interval I = (T−, T+) where the scalae factor a is bounded is finite or infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' From the results above it follows that using the anti-Gronwall condition (9) (a be bounded below) together with the trans-plankian bound we find that the norm |H(t)| will be bounded for all time, not just H, so that the interval I can be infinite (to the left, right, or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' This is so because according to the completeness theorem of [2] mentioned above, the integrability of |H|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=', |H| is bounded by the integrable function H0 as in (1)) is also a sufficient condition for geodesic completeness (the others being that spacetime is globally and regularly hyperbolic) to the past, future, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' 6 We note here that this argument is independent of the the usual assumption on the Ricci tensor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=', RµνXµXν ≥ 0 for non-spacelike vector fields) because the positive convergence condition is an independent hypothesis, and leads to the absence of past or future singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' For a Friedman universe in particular, geodesic completeness can only fail if there exists a time ti such that the norm of the Hubble parameter |H| becomes non-integrable in the interval [ti, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' The non-integrability of |H| provides the only necessary condition for a Friedman universe to be singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' There are different ways for this non-integrability to arise, and an exhaustive classification of the nature of possible singularities that occur this way was presented in [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Therefore we are led to conclude that using the completeness theorem, the trans- planckian bounds, and the anti-Gronwall assumption, there is a way out of the inevitabil- ity of the singular nature of Friedman universes either in the past or future, by providing conditions for the norm of the Hubble parameter to be bounded and so be integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' This argument also explains why the trans-planckian censorship conjecture favors scenarios like the ekpyrotic universe where the scale factor is bounded below, or the emergent universe scenarios [8] where H is not only integrable but in fact it is asymp- totically vanishing [6], rather than an inflationary universe where there is a singularity with a finite H, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' [9, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Therefore we conclude that future (or past) geodesic completeness and the associ- ated absence of future (past) singularities is a necessary consequence of trans-planckian bounds in any scenario in which the universe satisfies the anti-Gronwall assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' 4 Examples As an application of the previous results, we consider here a few representative examples that illustrate some of the features of the use of trans-planckian bounds in proving geodesic completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' 7 A a first example, let us consider the emergent universe scenario of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' For this model, the Gronwall hypothesis, namely that the expansion is sub-Hubblian, together with the trans-planckian bound (4) implies that the initial (Einstein static universe) scale factor a(ti) is bounded from below, avoiding the usual fine-tuning issues associated with the emergent scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' In addition, the anti-Gronwall bound on the Hubble parameter (9) implies a large classical expanding universe with scale factor given by (10) at late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' This universe is also future geodesically complete because the Hubble parameter is not only bounded by asymptotically vanishing, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' In fact it is not difficult to devise universes with an asymptotically vanishing Hubble parameter, thus signalling future geodesic completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' As an example, in any flat or negatively curved FRW model filled with a perfect fluid and scalar field with a positive, bounded potential, one can show that not only H but also the fluid density are future asymptotically vanishing, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' [10], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Hence in any model with logarithmic or generalized potentials of the form studied in [11, 12] the trans-planckian bound together with the anti-Gronwall hypothesis imply a singularity-free evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' 5 Discussion In this paper we have discussed the role of trans-planckian bounds in relation to the for- mation of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' We have first shown that such bounds can be naturally deduced from the Gronwall hypothesis which provides upper bounds to the Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' This leads to a new criterion for the absence of diverging cosmological solutions either at a finite time or at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Further we have shown that trans-planckian bounds, when combined with the condi- tion that the Hubble parameter is bounded away from zero, lead to geodesically complete universes satisfying the usual causality assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' We therefore conclude that trans-planckian bounds provide a way to singularity-free universes if the Hubble parameter is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' This opens the way to constructing 8 singularity-free cosmologies starting from a trans-planckian bound and examining the integrability of the expansion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' This depends on the kind of matter content of the universe and may lead to selection rules for non-singular cosmologies, from suitable restrictions on the fluid or other parameters of the matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' We shall consider this problem elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' Acknowledgments S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' is grateful to Robert Brandenberger for valuable comments on an earlier version of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9AyT4oBgHgl3EQfV_fO/content/2301.00156v1.pdf'} +page_content=' W.' metadata={'source': 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Tang3 +Changsheng Xu2,4,5 +1Nanjing University of Posts and Telecommunications +2Peng Cheng Laboratory +3CVL, ETH Z¨urich +4University of Chinese Academy of Sciences +5NLPR, Institute of Automation, CAS +Abstract +Synthesizing high-fidelity complex images from text is +challenging. Based on large pretraining, the autoregres- +sive and diffusion models can synthesize photo-realistic im- +ages. +Although these large models have shown notable +progress, there remain three flaws. +1) These models re- +quire tremendous training data and parameters to achieve +good performance. +2) The multi-step generation design +slows the image synthesis process heavily. +3) The syn- +thesized visual features are difficult to control and require +delicately designed prompts. To enable high-quality, effi- +cient, fast, and controllable text-to-image synthesis, we pro- +pose Generative Adversarial CLIPs, namely GALIP. GALIP +leverages the powerful pretrained CLIP model both in the +discriminator and generator. +Specifically, we propose a +CLIP-based discriminator. The complex scene understand- +ing ability of CLIP enables the discriminator to accurately +assess the image quality. Furthermore, we propose a CLIP- +empowered generator that induces the visual concepts from +CLIP through bridge features and prompts. +The CLIP- +integrated generator and discriminator boost training ef- +ficiency, and as a result, our model only requires about 3% +training data and 6% learnable parameters, achieving com- +parable results to large pretrained autoregressive and diffu- +sion models. Moreover, our model achieves ∼120×faster +synthesis speed and inherits the smooth latent space from +GAN. The extensive experimental results demonstrate the +excellent performance of our GALIP. Code is available at +https://github.com/tobran/GALIP. +1. Introduction +Over the last few years, we have witnessed the great suc- +cess of generative models for various applications [4, 45]. +Among them, text-to-image synthesis [3, 15, 18–21, 25, 28, +29, 33, 46, 48–50, 57] is one of the most appealing applica- +tions. It generates high-fidelity images according to given +*Corresponding Author +Figure 1. (a) Existing text-to-image GANs conduct adversarial +training from scratch. (b) Our proposed GALIP conducts adver- +sarial training based on the integrated CLIP model. +language guidance. Owing to the convenience of language +for users, text-to-images synthesis has attracted many re- +searchers and become an active research area. +Based on a large scale of data collections, model size, +and pretraining, recently proposed large pretrained autore- +gressive and diffusion models, e.g., DALL-E [33] and +LDM [35], show the impressive generative ability to syn- +thesize complex scenes and outperform the previous text-to- +image GANs significantly. Although these large pretrained +generative models have achieved significant advances, they +still suffer from three flaws. +First, these models require +tremendous training data and parameters for pretraining. +The large data and model size brings an extremely high +computing budget and hardware requirements, making it +inaccessible to many researchers and users. Second, the +generation of large models is much slower than GANs. +The token-by-token generation and progressive denoising +require hundreds of inference steps and make the generated +results lag the language inputs seriously. Third, there is no +intuitive smooth latent space as GANs, which maps mean- +ingful visual attributes to the latent vector. The multi-step +generation design breaks the synthesis process and scatters +the meaningful latent space. It makes the synthesis process +require delicately designed prompts to control. +To address the above limitations, we rethink Generative +Adversarial Networks (GAN). GANs are much faster than +autoregressive and diffusion models and have smooth latent +arXiv:2301.12959v1 [cs.CV] 30 Jan 2023 + +→Loss +Generator +Discriminator +a +Actor and his +Text + wife attend the +Encoder +premiere. +CLIP-ViT +CLIP-ViT +→LosS +(b) +Mate-G +Mate-D +Actor and his +Text +I wife attend the +Encoder +premiere.Figure 2. Comparing with Latent Diffusion Models (LDM) [35], our GALIP achieves comparable zero-shot Fr´echet Inception Distance +(ZS-FID) with measly 320M parameters (0.08B trainable parameters + 0.24B frozen CLIP parameters) and 12M training data. Furthermore, +our GALIP only requires 0.04s to synthesize one image which is ∼120×faster than LDM. Speed is calculated on NVIDIA 3090 GPU and +Intel Xeon Silver 4314 CPU. +space, which enables more controllable synthesis. How- +ever, GAN models are known for potentially unstable train- +ing and less diversity in the generation [6]. It makes current +text-to-image GANs suffer from unsatisfied synthesis qual- +ity under complex scenes. +In this work, we introduce the pretrained CLIP [30] into +text-to-image GANs. The large pretraining of CLIP brings +two advantages. First, it enhances the complex scene un- +derstanding ability. The pretraining dataset has many com- +plex images under different scenes. Armed with the Vision +Transformer (ViT) [8], the image encoder can extract in- +formative and meaningful visual features from complex im- +ages to align the corresponding text descriptions after ade- +quate pretraining. Second, the large pretraining dataset also +enables excellent domain generalization ability. It contains +various kinds of images, e.g., photos, drawings, cartoons, +and sketches, collected from a variety of publicly available +sources. The various images make the CLIP model can map +different kinds of images to the shared concepts and en- +able impressive domain generalization and zero-shot trans- +fer ability. These two advantages of CLIP, complex scene +understanding and domain generalization ability, motivate +us to build a more powerful text-to-image model. +We propose a novel text-to-image generation framework +named Generative Adversarial CLIPs (GALIP). As shown +in Figure 1, the GALIP integrates the CLIP model [30] in +both the discriminator and generator. To be specific, we pro- +pose the CLIP-based discriminator and CLIP-empowered +generator. The CLIP-based discriminator inherits the com- +plex scene understanding ability of CLIP [30]. It is com- +posed of a frozen ViT-based CLIP image encoder (CLIP- +ViT) and a learnable mate-discriminator (Mate-D). The +Mate-D is mated to the CLIP-ViT for adversarial training. +To retain the knowledge of complex scene understanding +in the CLIP-ViT, we freeze its weights and collect the pre- +dicted CLIP image features from different layers. Then, +the Mate-D further extracts informative visual features from +collected CLIP features to distinguish the synthesized and +real images. Based on the complex scene understanding +ability of CLIP-ViT and the continuous analysis of Mate- +D, the CLIP-based discriminator can assess the quality of +generated complex images more accurately. +Furthermore, we propose the CLIP-empowered genera- +tor, which exerts the domain generalization ability of CLIP +[30]. It is hard for the generator to synthesize complex im- +ages directly. +Some works employ sketch [10] and lay- +out [20, 22] as bridge domains to alleviate the difficulty. +However, such a design requires additional labeled data. +Different from these works, the excellent domain general- +ization of CLIP [30] motivates us that there may be an im- +plicit bridge domain, which is easier to synthesize but can +be mapped to the same visual concepts through the CLIP- +ViT. Thus, we design the CLIP-empowered generator. It +is composed of a frozen CLIP-ViT and a learnable mate- +generator (Mate-G). The Mate-G first predicts the implicit +bridge features from text and noise. Then the bridge feature +will be mapped to the visual concepts through CLIP-ViT. +Furthermore, we add some text-conditioned prompts to the +CLIP-ViT for task adaptation. The predicted visual con- +cepts close the gap between text features and target images +which enhances the complex image synthesis ability. +As shown in Figure 2, the proposed GALIP achieves +∼120×faster synthesis speed and comparable synthesis +ability based on significantly smaller trainable parameters +and training data. +Overall, our contributions can be summarized as follows: +• We propose an efficient, fast, and more controllable +model for text-to-image synthesis that can synthesize +high-quality complex images. +• We propose the CLIP-based discriminator, which as- +sesses the quality of generated complex images more +accurately. +• We propose the CLIP-empowered generator, which +synthesizes images based on text features and pre- +dicted CLIP visual features. +• Extensive experiments demonstrate that the proposed +GALIP can achieve comparable performance with +large pertaining models based on significantly smaller +computational costs. + +1.45B +1.5 +13 +400M +4.83s +GPU +12.63 +400 +12.54 +a size (Million) + CPU +illion) +4 - +300 +1.0 +12 +speed ( +3- +200 + Data +S +0.5 - +11- +0.32B +100- +11 +12M +0.04s 0.20s +0.0 +0 +0 +10 +LDM +GALIP +LDM +GALIP +LDM +GALIP +LDM +GALIPFigure 3. The architecture of the proposed GALIP for text-to-image synthesis. Armed with the CLIP-based discriminator and CLIP- +empowered generator, our model can synthesize more realistic complex images. +2. Related Work +Text-to-Image GANs. GAN-INT-CLS [34] first adopted +conditional GANs to synthesize images from text descrip- +tions. +To enable higher resolution synthesis, the Stack- +GAN [54, 55], AttnGAN [48], and DM-GAN [57] stacks +multiple generators and discriminators. +Tao et al. [42] +proposed a simpler yet effective text-to-image framework +called DF-GAN that enables one-stage high-resolution gen- +eration. +LAFITE [56] introduces CLIP text-image con- +trastive loss for text-to-image training and shows large im- +provements on CC3M [40]. +Text-to-Image Large Models. Recently, large pretrained +autoregressive and diffusion models have shown impres- +sive performance on text-to-image synthesis. DALL-E [33], +CogView [6], and M6 [23] leverage VQ-VAE [43] or VQ- +GAN [9] to tokenize the images into discrete image tokens. +Then they take the word tokens and image tokens together +to pre-train a large unidirectional transformer for an au- +toregressive generation. Parti [51] proposes a sequence-to- +sequence autoregressive model to treat text-to-image syn- +thesis as a translation task. Cogview2 [7] employs hierar- +chical transformers and local parallel autoregressive gen- +eration for faster autoregressive image generation. Some +works try to employ the diffusion model [5,13,14,26,41] to +overcome the slow generation defect of the autoregressive +model. VQ-Diffusion [11] combines the VQ-VAE [43] and +diffusion model [14,26] to eliminate the unidirectional bias +and avoids accumulated prediction errors. GLIDE [27] ap- +plies guided diffusion to the problem of text-conditional im- +age synthesis. DALL-E2 [32] combines the CLIP represen- +tation and diffusion model to make a CLIP decoder. Latent +Diffusion Models (LDM) [35] apply the diffusion model in +the latent space to enable the training on limited compu- +tational resources while retaining image quality. The par- +ticular text-to-image LDM is Stable Diffusion [36], which +is a favorite open-source project and provides an easy-to- +use interface. Imagen [38] introduces the large language +model [31] to provide high-quality text features and pro- +poses an Efficient U-Net for diffusion models. +3. Generative Adversarial CLIPs +In this paper, we propose a novel framework for text- +to-image synthesis named Generative Adversarial CLIPs +(GALIP). To synthesize high-quality complex images, we +propose: (i) a novel CLIP-based discriminator that inherits +the complex scene understanding ability of CLIP [30] for +more accurate image quality assessment. (ii) a novel CLIP- +empowered generator that exerts the domain generalization +ability of CLIP [30] and induces the CLIP visual concepts +to close the gap between text and image features. In the fol- +lowing of this section, we first present the overall structure +of our GALIP. Then, we introduce the CLIP-based discrim- +inator and CLIP-empowered generator in detail. +3.1. Model Overview +As shown in Figure 3, the proposed GALIP is composed +of a CLIP text encoder, a CLIP-based discriminator, and a +CLIP-empowered generator. The pretrained CLIP text en- +coder takes the text description and yields a global sentence +vector T . After the text-encoder is the CLIP-empowered + +Z : noise~N(0,1) +I : CLIP-ViT +p" +T : Sentence Vector +I : Mate-G +P : Prompt +TransBlock +TransBlock +TransBlock +TransBlock +TransBlock +TransBlock +TransBlock +Reshape +Reshape +: Mate-D +G : Generator +Visual +D : Discriminator +Concepts +Z +Actor and his wife +Text +Image +Bridge +T +Bridge-FP + attend the premiere. +Generator +Encoder +Feature +G: +D: +TransBlock +TransBlock +TransBlock +lock +lock +Reshape +Conv +TransBlc +TransBl +Quality +B +CLIP-FE +. +. +TransE +Assessor +Adversarial +lossFigure 4. The architecture of the proposed Mate-D for text-to-image synthesis. It further extracts informative visual features from collected +CLIP features and assesses the image quality more accurately. +generator and CLIP-based discriminator under the GAN +framework. The CLIP-empowered generator is composed +of a frozen CLIP-ViT and a mate generator (Mate-G). There +are three main modules in the Mate-G, the bridge feature +predictor (Bridge-FP), the prompt predictor, and the image +generator. The CLIP-empowered generator has two inputs, +the sentence vector T encoded from the text encoder and +the noise vector Z sampled from the Gaussian distribution. +The noise vector ensures the diversity of the synthesized im- +ages. In the CLIP-empowered generator, the sentence vec- +tor and noise are first fed into the bridge feature predictor. +The bridge feature predictor translates the sentence vector +and noise to the bridge feature for the CLIP-ViT. Further- +more, we add several text-conditioned prompts to the trans- +former blocks (TransBlock) in CLIP-ViT for task adapta- +tion. Finally, the image generator takes the predicted visual +concepts, bridge features, sentences, and noise vectors to +synthesize high-quality images. +The CLIP-based discriminator is composed of a frozen +CLIP-ViT and a mate discriminator (Mate-D). The CLIP- +ViT converts images into image feature through a convolu- +tion layer and a series of transformer blocks. The CLIP fea- +ture extractor (CLIP-FE) in Mate-D collects the image fea- +tures from different layers in CLIP-ViT. Then it further ex- +tracts informative visual features from collected CLIP fea- +tures for the quality assessor. Lastly, an adversarial loss will +be predicted by the quality assessor based on the extracted +informative features and sentence vectors. By distinguish- +ing synthesized images from real ones, the discriminator +promotes the generator to synthesize higher-quality images. +3.2. CLIP-based Discriminator +In this section, we detailed the proposed CLIP-based dis- +criminator, which is composed of a frozen CLIP-ViT and +a Mate-D. The CLIP-based discriminator inherits the com- +plex scene understanding ability from the frozen CLIP-ViT. +Furthermore, we propose the Mate-D, which is mated to +the CLIP-ViT to further extract informative visual features +and distinguish real and synthesized images. The CLIP-ViT +and Mate-D enable the discriminator to assess the quality of +generated complex images more accurately. +As shown in Figure 4, the Mate-D consists of a CLIP- +FE and a quality assessor. To fully utilize the knowledge +of complex scene understanding in CLIP-ViT, the CLIP-FE +takes the CLIP image features from multilayers. There are +N CLIP features collected for the CLIP-FE. We name them +CLIP Feature 1 to N, which are collected from shallow to +deep layers in CLIP-ViT. To further extract informative vi- +sual features from these CLIP features, we design a CLIP- +FE. It contains a sequence of extraction blocks, and each +block contains two convolution layers and two ReLU active +functions. And the extracted image feature is summed with +the shortcut and the next CLIP feature. There are N − 1 +extraction blocks stacked in CLIP-FE. Since the CLIP fea- +ture N is only added to the processed image features in the +last extraction block. To fuse the CLIP feature N, we ap- +pend two convolution layers without the CLIP feature addi- +tion behind. The CLIP-FE extracts informative visual fea- +tures for the quality assessor. Then the sentence vector is +replicated and concatenated with the extracted image fea- +tures. An adversarial loss is predicted by two convolution +layers to evaluate the image quality. Furthermore, to stabi- +lize the adversarial learning process of Mate-D, we apply +the matching-aware gradient penalty (MAGP) [42] on the +collected CLIP features and corresponding text features. +Based on the complex scene understanding ability of +CLIP-ViT, the CLIP-based discriminator can extract more +informative visual features from complex images. +The +higher-quality extracted visual features make it easier for +the discriminator to detect unreal image parts, which im- +proves the discriminative efficiency, thus prompting the +generator to generate more realistic images. +3.3. CLIP-empowered Generator +In this section, we detail the proposed CLIP-empowered +generator, which is composed of a frozen CLIP-ViT and +a Mate-G. The CLIP-empowered generator exerts the do- +main generalization ability of the CLIP-ViT. Furthermore, +we propose the Mate-G, which is mated to the CLIP-ViT to +induce useful visual features from the CLIP-ViT and gen- + +T +CLIP +CLIP +Feature +Feature +2 +Replicate +ReLU +ReLU +Conv +Conv +Conv +Conv +CLIP +ReLU +ReLU +Conv +Conv +ReLU +XN-1 +Extracted +Feature +Feature +Adversarial +loss + : CLIP Feature Extractor +: Quality Assessor + : Extraction BlockFigure 5. The architecture of the proposed CLIP-empowered generator for text-to-image synthesis. Armed with bridge feature predictor +and prompt predictor, it can induce meaningful visual concepts from the frozen CLIP-ViT for image synthesis. +erate images from text and induced visual features. The +Mate-G consists of a Bridge Feature Predictor (Bridge-FP), +a prompt predictor, a frozen CLIP-ViT, and an image gen- +erator (see Figure 3). We detail them next. +Bridge Feature Predictor. The structure of the Bridge- +FP is shown in Figure 5, as highlighted by the red dashed +box. The Bridge-FP consists of an FC (Fully-Connected) +layer and M fusion blocks (F-BLKs). The input noise is +fed into the FC layer and reshaped to (7, 7, 64) as an ini- +tial bridge feature. The initial bridge feature output by the +FC layer still contains a lot of noise. Therefore, we apply +a sequence of F-BLKs to fuse text information and make it +more meaningful. The F-BLK is composed of two convo- +lution layers (Conv) and two deep text-image fusion blocks +(DFBlock) [42]. The DFBlock has shown its effectiveness +in fusing text and image features through stacked affine +transformations. +Thus, we adopt it to fuse text features +and intermediate bridge features. There is a shortcut ad- +dition in F-BLK for effective information propagation and +gradient back-propagation. Through the Bridge-FP, the sen- +tence and noise vectors will be translated to the bridge fea- +ture, which is adjusted to induce meaningful visual concepts +from CLIP-ViT. +Prompt Predictor. The CLIP-ViT is pretrained to predict +visual features from image data. There is a large gap be- +tween text and image data. To alleviate the difficulty of +bridge feature translation from text features, we employ +prompt tuning [16], which has shown effectiveness on do- +main transferring for ViT. We design a prompt predictor, +which predicts prompts based on sentence and noise vec- +tors through an FC layer. The predicted text-conditioned +prompts are appended behind the visual patch embeddings +in CLIP-ViT. Furthermore, we find that it is better not to +add prompts to the last few layers in CLIP-ViT. The last +few layers summarize the visual features and output the last +image representations. The prompt predicted from text and +noise in the last few layers may defect its performance. +Image Generator. +The image generator consists of K +generation blocks (G-BLKs). +We sum the predicted vi- +sual concepts and bridge features through shortcut addition +for effective information propagation and gradient back- +propagation. The image generator receives the summed vi- +sual features as input and fuses sentence and noise vectors +through the DFBlocks [42] in each G-BLK. The interme- +diate image features grow larger during the generation pro- +cess by the upsample layers. Finally, the image features are +converted into high-resolution RGB images. +3.4. Objective Functions +To stabilize the training process of adversarial learning, +we employ the hinge loss [52] and one-way discriminator +[42]. Finally, the whole formulation of our GALIP is shown +as follows: +LD = − Ex∼Pr[min(0, −1 + D(C(x), e))] +− (1/2)EG(z,e)∼Pg[min(0, −1 − D(C(G(z, e)), e))] +− (1/2)Ex∼Pmis[min(0, −1 − D(C(x), e))] ++ kEx∼Pr[(∥∇C(x)D(C(x), e)∥ + ∥∇eD(C(x), e)∥)p], +LG = − EG(z,e)∼Pg[D(C(G(z, e)), e)] +− λEG(z,e)∼Pg[S(G(z, e), e)], +(1) +where z is the noise vector sampled from Gaussian distri- +bution; e is the sentence vector; G is the CLIP-empowered +generator; D is the Mate-D; C is the frozen CLIP-ViT in +CLIP-based discriminator; S represents the cosine similar- +ity between the encoded visual and text features of CLIP; +k and p are two hyper-parameters of gradient penalty; λ is +the coefficients of the text-image similarity;Pg, Pr, Pmis +denote the synthetic data distribution, real data distribution, +and mismatching data distribution, respectively. +4. Experiments +In this section, we introduce the datasets, training details, +and evaluation metrics employed in our experiments, then +evaluate our proposed GALIP and its variants quantitatively +and qualitatively. +Datasets. +We conduct experiments on four challenging +datasets: CUB bird [44], COCO [24], CC3M [40], and +CC12M [2]. For the CUB bird dataset, there are 11,788 +images belonging to 200 bird species, with each image cor- +responding to ten language descriptions. The train and vali- + + T +FC +z +P,-..Pn.P. +Reshape +DFBlock +DFBlock +DFBlock +Conv +Conv +Conv +Conv +FC ++ +Bridge +CLIP-ViT +xM +XK +Feature +Concepts +: Bridge Feature Predictor +: Prompt Predictor + : Generation Block +: Image Generator +: Fusion BlockFigure 6. Examples of images synthesized by LAFITE [56], VQ-Diffusion [11], and our proposed GALIP conditioned on text descriptions +from the test set of CUB and COCO datasets. +dation splits of the CUB bird dataset are implied as previous +works did [42,48,54,55,57]. Since there are various shapes, +colors, and postures of birds in the CUB dataset, it is always +employed to evaluate the performance of fine-grained con- +tent synthesis. For COCO dataset, it contains 80k images +for training and 40k images for testing. Each image corre- +sponds to 5 language descriptions. The image in the COCO +dataset is complex and always contains multiple objects un- +der different scenes. The COCO dataset is always employed +in recent works to evaluate the performance of complex im- +age synthesis. For CC3M and CC12M datasets, they are +two large datasets that contain about 3 and 12 million text- +image pairs. It is always adopted for pretraining and to eval- +uate the zero-shot performance of the text-to-image model. +Training and Evaluation Details. We choose the ViT-B/32 +[30] model as the CLIP model in our GALIP. In the CLIP- +based discriminator, the CLIP-FE collects the CLIP feature +from 2nd, 5th, 9th layers in CLIP-ViT. There are two extrac- +tion blocks stacked in CLIP-FE. In the CLIP-empowered +generator, the Bridge-FP contains 4 Fusion Blocks, and the +image generator contains 6 generation blocks for 224×224 +image synthesis. The prompt predictor predicts 8 prompts +for TransBlocks 2 to 10 in CLIP-ViT. We conduct some +ablation studies on these designs. +The hyper-parameters +of the discriminator k and p are set to 2 and 6 as [42]. +The hyper-parameters of the generator λ are set to 4 for +all the datasets. Furthermore, we employ the Adam opti- +mizer [17] with β1=0.0 and β2=0.9 to train our model. Ac- +cording to the two timescale update rule (TTUR) [12], the +learning rate is set to 0.0001 for the generator and 0.0004 +for the discriminator. Following the previous text-to-image +works [42, 47, 48, 57], we adopt the Fr´echet Inception Dis- +tance (FID) [12] and CLIPSIM [47] to evaluate the image +fidelity and text-image semantic consistency. All GALIP +models are trained on 8×3090 GPUs. We train our GALIP +for 0.5, 1.5, 2, and 3 days on CUB, COCO, CC3M, and +CC12M datasets, respectively. +Table 1. The results of FID and CLIPSIM (CS) compared with the +state-of-the-art methods on the test set of CUB and COCO. +Model +CUB +COCO +FID ↓ +CS ↑ +FID ↓ +CS ↑ +DM-GAN [57] +16.09 +- +32.64 +- +XMC-GAN [53] +- +- +9.30 +- +DAE-GAN [37] +15.19 +- +28.12 +- +DF-GAN [42] +14.81 +0.2920 +19.32 +0.2972 +LAFITE [56] +14.58 +0.3125 +8.21 +0.3335 +VQ-Diffusion [11] +10.32 +- +13.86 +- +GALIP (Ours) +10.08 +0.3164 +5.85 +0.3338 +4.1. Quantitative Evaluation +To evaluate the performance of our GALIP, we com- +pare the proposed model with several state-of-the-art meth- +ods [11,37,42,53,56,57], which have achieved impressive +results in text-to-image synthesis. The results are shown in +Table 1. Compared with other leading models, our GALIP +has a significant improvement on both CUB and COCO +datasets. Especially, compared with the recently proposed +LAFITE [56], which employs CLIP text-image contrastive +loss for text-to-image training, our GALIP decreases the +FID metric from 14.58 to 10.08 and improves the CLIPSIM +(CS) from 0.3125 to 0.3164 on the CUB dataset. Further- +more, our GALIP decreases the FID of COCO from 8.21 +to 5.85 significantly. Compared with VQ-diffusion [11], +which adopts diffusion models for text-to-image synthesis, +our GALIP also decreases FID from 10.32 to 10.08 on the +CUB dataset and decreases the FID of COCO from 13.86 +to 5.85 remarkably. The quantitative comparisons on CUB +and COCO datasets demonstrate that our GALIP is more +effective in synthesizing high-fidelity images, especially for +complex image generation. +Moreover, we evaluate the zero-shot text-to-image syn- +thesis ability of our GALIP. The results are shown in Ta- +ble 2. Compared with LAFITE [56] trained on CC3M, our +GALIP (CC3M) decreases FID from 26.94 to 16.12 signif- + +this tiny bird has a +a small brown +this bird has a +this bird has +this bird has +this bird has blue + A blue and red +Three children +A tray of cupcakes is + Three people smile +A woman standing there is a stuffed +in front of an oven +as they pose standing +bird with a lighter +set on top of a +yellow bill with a +wings that are +wings that are +train leaving the +showing off toy +very small bill, a + crown, a short +bear sitting on a + kitchen counter. +together. + shade of brown on +belly covered with +black with +white throat and + brown and has a + bill, and a +station. +cell phones while + book shelf +light brown + yellow lines and +rounded belly +white and delicate +its belly and +long neck +sitting in the grass. +a small beak +feathers and has a set + breast and a light +coverts. +of black rounded eyes. orange bill. +LAFITE +VQ-Diffusion + GALIPFigure 7. Text-to-Image samples from GALIP (CC12M) and Latent Diffusion (LAION-400M) [35,36]. We sample 16 images from each +given text description, and randomly select one as the final generation result +Table 2. We compare the performance of large pretrained autore- +gressive models (AR), diffusion models (DF), and GANs under +zero-shot setting on the COCO test dataset. +Model +Type +Param [B] +Data size [M] +ZS-FID ↓ +DALL-E [33] +AR +12 +250 +27.5 +Cogview [6] +AR +4 +30 +27.1 +Cogview2 [7] +AR +6 +30 +24.0 +Parti-350M [51] +AR +0.35 +>800 +14.10 +Parti-20B [51] +AR +20 +>800 +7.23 +GLIDE [27] +DF +5 +250 +12.24 +LDM [35] +DF +1.45 +400 +12.63 +DALL·E 2 [32] +DF +6.5 +250 +10.39 +Imagen [38] +DF +7.9 +860 +7.27 +eDiff-I [1] +DF +9.1 +1000 +6.95 +LAFITE [56] +GAN +0.15+0.08 +3 +26.94 +GALIP (CC3M) +GAN +0.24+0.08 +3 +16.12 +GALIP (CC12M) +GAN +0.24+0.08 +12 +12.54 +icantly. It demonstrates that integrating the CLIP model +in the generator and discriminator is more effective than +only introducing the CLIP loss for the GAN model. Com- +pared with autoregressive models (AR) and diffusion mod- +els (DF) which are pretrained with much larger model sizes +and datasets, our GALIP also achieves competitive perfor- +mance. Especially, compared with LDM [35] which is one +of the most important open-source large pretrained models, +our GALIP achieves better performance even with much +smaller model parameters and data size. Furthermore, as +shown in Figure 2, our GALIP only requires 0.04s to gen- +erate one image which is ∼120×faster than LDM [35]. Be- +sides, our GALIP can be inference on the CPU fastly with- +out other acceleration settings. This significantly reduces +the hardware requirements of users. In addition, the compu- +tational cost to pretrain our GALIP is quite less than these +large pretrained autoregressive and diffusion models. The +GALIP of CC12M is only pretrained on 8×3090 GPUs for +3 days. But these models require hundreds of GPUs and +many weeks to pre-train. +4.2. Qualitative Evaluation +To evaluate the visual quality of synthesized images, we +first compare the images synthesized by LAFITE [56], VQ- +Diffusion [11], and our GALIP which are trained on COCO +in Figure 6. Then, we compare our GALIP (CC12M) with +LDM (LAION-400M) [35,36] in Figure 7. +As shown in the 1st, 2nd, 4th and 5th columns of Fig- +ure 6, the birds synthesized by LAFITE [56] and VQ- +Diffusion [11] contain break or wrong shapes. Moreover, +both LAFITE [56] and VQ-Diffusion [11] lose some fine- +grained visual features (e.g., 1st, 2nd, 5th and 6th columns), +which makes the synthesized images lack details and look +unreal. However, the images synthesized by our GALIP +have correct object shapes and clear fine-grained contents. +The superiority is more obvious in complex COCO im- +ages, which contain various shapes and multiple objects. As +the results are shown in the 7th, 8th, 9th, 10th columns of +Figure 7, the LAFITE [56] and VQ-Diffusion [11] mod- +els cannot synthesize the right shape of “train”, “children”, +“woman”, and “stuffed bear”. Furthermore, they also can- +not synthesize the right visual concept of “showing off toy +cell phone” and “sitting on a book shelf”. However, armed +with the proposed CLIP-based D and CLIP-empowered +G, our GALIP can cope with more strict visual require- +ments and synthesize various shapes of different objects +(see 8th, 9th, 10th and 12th columns) and present the right +visual concepts in synthesized images. We also observe that +LAFITE [56] and VQ-Diffusion [11] also can not synthe- +size correct human facial features. For example, as shown +in the 8th, 9th, 12th, they can not synthesize realistic hu- +man faces. But our GALIP can synthesize these features +correctly. +Moreover, we compare the images synthesized by the +LDM (LAION-400M) [35, 36] and our GALIP (CC12M) +in Figure 7. As the results are shown in the 1st, 4th, 5th, +8th, 11th columns of Figure 7, the LDM does not gener- +ate the objects (“ghost”, “teddy bear”, “modem”, “person”, +“model”) described in the texts, but our GALIP can synthe- +size these objects correctly. Also, our model can generate +correct visual features such as “shining eyes”, “Blue Light- +house”, “smiling statue”, and “surprised girl” in the 3rd, +6th, 7th, 10th columns. Furthermore, as shown in the 9th, +10th, and 12th columns of Figure 7, our GALIP keeps the +superior performance of human face synthesis. The exten- +sive quantitative evaluation results demonstrate the superi- + +A 6'1" model is + newly - weds +line drawing +A smiling statue +person shining a +A girl with a laptop Portrait of a +Cute cell cartoon + Cute cartoon little +little teddy bear +The 4G LTE UE Watercolor of Blue +with matching + to be found in + surprised girl in +illustration of a +character showing owl with big +wearing a size M +after the wedding +modem +Lighthouse. Lovely +light into the stars + ceremony + a falling-in-love +shining eyes sitting + vest and skirt set +blue sky and dark +the Pavilion +dress and +classic oxford-style +kawaii cute ghost + on a tree branch at + blue light house on a +eyeglasses +Short-Sleeve Shirt in +face. Vector +illustration. + starry night +hillside. +Sky Blue +4GFigure 8. Images synthesized by interpolating four-sentence em- +beddings. Our GALIP supports gradual changes when interpo- +lating sentence embeddings describing different image styles. It +makes the degree of stylization of the image controllable and cre- +ates new styles by blending different styles. +ority and effectiveness of our proposed GALIP, which is +able to generate high-fidelity, creative and complex images +with various shapes and multiple objects. +Additionally, we conduct some experiments to show the +smooth latent space of our GALIP. Current autoregressive +and diffusion models are sensitive to input sentences. This +instability makes users need to try a lot of prompts to get sat- +isfied images. Differently, our GALIP inherits the smooth +latent space from GAN, it enables gradual and smooth +changes along with text changes. As shown in Figure 8, +there is a smooth transition of synthesized images from top +to bottom, left to right. The smooth latent space makes the +degree of stylization of the image controllable. The users +can fine-tune synthesized image styles like a style knob, and +it also enables the users to create new styles by blending dif- +ferent image styles, as highlighted by the red dashed lines. +4.3. Ablation Study +To verify the effectiveness of each component in the pro- +posed GALIP, we conduct ablation studies on the test set +of the COCO dataset. +The components being evaluated +in this subsection include CLIP-based D (CD) and CLIP- +empowered G (CG). We also further conduct ablation stud- +ies on Bridge-FP (BFP) and Prompt Predictor (PP) in CLIP- +empowered G, and CLIP-FE (CFE) in CLIP-based D. Fur- +thermore, we compare our CLIP-FE with CCM&CSM of +Table 3. The performance of different components of our model +on the test set of COCO. +Architecture +FID ↓ +CS ↑ +Baseline +17.31 +0.2996 +Baseline w/ CD w/ CFE +7.92 +0.3221 +Baseline w/ CD w/ CCM&CSM +10.77 +0.3123 +Baseline w/ CD w/ BFP +6.52 +0.3301 +Baseline w/ CD w/ BFP w/ PP (GALIP) +5.85 +0.3338 +GALIP w/ CFE (2nd) +13.41 +0.3015 +GALIP w/ CFE (5th) +8.60 +0.3145 +GALIP w/ CFE (12th) +10.72 +0.3104 +GALIP w/ CFE (2nd,5th) +6.70 +0.3301 +GALIP w/ CFE (2nd,5th,12th) +6.61 +0.3305 +GALIP w/ CFE (2nd,5th,9th) +5.85 +0.3338 +GALIP w/ CFE (2nd,5th,8th,9th) +6.01 +0.3305 +GALIP w/ PP (1st-12th) +6.24 +0.3320 +GALIP w/ PP (1st-9th) +5.85 +0.3338 +GALIP w/ PP (1st-6th) +6.40 +0.3310 +GALIP w/ PP (1st-3th) +6.52 +0.3305 +Projected GAN [39], which yields a U-Net architecture to +enable multi-scale feedback. In addition, we investigate the +layer choice strategy of CLIP-FE and Prompt Predictor. The +results on the COCO dataset are shown in Table 3. +Baseline. Our baseline is a one-stage text-to-image GAN +[42]. It is composed of a CLIP text encoder and CNN-based +generator and discriminator. And it generates complex im- +ages from sentence vectors directly. +Effect of CLIP-based D and CLIP-FE. The CLIP-based +D decreases FID from 17.31 to 7.92 and improves CLIM- +SIM (CS) from 0.2996 to 0.3221. +The results demon- +strate that the complex scene understanding ability of CLIP- +ViT promotes the complex image synthesis ability signifi- +cantly. Furthermore, we compared our CLIP-FE (CFE) with +CCM&CSM [39]. Our CLIP-FE achieves better FID and +CLIPSIM. It shows that our CLIP-FE is more effective in +extracting informative visual features from CLIP-ViT. +Effect of CLIP-empowered G and Bridge-FP. The CLIP- +empowered G with Bridge-FP further decreases FID from +7.92 to 6.52 and improves CLIPSIM from 0.3221 to 0.3301. +It demonstrates that predicted bridge features and CLIP-ViT +can enhance the complex image synthesis ability effectively. +Effect of Prompt Predictor. The proposed Prompt Predic- +tor (PP) also decreases FID from 6.52 to 5.85 and improves +CLIPSIM from 0.3301 to 0.3338. The result demonstrates +that the Prompt Predictor makes the CLIP-ViT more suit- +able for generation tasks and induces more meaningful fea- +tures from CLIP-ViT to improve the generative ability. +CLIP Layer Selection. +We find that the last few lay- +ers of CLIP-ViT defect the performance of CLIP-based D. +The reason may be that the first layers of CLIP-ViT ex- +tract useful visual features and understand complex images, +and the last layers focus on generalization ability to align + +interpolation +A beautiful home +A beautiful home +with a large yard +with a large yard +Watercolor +interpolation +A beautiful home +interpolation +A beautiful home +with a large yard. +with a large yard +Oil painting +Vector illustrationFigure 9. +Illustration of failure cases. +It is still hard for cur- +rent GALIP to synthesize some imaginary images. Enlarging the +model size and training data may improve image quality. +with high-level concepts in text features. The generalization +ability may defect the performance of CLIP-based D be- +cause it reduces the differences between synthetic and real +images and weakens the discriminator. Conversely, since +CLIP-empowered G requires the generalization ability to +map the bridge feature to meaningful visual features, adding +prompts in the last few layers may defect the generalization +ability. So we extract the CLIP features from 2nd,5th,and +9th layers in CLIP-based D, and add prompts to 1st-9th lay- +ers. And we find that extracting more CLIP features does +not lead to better performance. +4.4. Limitations +Our GALIP shows superiority in text-to-image synthe- +sis, but some limitations should be considered in future +studies. First, our model employs the CLIP model to pro- +vide text features for the generator and discriminator. How- +ever, current models [38] show that the generic large lan- +guage models [31] (e.g., T5) improve the performance of +text-to-image synthesis effectively. +Replacing the CLIP +text encoder with T5 may further improve the performance. +Second, the model size and pretraining dataset are much +smaller than other large pretrained models [1,32,35,38,51], +it limits the synthesis ability of imaginary images (see Fig- +ure 9). Pretraining on a larger dataset with a larger model +size may benefit the performance. We will try to address +these limitations in our future work. +5. Conclusion +In this paper, we propose a novel Generative Adversar- +ial CLIPs (GALIP) for text-to-image synthesis. Compared +with previous models, our GALIP can synthesize higher- +quality complex images. Moreover, we propose a CLIP- +based discriminator and CLIP-empowered generator, which +exerts the complex scene understanding and domain gener- +alization ability of CLIP. 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In CVPR, 2019. 1, 3, 6 + diff --git a/VdFOT4oBgHgl3EQf6jSQ/content/tmp_files/load_file.txt b/VdFOT4oBgHgl3EQf6jSQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f98a4e3c470abe54bda7c8d8d70185bec15c3e3 --- /dev/null +++ b/VdFOT4oBgHgl3EQf6jSQ/content/tmp_files/load_file.txt @@ -0,0 +1,677 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf,len=676 +page_content='GALIP: Generative Adversarial CLIPs for Text-to-Image Synthesis Ming Tao1 Bing-Kun Bao1,2* Hao Tang3 Changsheng Xu2,4,5 1Nanjing University of Posts and Telecommunications 2Peng Cheng Laboratory 3CVL, ETH Z¨urich 4University of Chinese Academy of Sciences 5NLPR, Institute of Automation, CAS Abstract Synthesizing high-fidelity complex images from text is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Based on large pretraining, the autoregres- sive and diffusion models can synthesize photo-realistic im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Although these large models have shown notable progress, there remain three flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 1) These models re- quire tremendous training data and parameters to achieve good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 2) The multi-step generation design slows the image synthesis process heavily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 3) The syn- thesized visual features are difficult to control and require delicately designed prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' To enable high-quality, effi- cient, fast, and controllable text-to-image synthesis, we pro- pose Generative Adversarial CLIPs, namely GALIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' GALIP leverages the powerful pretrained CLIP model both in the discriminator and generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Specifically, we propose a CLIP-based discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The complex scene understand- ing ability of CLIP enables the discriminator to accurately assess the image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Furthermore, we propose a CLIP- empowered generator that induces the visual concepts from CLIP through bridge features and prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The CLIP- integrated generator and discriminator boost training ef- ficiency, and as a result, our model only requires about 3% training data and 6% learnable parameters, achieving com- parable results to large pretrained autoregressive and diffu- sion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Moreover, our model achieves ∼120×faster synthesis speed and inherits the smooth latent space from GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The extensive experimental results demonstrate the excellent performance of our GALIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='com/tobran/GALIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Introduction Over the last few years, we have witnessed the great suc- cess of generative models for various applications [4, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Among them, text-to-image synthesis [3, 15, 18–21, 25, 28, 29, 33, 46, 48–50, 57] is one of the most appealing applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It generates high-fidelity images according to given Corresponding Author Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' (a) Existing text-to-image GANs conduct adversarial training from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' (b) Our proposed GALIP conducts adver- sarial training based on the integrated CLIP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' language guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Owing to the convenience of language for users, text-to-images synthesis has attracted many re- searchers and become an active research area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Based on a large scale of data collections, model size, and pretraining, recently proposed large pretrained autore- gressive and diffusion models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=', DALL-E [33] and LDM [35], show the impressive generative ability to syn- thesize complex scenes and outperform the previous text-to- image GANs significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Although these large pretrained generative models have achieved significant advances, they still suffer from three flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' First, these models require tremendous training data and parameters for pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The large data and model size brings an extremely high computing budget and hardware requirements, making it inaccessible to many researchers and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Second, the generation of large models is much slower than GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The token-by-token generation and progressive denoising require hundreds of inference steps and make the generated results lag the language inputs seriously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Third, there is no intuitive smooth latent space as GANs, which maps mean- ingful visual attributes to the latent vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The multi-step generation design breaks the synthesis process and scatters the meaningful latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It makes the synthesis process require delicately designed prompts to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' To address the above limitations, we rethink Generative Adversarial Networks (GAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' GANs are much faster than autoregressive and diffusion models and have smooth latent arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='12959v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='CV] 30 Jan 2023 →Loss Generator Discriminator a Actor and his Text wife attend the Encoder premiere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' CLIP-ViT CLIP-ViT →LosS (b) Mate-G Mate-D Actor and his Text I wife attend the Encoder premiere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Comparing with Latent Diffusion Models (LDM) [35], our GALIP achieves comparable zero-shot Fr´echet Inception Distance (ZS-FID) with measly 320M parameters (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='08B trainable parameters + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='24B frozen CLIP parameters) and 12M training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Furthermore, our GALIP only requires 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='04s to synthesize one image which is ∼120×faster than LDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Speed is calculated on NVIDIA 3090 GPU and Intel Xeon Silver 4314 CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' space, which enables more controllable synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' How- ever, GAN models are known for potentially unstable train- ing and less diversity in the generation [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It makes current text-to-image GANs suffer from unsatisfied synthesis qual- ity under complex scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' In this work, we introduce the pretrained CLIP [30] into text-to-image GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The large pretraining of CLIP brings two advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' First, it enhances the complex scene un- derstanding ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The pretraining dataset has many com- plex images under different scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Armed with the Vision Transformer (ViT) [8], the image encoder can extract in- formative and meaningful visual features from complex im- ages to align the corresponding text descriptions after ade- quate pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Second, the large pretraining dataset also enables excellent domain generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It contains various kinds of images, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=', photos, drawings, cartoons, and sketches, collected from a variety of publicly available sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The various images make the CLIP model can map different kinds of images to the shared concepts and en- able impressive domain generalization and zero-shot trans- fer ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' These two advantages of CLIP, complex scene understanding and domain generalization ability, motivate us to build a more powerful text-to-image model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We propose a novel text-to-image generation framework named Generative Adversarial CLIPs (GALIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' As shown in Figure 1, the GALIP integrates the CLIP model [30] in both the discriminator and generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' To be specific, we pro- pose the CLIP-based discriminator and CLIP-empowered generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The CLIP-based discriminator inherits the com- plex scene understanding ability of CLIP [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It is com- posed of a frozen ViT-based CLIP image encoder (CLIP- ViT) and a learnable mate-discriminator (Mate-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The Mate-D is mated to the CLIP-ViT for adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' To retain the knowledge of complex scene understanding in the CLIP-ViT, we freeze its weights and collect the pre- dicted CLIP image features from different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Then, the Mate-D further extracts informative visual features from collected CLIP features to distinguish the synthesized and real images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Based on the complex scene understanding ability of CLIP-ViT and the continuous analysis of Mate- D, the CLIP-based discriminator can assess the quality of generated complex images more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Furthermore, we propose the CLIP-empowered genera- tor, which exerts the domain generalization ability of CLIP [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It is hard for the generator to synthesize complex im- ages directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Some works employ sketch [10] and lay- out [20, 22] as bridge domains to alleviate the difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' However, such a design requires additional labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Different from these works, the excellent domain general- ization of CLIP [30] motivates us that there may be an im- plicit bridge domain, which is easier to synthesize but can be mapped to the same visual concepts through the CLIP- ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Thus, we design the CLIP-empowered generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It is composed of a frozen CLIP-ViT and a learnable mate- generator (Mate-G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The Mate-G first predicts the implicit bridge features from text and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Then the bridge feature will be mapped to the visual concepts through CLIP-ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Furthermore, we add some text-conditioned prompts to the CLIP-ViT for task adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The predicted visual con- cepts close the gap between text features and target images which enhances the complex image synthesis ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' As shown in Figure 2, the proposed GALIP achieves ∼120×faster synthesis speed and comparable synthesis ability based on significantly smaller trainable parameters and training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Overall, our contributions can be summarized as follows: We propose an efficient, fast, and more controllable model for text-to-image synthesis that can synthesize high-quality complex images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We propose the CLIP-based discriminator, which as- sesses the quality of generated complex images more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We propose the CLIP-empowered generator, which synthesizes images based on text features and pre- dicted CLIP visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Extensive experiments demonstrate that the proposed GALIP can achieve comparable performance with large pertaining models based on significantly smaller computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='45B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='5 13 400M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='83s GPU 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='63 400 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='54 a size (Million) CPU illion) 4 - 300 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='0 12 speed ( 3- 200 Data S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='5 - 11- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='32B 100- 11 12M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='04s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='20s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='0 0 0 10 LDM GALIP LDM GALIP LDM GALIP LDM GALIPFigure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The architecture of the proposed GALIP for text-to-image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Armed with the CLIP-based discriminator and CLIP- empowered generator, our model can synthesize more realistic complex images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Related Work Text-to-Image GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' GAN-INT-CLS [34] first adopted conditional GANs to synthesize images from text descrip- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' To enable higher resolution synthesis, the Stack- GAN [54, 55], AttnGAN [48], and DM-GAN [57] stacks multiple generators and discriminators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' [42] proposed a simpler yet effective text-to-image framework called DF-GAN that enables one-stage high-resolution gen- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' LAFITE [56] introduces CLIP text-image con- trastive loss for text-to-image training and shows large im- provements on CC3M [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Text-to-Image Large Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Recently, large pretrained autoregressive and diffusion models have shown impres- sive performance on text-to-image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' DALL-E [33], CogView [6], and M6 [23] leverage VQ-VAE [43] or VQ- GAN [9] to tokenize the images into discrete image tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Then they take the word tokens and image tokens together to pre-train a large unidirectional transformer for an au- toregressive generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Parti [51] proposes a sequence-to- sequence autoregressive model to treat text-to-image syn- thesis as a translation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Cogview2 [7] employs hierar- chical transformers and local parallel autoregressive gen- eration for faster autoregressive image generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Some works try to employ the diffusion model [5,13,14,26,41] to overcome the slow generation defect of the autoregressive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' VQ-Diffusion [11] combines the VQ-VAE [43] and diffusion model [14,26] to eliminate the unidirectional bias and avoids accumulated prediction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' GLIDE [27] ap- plies guided diffusion to the problem of text-conditional im- age synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' DALL-E2 [32] combines the CLIP represen- tation and diffusion model to make a CLIP decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Latent Diffusion Models (LDM) [35] apply the diffusion model in the latent space to enable the training on limited compu- tational resources while retaining image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The par- ticular text-to-image LDM is Stable Diffusion [36], which is a favorite open-source project and provides an easy-to- use interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Imagen [38] introduces the large language model [31] to provide high-quality text features and pro- poses an Efficient U-Net for diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Generative Adversarial CLIPs In this paper, we propose a novel framework for text- to-image synthesis named Generative Adversarial CLIPs (GALIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' To synthesize high-quality complex images, we propose: (i) a novel CLIP-based discriminator that inherits the complex scene understanding ability of CLIP [30] for more accurate image quality assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' (ii) a novel CLIP- empowered generator that exerts the domain generalization ability of CLIP [30] and induces the CLIP visual concepts to close the gap between text and image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' In the fol- lowing of this section, we first present the overall structure of our GALIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Then, we introduce the CLIP-based discrim- inator and CLIP-empowered generator in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Model Overview As shown in Figure 3, the proposed GALIP is composed of a CLIP text encoder, a CLIP-based discriminator, and a CLIP-empowered generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The pretrained CLIP text en- coder takes the text description and yields a global sentence vector T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' After the text-encoder is the CLIP-empowered Z : noise~N(0,1) I : CLIP-ViT p" T : Sentence Vector I : Mate-G P : Prompt TransBlock TransBlock TransBlock TransBlock TransBlock TransBlock TransBlock Reshape Reshape : Mate-D G : Generator Visual D : Discriminator Concepts Z Actor and his wife Text Image Bridge T Bridge-FP attend the premiere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Generator Encoder Feature G: D: TransBlock TransBlock TransBlock lock lock Reshape Conv TransBlc TransBl Quality B CLIP-FE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' TransE Assessor Adversarial lossFigure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The architecture of the proposed Mate-D for text-to-image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It further extracts informative visual features from collected CLIP features and assesses the image quality more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' generator and CLIP-based discriminator under the GAN framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The CLIP-empowered generator is composed of a frozen CLIP-ViT and a mate generator (Mate-G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' There are three main modules in the Mate-G, the bridge feature predictor (Bridge-FP), the prompt predictor, and the image generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The CLIP-empowered generator has two inputs, the sentence vector T encoded from the text encoder and the noise vector Z sampled from the Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The noise vector ensures the diversity of the synthesized im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' In the CLIP-empowered generator, the sentence vec- tor and noise are first fed into the bridge feature predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The bridge feature predictor translates the sentence vector and noise to the bridge feature for the CLIP-ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Further- more, we add several text-conditioned prompts to the trans- former blocks (TransBlock) in CLIP-ViT for task adapta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Finally, the image generator takes the predicted visual concepts, bridge features, sentences, and noise vectors to synthesize high-quality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The CLIP-based discriminator is composed of a frozen CLIP-ViT and a mate discriminator (Mate-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The CLIP- ViT converts images into image feature through a convolu- tion layer and a series of transformer blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The CLIP fea- ture extractor (CLIP-FE) in Mate-D collects the image fea- tures from different layers in CLIP-ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Then it further ex- tracts informative visual features from collected CLIP fea- tures for the quality assessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Lastly, an adversarial loss will be predicted by the quality assessor based on the extracted informative features and sentence vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' By distinguish- ing synthesized images from real ones, the discriminator promotes the generator to synthesize higher-quality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' CLIP-based Discriminator In this section, we detailed the proposed CLIP-based dis- criminator, which is composed of a frozen CLIP-ViT and a Mate-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The CLIP-based discriminator inherits the com- plex scene understanding ability from the frozen CLIP-ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Furthermore, we propose the Mate-D, which is mated to the CLIP-ViT to further extract informative visual features and distinguish real and synthesized images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The CLIP-ViT and Mate-D enable the discriminator to assess the quality of generated complex images more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' As shown in Figure 4, the Mate-D consists of a CLIP- FE and a quality assessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' To fully utilize the knowledge of complex scene understanding in CLIP-ViT, the CLIP-FE takes the CLIP image features from multilayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' There are N CLIP features collected for the CLIP-FE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We name them CLIP Feature 1 to N, which are collected from shallow to deep layers in CLIP-ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' To further extract informative vi- sual features from these CLIP features, we design a CLIP- FE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It contains a sequence of extraction blocks, and each block contains two convolution layers and two ReLU active functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' And the extracted image feature is summed with the shortcut and the next CLIP feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' There are N − 1 extraction blocks stacked in CLIP-FE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Since the CLIP fea- ture N is only added to the processed image features in the last extraction block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' To fuse the CLIP feature N, we ap- pend two convolution layers without the CLIP feature addi- tion behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The CLIP-FE extracts informative visual fea- tures for the quality assessor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Then the sentence vector is replicated and concatenated with the extracted image fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' An adversarial loss is predicted by two convolution layers to evaluate the image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Furthermore, to stabi- lize the adversarial learning process of Mate-D, we apply the matching-aware gradient penalty (MAGP) [42] on the collected CLIP features and corresponding text features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Based on the complex scene understanding ability of CLIP-ViT, the CLIP-based discriminator can extract more informative visual features from complex images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The higher-quality extracted visual features make it easier for the discriminator to detect unreal image parts, which im- proves the discriminative efficiency, thus prompting the generator to generate more realistic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' CLIP-empowered Generator In this section, we detail the proposed CLIP-empowered generator, which is composed of a frozen CLIP-ViT and a Mate-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The CLIP-empowered generator exerts the do- main generalization ability of the CLIP-ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Furthermore, we propose the Mate-G, which is mated to the CLIP-ViT to induce useful visual features from the CLIP-ViT and gen- T CLIP CLIP Feature Feature 2 Replicate ReLU ReLU Conv Conv Conv Conv CLIP ReLU ReLU Conv Conv ReLU XN-1 Extracted Feature Feature Adversarial loss : CLIP Feature Extractor : Quality Assessor : Extraction BlockFigure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The architecture of the proposed CLIP-empowered generator for text-to-image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Armed with bridge feature predictor and prompt predictor, it can induce meaningful visual concepts from the frozen CLIP-ViT for image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' erate images from text and induced visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The Mate-G consists of a Bridge Feature Predictor (Bridge-FP), a prompt predictor, a frozen CLIP-ViT, and an image gen- erator (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We detail them next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Bridge Feature Predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The structure of the Bridge- FP is shown in Figure 5, as highlighted by the red dashed box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The Bridge-FP consists of an FC (Fully-Connected) layer and M fusion blocks (F-BLKs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The input noise is fed into the FC layer and reshaped to (7, 7, 64) as an ini- tial bridge feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The initial bridge feature output by the FC layer still contains a lot of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Therefore, we apply a sequence of F-BLKs to fuse text information and make it more meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The F-BLK is composed of two convo- lution layers (Conv) and two deep text-image fusion blocks (DFBlock) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The DFBlock has shown its effectiveness in fusing text and image features through stacked affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Thus, we adopt it to fuse text features and intermediate bridge features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' There is a shortcut ad- dition in F-BLK for effective information propagation and gradient back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Through the Bridge-FP, the sen- tence and noise vectors will be translated to the bridge fea- ture, which is adjusted to induce meaningful visual concepts from CLIP-ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Prompt Predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The CLIP-ViT is pretrained to predict visual features from image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' There is a large gap be- tween text and image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' To alleviate the difficulty of bridge feature translation from text features, we employ prompt tuning [16], which has shown effectiveness on do- main transferring for ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We design a prompt predictor, which predicts prompts based on sentence and noise vec- tors through an FC layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The predicted text-conditioned prompts are appended behind the visual patch embeddings in CLIP-ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Furthermore, we find that it is better not to add prompts to the last few layers in CLIP-ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The last few layers summarize the visual features and output the last image representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The prompt predicted from text and noise in the last few layers may defect its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Image Generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The image generator consists of K generation blocks (G-BLKs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We sum the predicted vi- sual concepts and bridge features through shortcut addition for effective information propagation and gradient back- propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The image generator receives the summed vi- sual features as input and fuses sentence and noise vectors through the DFBlocks [42] in each G-BLK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The interme- diate image features grow larger during the generation pro- cess by the upsample layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Finally, the image features are converted into high-resolution RGB images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Objective Functions To stabilize the training process of adversarial learning, we employ the hinge loss [52] and one-way discriminator [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Finally, the whole formulation of our GALIP is shown as follows: LD = − Ex∼Pr[min(0, −1 + D(C(x), e))] − (1/2)EG(z,e)∼Pg[min(0, −1 − D(C(G(z, e)), e))] − (1/2)Ex∼Pmis[min(0, −1 − D(C(x), e))] + kEx∼Pr[(∥∇C(x)D(C(x), e)∥ + ∥∇eD(C(x), e)∥)p], LG = − EG(z,e)∼Pg[D(C(G(z, e)), e)] − λEG(z,e)∼Pg[S(G(z, e), e)], (1) where z is the noise vector sampled from Gaussian distri- bution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' e is the sentence vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' G is the CLIP-empowered generator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' D is the Mate-D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' C is the frozen CLIP-ViT in CLIP-based discriminator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' S represents the cosine similar- ity between the encoded visual and text features of CLIP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' k and p are two hyper-parameters of gradient penalty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' λ is the coefficients of the text-image similarity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='Pg, Pr, Pmis denote the synthetic data distribution, real data distribution, and mismatching data distribution, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Experiments In this section, we introduce the datasets, training details, and evaluation metrics employed in our experiments, then evaluate our proposed GALIP and its variants quantitatively and qualitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We conduct experiments on four challenging datasets: CUB bird [44], COCO [24], CC3M [40], and CC12M [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' For the CUB bird dataset, there are 11,788 images belonging to 200 bird species, with each image cor- responding to ten language descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The train and vali- T FC z P,-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='.Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Reshape DFBlock DFBlock DFBlock Conv Conv Conv Conv FC + Bridge CLIP-ViT xM XK Feature Concepts : Bridge Feature Predictor : Prompt Predictor : Generation Block : Image Generator : Fusion BlockFigure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Examples of images synthesized by LAFITE [56], VQ-Diffusion [11], and our proposed GALIP conditioned on text descriptions from the test set of CUB and COCO datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' dation splits of the CUB bird dataset are implied as previous works did [42,48,54,55,57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Since there are various shapes, colors, and postures of birds in the CUB dataset, it is always employed to evaluate the performance of fine-grained con- tent synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' For COCO dataset, it contains 80k images for training and 40k images for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Each image corre- sponds to 5 language descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The image in the COCO dataset is complex and always contains multiple objects un- der different scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The COCO dataset is always employed in recent works to evaluate the performance of complex im- age synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' For CC3M and CC12M datasets, they are two large datasets that contain about 3 and 12 million text- image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It is always adopted for pretraining and to eval- uate the zero-shot performance of the text-to-image model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Training and Evaluation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We choose the ViT-B/32 [30] model as the CLIP model in our GALIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' In the CLIP- based discriminator, the CLIP-FE collects the CLIP feature from 2nd, 5th, 9th layers in CLIP-ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' There are two extrac- tion blocks stacked in CLIP-FE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' In the CLIP-empowered generator, the Bridge-FP contains 4 Fusion Blocks, and the image generator contains 6 generation blocks for 224×224 image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The prompt predictor predicts 8 prompts for TransBlocks 2 to 10 in CLIP-ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We conduct some ablation studies on these designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The hyper-parameters of the discriminator k and p are set to 2 and 6 as [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The hyper-parameters of the generator λ are set to 4 for all the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Furthermore, we employ the Adam opti- mizer [17] with β1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='0 and β2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='9 to train our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Ac- cording to the two timescale update rule (TTUR) [12], the learning rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='0001 for the generator and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='0004 for the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Following the previous text-to-image works [42, 47, 48, 57], we adopt the Fr´echet Inception Dis- tance (FID) [12] and CLIPSIM [47] to evaluate the image fidelity and text-image semantic consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' All GALIP models are trained on 8×3090 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We train our GALIP for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='5, 2, and 3 days on CUB, COCO, CC3M, and CC12M datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The results of FID and CLIPSIM (CS) compared with the state-of-the-art methods on the test set of CUB and COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Model CUB COCO FID ↓ CS ↑ FID ↓ CS ↑ DM-GAN [57] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='09 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='64 XMC-GAN [53] 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='30 DAE-GAN [37] 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='19 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='12 DF-GAN [42] 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='2920 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='2972 LAFITE [56] 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3125 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3335 VQ-Diffusion [11] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='32 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='86 GALIP (Ours) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3164 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3338 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Quantitative Evaluation To evaluate the performance of our GALIP, we com- pare the proposed model with several state-of-the-art meth- ods [11,37,42,53,56,57], which have achieved impressive results in text-to-image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Compared with other leading models, our GALIP has a significant improvement on both CUB and COCO datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Especially, compared with the recently proposed LAFITE [56], which employs CLIP text-image contrastive loss for text-to-image training, our GALIP decreases the FID metric from 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='58 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='08 and improves the CLIPSIM (CS) from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3125 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3164 on the CUB dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Further- more, our GALIP decreases the FID of COCO from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='21 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='85 significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Compared with VQ-diffusion [11], which adopts diffusion models for text-to-image synthesis, our GALIP also decreases FID from 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='32 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='08 on the CUB dataset and decreases the FID of COCO from 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='86 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='85 remarkably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The quantitative comparisons on CUB and COCO datasets demonstrate that our GALIP is more effective in synthesizing high-fidelity images, especially for complex image generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Moreover, we evaluate the zero-shot text-to-image syn- thesis ability of our GALIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The results are shown in Ta- ble 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Compared with LAFITE [56] trained on CC3M, our GALIP (CC3M) decreases FID from 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='94 to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='12 signif- this tiny bird has a a small brown this bird has a this bird has this bird has this bird has blue A blue and red Three children A tray of cupcakes is Three people smile A woman standing there is a stuffed in front of an oven as they pose standing bird with a lighter set on top of a yellow bill with a wings that are wings that are train leaving the showing off toy very small bill, a crown, a short bear sitting on a kitchen counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' shade of brown on belly covered with black with white throat and brown and has a bill, and a station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' cell phones while book shelf light brown yellow lines and rounded belly white and delicate its belly and long neck sitting in the grass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' a small beak feathers and has a set breast and a light coverts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' of black rounded eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' orange bill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' LAFITE VQ-Diffusion GALIPFigure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Text-to-Image samples from GALIP (CC12M) and Latent Diffusion (LAION-400M) [35,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We sample 16 images from each given text description, and randomly select one as the final generation result Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We compare the performance of large pretrained autore- gressive models (AR), diffusion models (DF), and GANs under zero-shot setting on the COCO test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Model Type Param [B] Data size [M] ZS-FID ↓ DALL-E [33] AR 12 250 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='5 Cogview [6] AR 4 30 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='1 Cogview2 [7] AR 6 30 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='0 Parti-350M [51] AR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='35 >800 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='10 Parti-20B [51] AR 20 >800 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='23 GLIDE [27] DF 5 250 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='24 LDM [35] DF 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='45 400 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='63 DALL·E 2 [32] DF 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='5 250 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='39 Imagen [38] DF 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='9 860 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='27 eDiff-I [1] DF 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='1 1000 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='95 LAFITE [56] GAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='15+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='08 3 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='94 GALIP (CC3M) GAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='24+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='08 3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='12 GALIP (CC12M) GAN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='24+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='08 12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='54 icantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It demonstrates that integrating the CLIP model in the generator and discriminator is more effective than only introducing the CLIP loss for the GAN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Com- pared with autoregressive models (AR) and diffusion mod- els (DF) which are pretrained with much larger model sizes and datasets, our GALIP also achieves competitive perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Especially, compared with LDM [35] which is one of the most important open-source large pretrained models, our GALIP achieves better performance even with much smaller model parameters and data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Furthermore, as shown in Figure 2, our GALIP only requires 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='04s to gen- erate one image which is ∼120×faster than LDM [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Be- sides, our GALIP can be inference on the CPU fastly with- out other acceleration settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' This significantly reduces the hardware requirements of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' In addition, the compu- tational cost to pretrain our GALIP is quite less than these large pretrained autoregressive and diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The GALIP of CC12M is only pretrained on 8×3090 GPUs for 3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' But these models require hundreds of GPUs and many weeks to pre-train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Qualitative Evaluation To evaluate the visual quality of synthesized images, we first compare the images synthesized by LAFITE [56], VQ- Diffusion [11], and our GALIP which are trained on COCO in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Then, we compare our GALIP (CC12M) with LDM (LAION-400M) [35,36] in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' As shown in the 1st, 2nd, 4th and 5th columns of Fig- ure 6, the birds synthesized by LAFITE [56] and VQ- Diffusion [11] contain break or wrong shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Moreover, both LAFITE [56] and VQ-Diffusion [11] lose some fine- grained visual features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=', 1st, 2nd, 5th and 6th columns), which makes the synthesized images lack details and look unreal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' However, the images synthesized by our GALIP have correct object shapes and clear fine-grained contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The superiority is more obvious in complex COCO im- ages, which contain various shapes and multiple objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' As the results are shown in the 7th, 8th, 9th, 10th columns of Figure 7, the LAFITE [56] and VQ-Diffusion [11] mod- els cannot synthesize the right shape of “train”, “children”, “woman”, and “stuffed bear”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Furthermore, they also can- not synthesize the right visual concept of “showing off toy cell phone” and “sitting on a book shelf”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' However, armed with the proposed CLIP-based D and CLIP-empowered G, our GALIP can cope with more strict visual require- ments and synthesize various shapes of different objects (see 8th, 9th, 10th and 12th columns) and present the right visual concepts in synthesized images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We also observe that LAFITE [56] and VQ-Diffusion [11] also can not synthe- size correct human facial features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' For example, as shown in the 8th, 9th, 12th, they can not synthesize realistic hu- man faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' But our GALIP can synthesize these features correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Moreover, we compare the images synthesized by the LDM (LAION-400M) [35, 36] and our GALIP (CC12M) in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' As the results are shown in the 1st, 4th, 5th, 8th, 11th columns of Figure 7, the LDM does not gener- ate the objects (“ghost”, “teddy bear”, “modem”, “person”, “model”) described in the texts, but our GALIP can synthe- size these objects correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Also, our model can generate correct visual features such as “shining eyes”, “Blue Light- house”, “smiling statue”, and “surprised girl” in the 3rd, 6th, 7th, 10th columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Furthermore, as shown in the 9th, 10th, and 12th columns of Figure 7, our GALIP keeps the superior performance of human face synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The exten- sive quantitative evaluation results demonstrate the superi- A 6\'1" model is newly - weds line drawing A smiling statue person shining a A girl with a laptop Portrait of a Cute cell cartoon Cute cartoon little little teddy bear The 4G LTE UE Watercolor of Blue with matching to be found in surprised girl in illustration of a character showing owl with big wearing a size M after the wedding modem Lighthouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Lovely light into the stars ceremony a falling-in-love shining eyes sitting vest and skirt set blue sky and dark the Pavilion dress and classic oxford-style kawaii cute ghost on a tree branch at blue light house on a eyeglasses Short-Sleeve Shirt in face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Vector illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' starry night hillside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Sky Blue 4GFigure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Images synthesized by interpolating four-sentence em- beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Our GALIP supports gradual changes when interpo- lating sentence embeddings describing different image styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It makes the degree of stylization of the image controllable and cre- ates new styles by blending different styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' ority and effectiveness of our proposed GALIP, which is able to generate high-fidelity, creative and complex images with various shapes and multiple objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Additionally, we conduct some experiments to show the smooth latent space of our GALIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Current autoregressive and diffusion models are sensitive to input sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' This instability makes users need to try a lot of prompts to get sat- isfied images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Differently, our GALIP inherits the smooth latent space from GAN, it enables gradual and smooth changes along with text changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' As shown in Figure 8, there is a smooth transition of synthesized images from top to bottom, left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The smooth latent space makes the degree of stylization of the image controllable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The users can fine-tune synthesized image styles like a style knob, and it also enables the users to create new styles by blending dif- ferent image styles, as highlighted by the red dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Ablation Study To verify the effectiveness of each component in the pro- posed GALIP, we conduct ablation studies on the test set of the COCO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The components being evaluated in this subsection include CLIP-based D (CD) and CLIP- empowered G (CG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We also further conduct ablation stud- ies on Bridge-FP (BFP) and Prompt Predictor (PP) in CLIP- empowered G, and CLIP-FE (CFE) in CLIP-based D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Fur- thermore, we compare our CLIP-FE with CCM&CSM of Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The performance of different components of our model on the test set of COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Architecture FID ↓ CS ↑ Baseline 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='2996 Baseline w/ CD w/ CFE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3221 Baseline w/ CD w/ CCM&CSM 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3123 Baseline w/ CD w/ BFP 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3301 Baseline w/ CD w/ BFP w/ PP (GALIP) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3338 GALIP w/ CFE (2nd) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3015 GALIP w/ CFE (5th) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3145 GALIP w/ CFE (12th) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3104 GALIP w/ CFE (2nd,5th) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3301 GALIP w/ CFE (2nd,5th,12th) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3305 GALIP w/ CFE (2nd,5th,9th) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3338 GALIP w/ CFE (2nd,5th,8th,9th) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3305 GALIP w/ PP (1st-12th) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3320 GALIP w/ PP (1st-9th) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3338 GALIP w/ PP (1st-6th) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3310 GALIP w/ PP (1st-3th) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3305 Projected GAN [39], which yields a U-Net architecture to enable multi-scale feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' In addition, we investigate the layer choice strategy of CLIP-FE and Prompt Predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The results on the COCO dataset are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Our baseline is a one-stage text-to-image GAN [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It is composed of a CLIP text encoder and CNN-based generator and discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' And it generates complex im- ages from sentence vectors directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Effect of CLIP-based D and CLIP-FE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The CLIP-based D decreases FID from 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='31 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='92 and improves CLIM- SIM (CS) from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='2996 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The results demon- strate that the complex scene understanding ability of CLIP- ViT promotes the complex image synthesis ability signifi- cantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Furthermore, we compared our CLIP-FE (CFE) with CCM&CSM [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Our CLIP-FE achieves better FID and CLIPSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It shows that our CLIP-FE is more effective in extracting informative visual features from CLIP-ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Effect of CLIP-empowered G and Bridge-FP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The CLIP- empowered G with Bridge-FP further decreases FID from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='92 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='52 and improves CLIPSIM from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3221 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It demonstrates that predicted bridge features and CLIP-ViT can enhance the complex image synthesis ability effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Effect of Prompt Predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The proposed Prompt Predic- tor (PP) also decreases FID from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='52 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='85 and improves CLIPSIM from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3301 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='3338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The result demonstrates that the Prompt Predictor makes the CLIP-ViT more suit- able for generation tasks and induces more meaningful fea- tures from CLIP-ViT to improve the generative ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' CLIP Layer Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We find that the last few lay- ers of CLIP-ViT defect the performance of CLIP-based D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The reason may be that the first layers of CLIP-ViT ex- tract useful visual features and understand complex images, and the last layers focus on generalization ability to align interpolation A beautiful home A beautiful home with a large yard with a large yard Watercolor interpolation A beautiful home interpolation A beautiful home with a large yard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' with a large yard Oil painting Vector illustrationFigure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Illustration of failure cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' It is still hard for cur- rent GALIP to synthesize some imaginary images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Enlarging the model size and training data may improve image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' with high-level concepts in text features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' The generalization ability may defect the performance of CLIP-based D be- cause it reduces the differences between synthetic and real images and weakens the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Conversely, since CLIP-empowered G requires the generalization ability to map the bridge feature to meaningful visual features, adding prompts in the last few layers may defect the generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' So we extract the CLIP features from 2nd,5th,and 9th layers in CLIP-based D, and add prompts to 1st-9th lay- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' And we find that extracting more CLIP features does not lead to better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Limitations Our GALIP shows superiority in text-to-image synthe- sis, but some limitations should be considered in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' First, our model employs the CLIP model to pro- vide text features for the generator and discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' How- ever, current models [38] show that the generic large lan- guage models [31] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=', T5) improve the performance of text-to-image synthesis effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Replacing the CLIP text encoder with T5 may further improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Second, the model size and pretraining dataset are much smaller than other large pretrained models [1,32,35,38,51], it limits the synthesis ability of imaginary images (see Fig- ure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Pretraining on a larger dataset with a larger model size may benefit the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' We will try to address these limitations in our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Conclusion In this paper, we propose a novel Generative Adversar- ial CLIPs (GALIP) for text-to-image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Compared with previous models, our GALIP can synthesize higher- quality complex images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Moreover, we propose a CLIP- based discriminator and CLIP-empowered generator, which exerts the complex scene understanding and domain gener- alization ability of CLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Our GALIP achieves significant improvements on challenging datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' Furthermore, cur- rent large models are pretrained for generative or under- standing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} +page_content=' In this work, we integrate the understanding model (CLIP-ViT) into a generative model and achieve im- pressive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFOT4oBgHgl3EQf6jSQ/content/2301.12959v1.pdf'} 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Khedr +˙ID , Yifeng Xiong +˙ID , and Kun He +˙ID , Senior Member, IEEE +Abstract—Face recognition is known to be vulnerable to +adversarial face images. Existing works craft face adversarial +images by indiscriminately changing a single attribute without +being aware of the intrinsic attributes of the images. To this +end, we propose a new Semantic Adversarial Attack called SAA- +StarGAN that tampers with the significant facial attributes for +each image. We predict the most significant attributes by applying +the cosine similarity or probability score. The probability score +method is based on training a Face Verification model for an +attribute prediction task to obtain a class probability score for +each attribute. The prediction process will help craft adversarial +face images more easily and efficiently, as well as improve the +adversarial transferability. Then, we change the most significant +facial attributes, with either one or more of the facial attributes +for impersonation and dodging attacks in white-box and black- +box settings. Experimental results show that our method could +generate diverse and realistic adversarial face images meanwhile +avoid affecting human perception of the face recognition. SAA- +StarGAN achieves an 80.5% attack success rate against black- +box models, outperforming existing methods by 35.5% under the +impersonation attack. Concerning the black-box setting, SAA- +StarGAN achieves high attack success rates on various models. +The experiments confirm that predicting the most important +attributes significantly affects the success of adversarial attacks +in both white-box and black-box settings and could enhance the +transferability of the crafted adversarial examples. +Index Terms—Adversarial examples, face verification, image- +to-image translation, feature fusion, adversarial transferability. +I. INTRODUCTION +Face Recognition (FR) [1] is an important computer vision +task widely used in solving authentication problems. FR can be +categorized as Face Identification and Face Verification (FV). +Over the past decades, FV, which determines whether a pair of +face images belong to the same identity [2], has achieved great +achievements in various applications such as mobile payment, +military, finance, surveillance security, and border control. +However, Szegedy et al. [3] find that Deep Neural Net- +works (DNNs) are susceptible to adversarial examples. These +adversarial examples have tiny perturbations added to the +benign images that remain imperceptible to human vision but +Manuscript received October 19, 2022. +This work is supported by International Cooperation Foundation of Hubei +Province, China (2021EHB011) and National Natural Science Foundation +(62076105). Corresponding author: Kun He. +Yasmeen M. Khedr is with the School of Computer Science and Technol- +ogy, Huazhong University of Science and Technology, Wuhan 430074, China, +and also with the Faculty of Computers and Informatics, Zagazig University, +Zagazig 44519, Egypt (e-mail: yasmeenkhedr@hust.edu.cn). +Yifeng Xiong and Kun He are with the School of Computer Science and +Technology, Huazhong University of Science and Technology, Wuhan 430074, +China (e-mail: xiongyf@hust.edu.cn; brooklet60@hust.edu.cn). +could mislead DNN models to produce incorrect predictions. +Besides, some studies confirm the vulnerability of DNNs +to input variations [2]–[4]. Also, adversarial attacks can be +divided into categories with different goals and assumptions +on the attacker’s knowledge. White-box and black-box are +two main settings based on the assumption of the attacker’s +knowledge. The former supposes that the attacker can access +the model’s parameter values, architecture, training method, +inputs, outputs, and weights. Whereas the latter assumes that +the attacker only has access to the inputs and outputs of the +model but knows no information about the model [1] +There is a growing interest in adversarial studies for FR +models [5]–[8]. Adversarial studies seek to generate adversar- +ial face images to mislead facial recognition models. Methods +used to manipulate the facial content includes face synthesis, +identity swap, face morphing, face attribute manipulation, and +expression swap [9]. Face attributes are among the emerging +soft biometrics for modern security systems. Some studies +use face attribute manipulation for different goals. Rozsa et +al. [10], [11] propose the Fast Flipping Attribute technique +to mislead facial attribute recognition. Also, Mirjalili and +Ross +[12] use the face attribute to modify the face image +for a gender classifier. Recently, methods based on Generative +Adversarial Network (GAN) have appeared that are used to +manipulate facial attribute images, such as StarGAN [13], +STGAN [14], and AttGAN [15]. Joshi et al. [16] use AttGAN +to generate semantic attacks to deceive gender classifiers. +These studies are limited to classification problems instead +of facial recognition. +Meanwhile, Qiu et al. [6] craft adversarial examples to +mislead FR by changing the attribute individually and check- +ing whether the generated image is adversarial until they find +an adversarial example or failed after attempting the change. +But, they craft an adversarial face image by indiscriminately +distorting facial attributes without being aware of the signif- +icant facial attributes on each image. Hu et al. [17] propose +Adversarial Makeup Transfer GAN (AMT-GAN) to generate +adversarial face images, but it tends to produce high-quality +images of females due to an imbalance of gender in the +training dataset. These studies handle the attack on FR but they +either suffer from the weakness of transferability to black-box +models due to changing the face attribute randomly, or exhibit +bias on genders due to the imbalance of data. +Our work aims to mislead FR models depending on chang- +ing the significant facial attributes. So, we propose a new +attack method called the Semantic Adversarial Attack using +StarGAN (SAA-StarGAN), which effectively and easily crafts +semantic adversarial examples besides improving the attack +arXiv:2301.12046v1 [cs.CV] 28 Jan 2023 + +2 +transferability significantly by tampering with the significant +facial attributes for each input image. These attributes are sup- +posed to affect the decisions of different FV models, leading +to deceiving the FV models and enhancing the adversarial +transferability. In the white-box setting, we predict the most +significant attributes for each input image by using either the +cosine similarity (CS) or the probability score (PS) based on +the Target Face Verification (TFV) model. Then, we change +one or multiple via the StarGAN model in the feature space. +The Attention Feature Fusion (AFF) method is used to fuse the +features of inconsistent semantics to generate a realistic image +and produce β different values used for interpolation. In the +black-box setting, SAA-StarGAN depends on predicting the +most important attributes through the cosine similarity (CS) +method. These attributes are modified on the input image +according to their arrangement based on the prediction step +by making an iterative loop to alter them sequentially until +reaching the adversarial face images. +The empirical results confirm that predicting the most +significant attributes (that will be changed first) plays an im- +portant role in successful attacks. Our SAA-StarGAN method +outperforms other methods significantly on the attack success +rate in the black-box setting and also preserves high attack +success rates in the white-box setting for both impersonation +and dodging attacks. Our method provides perceptually real- +istic images that maintain the source image identity to avoid +confusing human perception. We also analyze the attention +map of the TFV model that is attacked by our adversarial face +images using gradient-weighted class activation. As a result, +our method focuses on trivial features instead of prominent +features. +The main contributions of this work are summarized as +follows: +• We propose a novel attack method called +Semantic Adversarial Attack using StarGAN +(SAA-StarGAN) that enhances the transferabil- +ity of adversarial face images by tampering with +the critical facial attributes for each input image. +• SAA-StarGAN generates semantic adversarial +face images easily and effectively in white-box +and black-box settings by predicting the most +significant facial attributes using two techniques, +cosine similarity or probability score, for imper- +sonation and dodging attacks. +• We propose modifications on SAA-StarGAN to +depend only on the output of the target model in +a black-box setting by applying a linear search +to find the optimal value of the interpolation co- +efficient that affects the generated face images. +• The empirical results confirm that predicting +the most significant attributes (which will be +changed first) plays a vital role in a successful +attack. Our SAA-StarGAN method outperforms +other methods considerably on the white-box +attack success rate and black-box adversarial +transferability. Also, it provides perceptually re- +alistic images that maintain the source image +identity to avoid confusing human perception. +II. RELATED WORK +In this section introduce the related work of generating +adversarial examples for both image classification and FR +models. +A. Adversarial Attacks on Images +Many adversarial example generation methods have been +proposed to mislead different image classification models. +Most studies focus on generating restricted adversarial ex- +amples by adding perturbations to the input images. Szegedy +et al. [3] first find the existence of adversarial examples for +image classification, which transforms an image by a small +amount to be undetectable and thereby changes how the image +is classified. Goodfellow et al. [4] propose a Fast Gradient +Sign Method (FGSM) that uses the gradients of the neural +network to generate adversarial examples. They just applied +a one-step gradient update along the direction of the gradient +sign at each pixel. Kurakin et al. [18] propose a Basic Iterative +Method (BIM), which applies FGSM perturbations of smaller +magnitude for multiple iterations to improve the attack success +rates. They clipped pixels in each iteration to avoid a large +change Besides, Projected Gradient Descent (PGD) [19] is +considered an extension of the BIM method to diversify the +synthesized adversarial examples. +Dong et al. [20] and Lin et al. [21] incorporate momentum +into the iterative FGSM to boost the attack transferbility. +Diverse methods have been proposed, such as designing an +efficient saliency adversarial map for seeking the adversarial +noise [22], image denoising attack [23], and adding perturba- +tion based on the attended regions and features [24]. +Meanwhile, GAN is also used in the construction of ad- +versarial examples due to its awesome ability to generate +images. Xiao et al. [25] propose AdvGAN, which constructs +a generator network based on encoder-decoder to generate +adversarial perturbation and then adds this perturbation to +the original image to mislead the model. On the other hand, +Jandial et al. [26] propose AdvGAN++, showing that latent +features achieve higher attack success rates than AdvGAN and +craft realistic images on CIFAR and MNIST datasets. Song et +al. [27] propose a method for generating unrestricted adver- +sarial examples based on the ACGAN from scratch instead of +adding small perturbations on a source image for the classifier. +Adversarial Transfer on Generative Adversarial Net (AT-GAN) +[28] generates non-constrained adversarial examples directly +from any input noise, which aims to learn the distribution of +the adversarial examples. +B. Adversarial Attacks on Face Recognition +Recently, many adversarial attacks have been proposed for +attacking the FR models, which can be divided into three cat- +egories: (1) adding adversarial perturbations, (2) manipulating +facial attributes, and (3) physical attacks. +One line of study focuses on changing the facial appear- +ance of input images by adding small perturbations in a +specific region to be imperceptible to human eyes. Deb et +al. [29] propose an automated adversarial face method called + +3 +AdvFaces that generate minimal perturbations in the salient +facial regions via GANs. Zhu et al. [30] hide the attack +information by the makeup effect to attack the eye regions +only. Yang et al. [5] propose Attentional Adversarial Attack +Generative Network (A3GN) to deceive the FR model under +impersonation attacks, where attention modules and variational +autoencoder are incorporated to learn semantic information +from the target. Zhong and Deng [31] propose Dropout Face +Attacking Network (DFANet) to improve the transferability by +integrating the dropout in convolution layers in the iterative +steps to generate adversarial examples. The second category +is based on manipulating facial attributes. Rozsa et al. [11] +propose Fast Flipping Attribute (FFA) technique which found +that the robustness of DNNs against adversarial attacks varies +highly between facial attributes. Kakizaki and Yoshida [7] +focus on the serious risks resulting from the ineffectiveness +of conventional certified defenses against adversarial examples +that are not restricted to small perturbations. They use image +translation techniques to generate unrestricted adversarial ex- +amples by translating the source image into any desired facial +appearance with large perturbations. Qiu et al. [6] introduce +SemanticAdv, which can generate unrestricted adversarial ex- +amples by altering a single facial attribute. +The physical attacks can be generated by a variety of +different tools, such as adding some adversarial face acces- +sories [32], natural makeup [33] and adversarial patches [34]. +These methods make the attacks more dangerous in the +physical world. +III. FACE RECOGNITION ATTACK +In this section, we first provide the problem definition. Then +we present a detailed description of our method in white-box +and black-box settings. Ultimately, we introduce a preliminary +method called Random Selection as a baseline method. +A. Problem Definition +The main purpose of the FR model is to recognize the input +image by training the model on dataset D(x, y), where x is a +face image sampled according to the latent distribution, and y +is the corresponding ground-truth label. According to f(x) : +x → y, the model can predict the label for each input face +image. The main goal is to generate an adversarial face image +xadv that is similar to the original image x but misleads the +FV model, i.e. FV (xadv) = y′ ̸= y. +On the other hand, adversarial attacks are divided into two +types: dodging and impersonation attacks [1]. The dodging +attack (untargeted attack) is developed to fool the target model +such that the output is a random identity excluding the original +one. In contrast, the impersonation attack (targeted attack) +misleads the target model by recognizing the adversarial face +image as a specified target identity. For the dodging attack, +we generate xadv, which is identified as not the same identity +as FV (xadv) ̸= y. For impersonation attack, we seek to make +the model recognize xadv as the same identity of another given +image such that FV (xadv) = ytgt. +B. Semantic Adversarial Attack (SAA-StarGAN) +We propose an efficient attack method that first predicts the +most significant facial attributes for each input image. Then, +we generate xadv in the intermediate layers instead of the +output layers because this increases internal feature distortion, +leading to improved performance. There are two steps in the +proposed framework for a white-box attack: 1) Predict the +most significant attributes for each input image; 2) Generate +xadv by modifying one or multiple most significant attributes. +These attributes are changed by using the StarGAN model. +StarGAN model [13] consists of a single generator G and a +discriminator D trained to learn to translate images from one +domain to another. +1) Significant Attribute Prediction: We propose using either +the cosine similarity (CS) or the probability score (PS) to +detect the significant attributes. The corresponding methods +are denoted as SAA-StarGAN-CS and SAA-StarGAN-PS, +respectively. We first apply CS or PS to predict the most +significant attributes and compare their results. Consequently, +we retrain the G of StarGAN on all the facial attributes to use +in the significant attribute prediction step. +a) Cosine Similarity (CS): +We utilize the TFV models [35], [36] to obtain the important +attributes by computing the cosine similarity between the out- +put features of the TFV model. More details of the algorithm +for the predicted most significant attributes using CS are shown +in Supplementary. For image attributes a = (a1, a2, a3, ..., aK) +where ai indicates the ith attribute, and K the total number of +attributes, we first use StarGAN to change each ai of the input +x to get the image synthesis x∗ +ai. Then, we extract the synthesis +image features fx∗ai , and the original image features fx by the +TFV model, and calculate their cosine similarity to get Sai, +that refers to the degree of change in the output of TFV by +changing attribute ai. We then sort the attributes in a in the +ascending order according to the similarity score values to get +the most important attributes C = (c1, c2, c3, ..., cK) where ci +is the ith sorted attribute. The less the cosine similarity, the +more the significance of the attribute. The cosine similarity, +as illustrated in Eq. (1), is a way to measure the similarity +between two vectors, ranging from 0 to 1. +Sai = CS(fx, fx∗ai ) = +fx · fx∗ai +∥fx∥ · ∥fx∗ai ∥. +(1) +b) Probability Score (PS): +The basic idea of the probability score method is to use the +TFV model as the Attribute Prediction model (Att-Pred) to +predict the important attributes of the input. For each attribute +ai in input x, we use the Att-Pred model to obtain the class +probability score (PS), as shown in Fig. 1. If a′ +i is found in the +original image x, we remove it in the synthesis image x∗ and +vice versa. Therefore, we use the StarGAN model to change +a′ +i to get the x∗ +a′ +i. +Moreover, each attribute in the input face image has a +different impact on the final decision. Thus, we need to +calculate the probability value Pa′ +i of a′ +i for x and x∗ to get +the degree of change. ∆Pa′ +i indicates the degree of change in + +4 +Young +Wearing_Lipstick +Wavy_Hair +Pointy_Nose +No_Beard +Heavy_Makeup +Blond_Hair +Attractive +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Fig. 1. Output from the attribute prediction model using FaceNet. +the probability value in image x before and after changing a′ +i +to determine the most significant attributes for each image x. +∆Pa′ +i = Pa′ +i(x) − Pa′ +i(x∗ +a′ +i), +(2) +where +x = a1a2...ai...ak, +x∗ +a′ +i = a1a2...a′ +i...ak. +Here x∗ +a′ +i indicates the generated image after changing attribute +a′ +i. +Finally, we sort the attributes in a in descending order using +∆Pa′ +i to obtain the most significant attributes C to help us +craft xadv. These attributes represent the best attack effect. +The algorithm of this step is illustrated in Supplementary. +2) Adversarial Face Image Generation: The second step +focuses on generating the adversarial face images, as illus- +trated in Fig. 2. To generate xadv, we apply two kinds of per- +turbations using single or multiple attributes. SAA-StarGAN- +CS-M and SAA-StarGAN-PS-M indicate the methods used for +multiple attributes in CS and PS techniques, respectively. +a) For single attribute, after completing the first step of +predicting C = (c1, c2, c3, ..., cK), we use the generator G of +StarGAN, which we have already trained on facial attributes in +the significant attribute prediction step. G is composed of an +encoder (GE) and a decoder (GD), as shown in Eq. (3). The +(GE) takes an input image x and the single significant attribute +c1, then we get the output feature in intermediate layers. (GD) +takes the feature as input, and outputs the synthesized image. +As illustrated in Fig. 2, we use the x as an input to the +GE with significant attribute c1. Then we extract the output +features from the conv layer f ∗ +conv and the Residual Block +layer f ∗ +res of the encoder, as shown in Eqs. (4) and (5). +G = GE· GD, +(3) +f ∗ +conv = GE(x, c, conv layer), +(4) +f ∗ +res = GE(x, c, res layer). +(5) +On the other hand, we exploit the Attention Feature Fusion +(AFF) method [37] to obtain the fused feature as an input to +the decoder GD. Usually, addition and concatenation methods +combine features. But recently, the attention mechanism has +succeeded in many computer vision applications [38]. So, +we use AFF to combine the features. AFF is a framework +that combines the features from different layers based on the +Multi-Scale channel attention (MS) module to overcome the +inconsistent semantics among the input features and generate a +more realistic image. More details about the AFF framework, +AFF +Original image +1 +1 +0 +0 +0 +0 +Most significant +attributes +𝒇𝒓𝒆𝒔 +∗ +𝒇𝒄𝒐𝒏𝒗 +∗ +𝒇∗ +TFV +Match or not +Target image +𝒙𝒂𝒅𝒗 +𝒙𝒂𝒅𝒗 +𝒙𝒕𝒈𝒕 +Adversarial loss +Attribute labels +Blond/Smiling/Eyeglasses/ +Chubby/Bangs/Young +Encoder (E) +Decoder (D) +MS +𝜷 +Fig. 2. The SAA-StarGAN attack framework. The original image and most +significant attributes are fed into Encoder (GE) to change these attributes and +then extract their features from different layers. The Attention Feature Fusion +(AFF) framework is then used to generate β and perform fusion between +features. After that, the fused features are sent to the decoder (GD) to obtain +the synthesis image. Finally, the (TFV ) model receives both the synthesis +image and target identity to calculate the adversarial loss and optimize the β +value at the feature level. +including the MS module, are presented in [37]. Eq. (6) +indicates the fusion weights β, where β ∈ [0, 1] is calculated +from the attention weights generated by the MS module in +AFF. As a result, we update the value of β until the model +is misleading. To get the better fused feature f ∗, we apply an +interpolation in the feature space, as shown in Eq. (7). Finally, +the decoder GD takes the fused feature f ∗ as input and gets +the synthesized face image x∗ as the output, as shown in Eq. +(8). +β = MS(f ∗ +conv, f ∗ +res) +(6) +f ∗ = β· f ∗ +conv + (1 − β)· f ∗ +res +(7) +x∗ = D(f ∗) +(8) +Our work obtains the xadv by modifying the significant +attribute c1 for each image through feature-level interpolation. +To achieve the impersonation attack, we use the L2 loss +function to minimize the distance between the face embedding +of xadv and the target image xtgt, as shown in Eq. (9). +xadv = Minx∗ ∥ TFV (x∗) − TFV (xtgt) ∥2 +2 +(9) +The adversarial face image is crafted for dodging the attack +by maximizing the distance between xadv and the x in the +feature space, as shown in the following: +xadv = Maxx∗ ∥ TFV (x∗) − TFV (x) ∥2 +2 +(10) +b) For multiple attributes, similarly, we use the same G of +StarGAN, which has been used to modify a single attribute, but +in this step, to change multiple significant attributes C in the +feature space. In our experiments, we change two significant +attributes, c1 and c2, represented as one-hot vectors. As a +result, we obtain the synthesized image with a bigger change. +After that, We study the effect of this change on attacking the +face images. We do not change more than two attributes as we + +Young +Wearing_Lipstick +Wavy_Hair +Pointy_Nose +No_Beard +Heavy_Makeup +Blond_Hair +Attractive +0.0 +0.2 +0.4 +0.6 +0.8 +1.05 +have tested using more significant attributes in C, but some +of the resulting images are not realistic. +3) SAA-StarGAN in Black-Box Setting: A black-box attack +differs from a white-box attack as the adversary has no access +to the model’s gradient or parameters. Therefore, we propose +modifications for our proposed method, SAA-StarGAN, to +depend only on the target model’s output in the black-box +setting. +To generate a successful attack, we need two steps: 1) +Predicting the most significant attributes for the target model. +This step is similar to that in the white-box setting, but we +only use the CS method as it does not need to train a target +model to predict the attributes. 2) Doing a linear search to +find the largest γ value that affects the generated face image +by changing the most significant attributes with γ for each +face image in an iterative loop until the output misleads the +model. +More details of the algorithm for the black-box attack +method are shown in Supplementary. To change the most +significant attributes C, we use GE from the StarGAN model +to get the features. But, we make an iterative loop to alter the +attributes according to the most important to the least until we +meet the adversarial condition. We use a bilinear interpolation +with a variable γ value to generate the fused features f ∗ +γi. +Therefore, we make a linear search to find the optimal γopt +value according to the change in the confidence score. We +create a vector γ that consists of 100 random values drawn +from [0, 1]. This vector is applied to study the effect of each +value γ in Eq. (11). +Then, the values are arranged according to the score change. +For an impersonation attack, we select the optimal γopt value +that decreases the distance between the face embedding of +the generated face image and the target face image. For +the dodging attack, we select the optimal γopt value that +maximizes the distance between the face embedding of the +generated face image and the input face image. Finally, we +substitute the optimal γopt value to generate the fused feature +and feed it to a decoder to get the x∗ by the following +equations: +f ∗ = γopt· f ∗ +1 + (1 − γopt)· f ∗ +ci, +(11) +x∗ = GD(f ∗). +(12) +To guarantee that the generated face image will preserve +semantic similarity from the original face image, we need to +measure the semantic similarity sim between the generated +face image and the input face image to filter out the face +images that are not realistic and control their quality. The +adversarial face image is found when the adversarial criterion +is achieved and the semantic similarity is above a threshold +th (sim > th). +The above steps will repeat to add the next significant +attribute in the ordered significant attributes until we find an +adversarial example (Lines 18 - 34). +C. Random Attribute Selection based Attack +This subsection presents a preliminary method , the Random +Selection, as a baseline method to compare with our SAA- +StarGAN method. We generate adversarial face images based +on randomly selecting a set of attributes. These attributes are +changed by using the StarGAN model. +Therefore, we make two copies of the generator from +StarGAN and re-train them separately using two different sets +of attributes representing each attribute a as a one-hot vector. +The first generator, G1, is trained by the first set S1.The second +generator, G2, is trained by the second set S2As a result, the +overall Random Selection system involves three components, +namely G1, G2, and the TFV model. +Firstly, G1 takes x with a specific attribute as1 from S1 to +generate translated image with dimensions H, W, and L for +height, width, and channels, respectively. The second image +synthesis x∗ +2 is then obtained using the translated image as +the input to G2 with another attribute as2 from S2, as shown +in Eq. (13). Note that the two attributes are chosen randomly +from the sets S1 and S2. Through permutations between the +attributes from the previous sets, we change two attributes +using the StarGAN model. +Finally, interpolation is applied between the pair of images +produced from G1 and G2 to generate xadv as shown in Eq. +(14). According to the presented procedure, 25 adversarial +face images are generated for each x due to the permutation +between different attributes from the two sets. +x∗ +2 = G2(G1(x, as1), as2) +(13) +xadv = α· x∗ +1 + (1 − α)· x∗ +2 +(14) +IV. EXPERIMENTAL SETUP +This section provide experimental setup, including dataset, +models, baselines, evaluation metrics and implementation de- +tails of our method. +A. Dataset +The proposed SAA-StarGAN method uses the CelebA +dataset, which has 202,599 face images with 40 facial at- +tributes and 10,177 identities [39], to generate semantic ad- +versarial face images. This dataset is the most popular type +used in the face recognition task. We use 40 facial attributes +to train the StarGAN model in our work. Also, we randomly +choose 5,000 different identities as the original images and +5,000 different identities as the target images. +B. Target Face Verification Models +To evaluate the effectiveness of the proposed SAA- +StarGAN, we choose ten state-of-the-art FV models, con- +taining different model architectures and training loss func- +tions. We use two of them as the white-box TFV models: +FaceNet +[35] and ArcFace [36]. The two models return +512-dimensional embeddings of the images. Besides, we use +two other publicly available models, SphereFace [40] and +CosFace [41], for evaluation. Then, we select different trained +models under different backbones and loss functions such as +ResNet-101 [42], IResNet50 [36], [43], MobileFace [44], and +ShuffleNet V2 [45] to demonstrate the effectiveness of our +SAA-StarGAN on different models. We compute the optimal +threshold T based on the false-positive rate (FPR) for each +FV model used in the evaluation. Details of each model are +presented in Supplementary materials. + +6 +C. Baselines +We adopt eight baseline methods to evaluate our attack +method, including the Random Selection method presented in +Section III-C, the typical one-step attack method of FGSM [4], +BIM [18], PGD [19], and MI-FGSM [20]. Besides, face +attack methods such as Sticker and Face mask attacks [46] +are included. The facial attack method of SemanticAdv [6] +is the most comparable method to ours, which modifies the +face attribute to generate adversarial face images. For the +Random Selection method, five facial attributes are applied +for G1: hair color (black-blond), heavy makeup, gender, and +pale skin; while for G2, smiling, mouth slightly open, bangs, +eyeglasses, and young attributes are applied. For FGSM, we +set the perturbation as ϵ = 0.2 for L2 attack and the number +of iterations as 20 for BIM. Also, we set the perturbation as +ϵ = 8 for PGD and MI-FGSM with pixel values in the range +[0, 255]. The number of iterations is 40, and the decay factor +µ = 1.0. +D. Evaluation Metrics +We use several evaluation matrices to estimate our attack +effectiveness over different baselines. We select the attack +success rate to evaluate the adversarial face images crafted by +SAA-StarGAN. To measure the similarity between images, we +use the cosine similarity and use a threshold T @ 0.1 % FPR +for each TFV model to decide whether the similarity between +two identity persons matches. We can obtain cosine threshold +Ts from the Euclidean threshold after normalizing features as +Ts = 1 − (T/2). +The attack success rate used for impersonation attack is +computed as follows: +success rate = (#ImagePairs(xadv, xtgt) ≥ Ts) +(#TotalImagePairs) +, +(15) +where ImagePairs consists of an adversarial face image gen- +erated by SAA-StarGAN and the matched target face. +The attack success rate used for dodging attack is computed +as follows: +success rate = (#ImagePairs(xadv, x) < Ts) +(#TotalImagePairs) +, +(16) +where ImagePairs consists of an adversarial face image gen- +erated by SAA-StarGAN and the input face image. +Finally, we compute the Mean Square Error (MSE) [8] +and the Structural Similarity Index Measure (SSIM) [47] to +evaluate the quality between the original face images x and +the adversarial face images xadv for different attacks. +E. Implementation Details +We use the Adam optimizer [48] with a fixed learning rate of +0.05 and up to 300 epochs for all experiments. SAA-StarGAN +is implemented using PyTorch v1.7.0. For the PS, we train the +TFV model for the attribute prediction task to have the class +probability score (PS) for each attribute. All experiments are +conducted on a single Titan X GPU to generate adversarial +face images. We perform impersonation attacks based on the +original and target images with different identities and dodging +attacks based on the pairs of original images with the same +identities. Firstly, all the selected face images pass through +an MTCNN detector [49] to detect the face image and align +images for the entire image. Then, we obtain the resized +images in 112 × 112 × 3, but inside each model it will carry +out the specific input size due to the diversity of model inputs. +V. EXPERIMENTAL RESULTS +To validate the effectiveness of SAA-StarGAN, we empir- +ically evaluate our adversarial face images and illustrate that +SAA-StarGAN achieves higher attack success rates against +different FV models. On the other hand, SAA-StarGAN boosts +the adversarial transferability with high efficiency. The black- +box setting demonstrates that SAA-StarGAN achieves high +attack success rates significantly and shows the importance +of significant attribute prediction. Besides, SAA-StarGAN can +generate realistic and diverse adversarial face images. Table I +and Table II list the attack success rates of various methods, +using FaceNet and ArcFace models to generate adversarial +examples respectively, for impersonation and dodging attacks. +A. Comparison for White-box Attacks +To validate the efficacy of our attack method, we fisrt +compare SAA-StarGAN with the baseline methods in the +white-box setting. For a fair comparison, we train these base- +line methods using FaceNet and ArcFace models on CelebA +dataset with T @ 0.1% FPR for both impersonation and +dodging attacks. Then we compute the attack success rate for +each method. The first column in Table I compares our SAA- +StarGAN and the baselines in the white-box setting on the +FaceNet model for the impersonation attack in (a) and dodging +attack in (b). All variants of the proposed SAA-StarGAN +method have reached a nearly 100 % attack success rate, +confirming that SAA-StarGAN can mislead the TFV models +successfully. The second column in Table II compares our +SAA-StarGAN and the baselines in the white-box setting on +the ArcFace model for the impersonation attack in (a) and +dodging attack in (b). For all the cases, it can be concluded +that our SAA-StarGAN can effectively craft adversarial face +images to fool TFV models in the white-box setting. +B. Comparison for Black-box Attacks +Most face recognition systems do not allow access to any +internal information of the neural networks. In the black-box +setting, the weights and network architectures are not included +in the training process. So, it is essential to evaluate the +vulnerability of FRs in both the transfer-based attack setting +and the attack based on score confidence. +1) Transferability Analysis: The transferability across dif- +ferent models is one of the most important properties of +adversarial examples. To demonstrate the transferability of the +adversarial face images generated by SAA-StarGAN under +two white-box models, we construct a dataset from successful +adversaries crafted by the FaceNet and ArcFace models. Then +we evaluate the attack success rate on nine different TFV +models. We can observe from Table I that for impersonation + +7 +attacks, our SAA-StarGAN achieves a high attack success rate +under different experimental conditions compared to various +baselines. In contrast, the attack success rate of the Sticker +and Face mask attacks has not exceeded 17 %. FGSM and +BIM methods exhibit the weakest transferability on all TFV +models after the Sticker and Face mask attacks. The PGD, MI- +FGSM, Random Selection, and SemanticAdv methods achieve +better transferability than FGSM and BIM. The attack success +rate of these attacks improves by 9.5 % ∼ 38.6 %, and our +SAA-StarGAN outperforms them by a clear margin. +Although the Random Selection and SemanticAdv methods +craft the adversaries by modifying the facial attributes, they +only depend on changing random attributes or a single fixed +attribute. We can see that the transferability of SAA-StarGAN- +CS for the single attribute surpasses that of other methods +in all cases. In addition, attacking the ShuffleNet V2 model +using adversarial face images generated by FaceNet for the +SAA-StarGAN-CS achieves the best attack success rate of +54.30 %, outperforming the baseline attacks by 15.7 % ∼ +43.1 %. Finally, we conclude that our SAA-StarGAN method +outperforms others on all models besides maintaining a high +attack success rate on the white-box setting. Similarly, our +attack outperforms the other baselines significantly in dodging +attack, as presented in Table I. We can observe that the ad- +versarial face images generated by SAA-StarGAN-CS against +the SphereFace exceed 14.1 % ∼ 53.7 % for the baseline +attacks. Face mask based on the grid level is effective than +PGD and MI-FGSM under dodging attacks. In addition, this +attack is considered effective for transferability after our SAA- +StarGAN. +0.388 +0.512 +0.59 +0.5 +0.64 +0.348 +0.51 +0.52 +0.68 +0.28 +0.554 +0.76 +0.748 +0.798 +0.77 +0.606 +0.75 +0.738 +0.82 +0.436 +0.45 +0.613 +0.644 +0.651 +0.702 +0.524 +0.631 +0.645 +0.705 +0.417 +0.997 +0.926 +0.908 +0.998 +0.801 +0.77 +0.8 +0.8 +0.914 +0.87 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) Dodging attack + SAA-StarGAN +Attack Sucess Rate (%) + ImSemanticAdv +FaceNet +ArcFace +ResNet-101 +CosFace +SphereFace +MobileFace +ShuffleNet V2 +IR50-Softmax +IR50-CosFace +IR50-SphereFace +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Attack Sucess Rate (%) +FaceNet +ArcFace +ResNet-101 +CosFace +SphereFace +MobileFace +ShuffleNet V2 +IR50-Softmax +IR50-CosFace +IR50-SphereFace +(a) Impersonation attack +Fig. 3. +The average attack success rate (%) for black-box attacks. SAA- +StarGAN outperforms ImSemanticAdv by a clear margin. +Original +Target +Adversarial +Single attribute +(+) Receding Hairline +(+) Wearing Lipstick +Adversarial +Multiple attributes +(+) Receding Hairline +(+) Chubby +(+)Wearing Lipstick +(+) Eyeglasses +(a) SAA-StarGAN-CS +FaceNet model +(-) Chubby +(+) Bangs +(-) Chubby +(+) Eyeglasses +(+) Bangs +(-) Mouth Slightly Open +(b) SAA-StarGAN-PS +ArcFace model +Fig. 4. +Adversarial face images generated by SAA-StarGAN in white-box +setting. It shows the success of SAA-StarGAN in producing realistic images. +Fig. 5. Adversarial face images generated by SAA-StarGAN in the black-box +setting based on the score confidence. The figure illustrates the realistic face +images, and the red borders indicate a few unrealistic adversarial face images +generated by SAA-StarGAN. +Table II illustrates that our SAA-StarGAN significantly im- +proves the transferability of adversarial face images crafted by +the ArcFace model against different models over the baseline +attacks. The main reason is that SAA-StarGAN depends on +manipulating the most important attributes that affect the de- +cision. As clear from Table II, SAA-StarGAN-CS outperforms +the baselines of MI-FGSM, PGD, Random Selection, Seman- +ticAdv, BIM, FGSM, Face Mask, and Sticker by 10.8, 18.8, +31.0, 35.5, 45.0, 47.5, 66.0, and 68.1 % respectively under the +impersonation attack, against IR50-CosFace model. To dodge +an attack, The results are illustrated in Table II. We conclude +that the main advantage of SAA-StarGAN-CS is that it does +not need to train the TFV models for the attribute prediction +tasks. We can directly apply different TFV models easily and +efficiently. Besides, we observe that the models within the +same backbone have good transferability. So, the generated +adversarial face images crafted on the ArcFace model against +IR50-Softmax, IR50-CosFace, and IR50-SphereFace have high +transferability. And ShuffleNet V2 and MobileFace have light +weights which are easily attacked by the adversarial face +images generated on ArcFace. In contrast, the FaceNet model +is challenging to transfer to other models, where it is trained +on IncepetionResNetV2 based on the softmax loss. + +8 +TABLE I +TRANSFERABILITY OF THE ADVERSARIAL EXAMPLES GENERATED BY SAA-STARGAN AND THE BASELINES AGAINST BLACK-BOX MODELS. WE USE +ADVERSARIAL EXAMPLES GENERATED ON FACENET MODEL TO ATTACK NINE DIFFERENT TFV MODELS FOR IMPERSONATION & DODGING ATTACKS. +VALUES REPRESENT THE ATTACK SUCCESS RATE (%). ∗ INDICATES WHITE-BOX ATTACKS. THE NUMBERS MARKED IN BOLD REPRESENT THE BEST +ATTACK SUCCESS RATE. +(a) Impersonation attack +Attack +FaceNet +ArcFace +ResNet-101 +CosFace +SphereFace +MobileFace +ShuffleNet V2 +IR50-Softmax +IR50-CosFace +IR50-SphereFace +FGSM +97.7∗ +11.6 +02.6 +12.4 +18.4 +11.6 +13.8 +14.9 +10.1 +10.3 +BIM +98.0∗ +13.0 +08.6 +20.1 +21.9 +19.4 +19.6 +14.8 +09.8 +11.8 +PGD +99.5∗ +12.5 +10.4 +24.6 +25.1 +27.5 +34.9 +17.4 +23.7 +20.7 +MI-FGSM +99.7∗ +13.4 +12.4 +29.1 +31.6 +28.3 +38.6 +21.7 +29.9 +21.8 +SemanticAdv +99.8∗ +32.4 +09.9 +22.1 +29.0 +28.1 +35.7 +16.2 +19.5 +14.7 +Random Selection +99.7∗ +31.6 +09.5 +19.9 +25.1 +27.7 +31.4 +17.2 +29.5 +16.2 +Sticker +100.0∗ +9.7 +7.4 +10.4 +11.5 +10.7 +11.2 +9.9 +10.5 +11.1 +Face Mask +95.4∗ +10.2 +9.6 +13.7 +17.0 +11.9 +15.7 +12.1 +11.4 +13.0 +SAA-StarGAN-CS +100.0∗ +46.8 +31.5 +40.3 +53.0 +48.2 +54.3 +37.8 +35.5 +36.2 +SAA-StarGAN-PS +100.0∗ +45.5 +31.3 +39.6 +50.0 +48.1 +53.2 +34.5 +36.5 +35.3 +SAA-StarGAN-CS-M +100.0∗ +44.3 +29.0 +39.9 +48.7 +46.7 +52.0 +34.9 +34.8 +36.0 +SAA-StarGAN-PS-M +100.0∗ +46.2 +30.5 +40.2 +50.6 +47.0 +53.3 +34.3 +35.7 +34.4 +(b) Dodging attack +Attack +FaceNet +ArcFace +ResNet-101 +CosFace +SphereFace +MobileFace +ShuffleNet V2 +IR50-Softmax +IR50-CosFace +IR50-SphereFace +FGSM +94.6∗ +26.4 +14.4 +21.5 +30.2 +26.0 +28.3 +29.0 +21.0 +21.4 +BIM +99.3∗ +28.0 +19.5 +32.4 +33.7 +29.6 +33.2 +30.2 +20.7 +25.8 +PGD +100.0∗ +50.0 +27.3 +62.4 +60.6 +49.9 +62.0 +41.5 +37.6 +42.9 +MI-FGSM +100.0∗ +51.1 +28.0 +60.3 +61.1 +50.1 +62.8 +43.2 +42.2 +48.4 +SemanticAdv +100.0∗ +45.2 +20.4 +41.0 +39.5 +31.0 +48.1 +32.4 +33.1 +27.6 +Random Selection +100.0 ∗ +43.2 +20.0 +40.2 +36.2 +30.8 +42.9 +36.2 +47.2 +28.0 +Sticker +100.0∗ +21.4 +9.8 +22.1 +23.5 +27.4 +26.4 +19.9 +21.4 +23.4 +Face Mask +100.0 ∗ +55.6 +31.2 +63.5 +63.1 +52.1 +63.4 +40.1 +38.4 +43.7 +SAA-StarGAN-CS +100.0∗ +65.3 +44.6 +65.1 +77.2 +53.6 +76.8 +49.5 +53.4 +55.4 +SAA-StarGAN-PS +100.0 ∗ +62.4 +43.5 +61.4 +71.8 +53.8 +69.4 +47.3 +49.5 +54.3 +SAA-StarGAN-CS-M +100.0 ∗ +51.5 +41.1 +61.2 +61.4 +54.9 +63.2 +42.8 +47.6 +59.2 +SAA-StarGAN-PS-M +100.0 ∗ +54.7 +42.8 +64.4 +66.76 +54.5 +66.0 +47.2 +48.9 +51.0 +TABLE II +TRANSFERABILITY OF THE ADVERSARIAL EXAMPLES GENERATED BY SAA-STARGAN AND THE BASELINES AGAINST BLACK-BOX MODELS. WE USE +ADVERSARIAL EXAMPLES GENERATED ON ARCFACE MODEL TO ATTACK NINE DIFFERENT TFV MODELS FOR IMPERSONATION & DODGING ATTACKS. +VALUES REPRESENT THE ATTACK SUCCESS RATE (%). ∗ INDICATES WHITE-BOX ATTACKS. THE NUMBERS MARKED IN BOLD REPRESENT THE BEST +ATTACK SUCCESS RATE. +(a) Impersonation attack +Attack +FaceNet +ArcFace +ResNet-101 +CosFace +SphereFace +MobileFace +ShuffleNet V2 +IR50-Softmax +IR50-CosFace +IR50-SphereFace +FGSM +20.1 +97.5∗ +12.4 +25.4 +23.4 +44.8 +41.4 +27.9 +33.0 +24.2 +BIM +21.8 +100.0∗ +16.4 +29.6 +27.1 +45.3 +44.6 +29.1 +35.5 +30.0 +PGD +26.3 +99.9∗ +25.4 +42.5 +47.5 +53.8 +49.4 +45.3 +61.7 +45.1 +MI-FGSM +29.3 +100.0∗ +25.8 +48.9 +56.3 +65.8 +56.7 +51.9 +69.7 +52.7 +SemanticAdv +30.4 +95.8∗ +17.5 +32.6 +45.6 +51.5 +47.3 +37.5 +45.0 +30.7 +Random Selection +30.6 +97.8∗ +19.4 +33.4 +38.9 +50.1 +48.7 +38.0 +49.5 +27.3 +Sticker +18.7 +100.0∗ +8.9 +14.0 +10.1 +21.4 +20.9 +15.8 +12.4 +13.2 +Face Mask +19.4 +100.0∗ +9.3 +16.5 +11.4 +22.6 +21.9 +17.0 +14.5 +13.9 +SAA-StarGAN-CS +37.2 +99.7∗ +27.6 +54.2 +62.1 +74.3 +65.6 +65.9 +80.5 +58.5 +SAA-StarGAN-PS +36.7 +99.8∗ +26.6 +52.0 +61.5 +73.5 +64.1 +64.7 +78.1 +57.5 +SAA-StarGAN-CS-M +34.2 +99.1∗ +24.0 +53.0 +59.4 +67.1 +57.0 +61.9 +73.0 +52.9 +SAA-StarGAN-PS-M +35.9 +99.2∗ +26.3 +53.7 +61.5 +68.5 +59.1 +64.4 +76.6 +56.4 +(b) Dodging attack +Attack +FaceNet +ArcFace +ResNet-101 +CosFace +SphereFace +MobileFace +ShuffleNet V2 +IR50-Softmax +IR50-CosFace +IR50-SphereFace +FGSM +34.1 +99.0∗ +32.7 +46.7 +44.7 +66.1 +62.7 +49.2 +54.3 +45.5 +BIM +35.6 +100.0∗ +35.8 +51.5 +49.0 +67.2 +66.5 +51.0 +57.4 +49.9 +PGD +47.7 +99.9∗ +39.2 +63.2 +70.6 +72.6 +66.9 +64.9 +67.7 +55.5 +MI-FGSM +48.2 +100.0∗ +41.5 +66.0 +73.0 +75.8 +70.0 +66.8 +69.8 +56.7 +SemanticAdv +44.4 +99.9∗ +37.4 +53.2 +64.6 +80.5 +76.3 +66.5 +64.0 +52.7 +Random Selection +45.6 +99.8∗ +38.9 +54.8 +60.3 +81.5 +77.1 +69.4 +61.9 +53.6 +Sticker +21.3 +100.0∗ +25.4 +27.8 +30.1 +40.2 +38.4 +24.7 +26.8 +27.4 +Face Mask +45.0 +100.0∗ +42.4 +64.7 +70.0 +74.5 +70.5 +68.7 +69.1 +58.4 +SAA-StarGAN-CS +53.2 +100.0∗ +48.0 +75.6 +86.4 +94.6 +89.1 +77.1 +89.3 +69.9 +SAA-StarGAN-PS +52.4 +100.0∗ +47.2 +73.3 +83.8 +93.9 +88.7 +76.0 +88.1 +64.8 +SAA-StarGAN-CS-M +49.2 +100.0∗ +44.6 +72.4 +80.4 +91.1 +86.0 +70.9 +72.1 +61.9 +SAA-StarGAN-PS-M +50.9 +100.0∗ +46.6 +72.7 +81.5 +92.5 +87.1 +73.4 +75.6 +65.4 + +9 +Original +Target +FGSM +SemanticAdv +SAA-StarGAN-CS +SAA-StarGAN-PS +(+) Eyeglasses +(+) Heavy Makeup +(+) Heavy Makeup +(+) Bangs +BIM +MI-FGSM +PGD +Sticker +Face Mask +(+) Blond Hair +Fig. 6. +The adversarial face images generated on FaceNet by different +methods. SAA-StarGAN can generate high quality adversarial face images +as compared with the baselines. +We provide an additional experiment described in the Sup- +plementary to confirm the effectiveness of the SAA-StarGAN. +2) Comparison of Attacks based on Score Confidence: We +also evaluate the performance of SAA-StarGAN of black-box +attacks based on score confidence to empirically demonstrate +that the proposed method in the black-box setting can generate +adversarial images that mislead different FV models. We have +improved the SemanticAdv method (denoted as ImSemanti- +cAdv) to add multiple attributes in black-box setting, based +on the random attribute selection. We compare SAA-StarGAN +with ImSemanticAdv that follows our method’s procedure +when changing attributes. This method is performed by mak- +ing a loop to change the random attributes until reaching the +adversary randomly. The main results of black-box attacks +based on score confidence for the impersonation and dodging +are shown in Fig. 3. We can observe that SAA-StarGAN +outperforms ImSemanticAdv across all different models by +a large margin for the impersonation and dodging black- +box attacks. Compared with ImSemanticAdv, SAA-StarGAN +improves the average attack success rate by 13 % ∼ 29.8 +%, and 9.9 % ∼ 54.7 % under impersonation and dodging +attacks, respectively. Therefore, predicting the most important +attributes of the target model affects the attack’s performance +significantly. +C. Visualization for SAA-StarGAN +The goal of adversarial face images is to mislead the +FV models Without fooling humans. Consequently, we illus- +trate samples of adversarial face images generated by SAA- +StarGAN-CS and SAA-StarGAN-PS on FaceNet and ArcFace +respectively in the white-box setting. As illustrated in Fig. 4, +the label of images, sign (+) indicates the addition of attribute +while, sign (-) denotes removing the attribute. We can see that +the proposed method achieves the goal of producing realistic +images and the change on the images is slight. +Fig. 5 presents a set of adversarial face images that are +generated on FaceNet in the black-box setting based on the +score confidence. We observe that a few images are not +realistic. But in general, the proposed method can generate +realistic and diverse images. The comparison between SAA- +StarGAN and baselines on FaceNet model is illustrated in Fig. +6. Although two attributes are applied to the proposed SAA- +StarGAN-CS and SAA-StarGAN-PS methods, the realistic of +the generated adversarial face images are close enough to those +produced by SemanticAdv with single attribute. On the other +hand, the adversarial face images generated by FGSM, BIM, +PGD, and MI-FGSM are not clear compared to our SAA- +StarGAN. The face images generated by Sticker and Face +Mask attacks are unrealistic and misleading to the human eye. +We can conclude that the goal has been successfully achieved +in the current study, as the presented faces are realistic and +can be easily identified. +D. Visualizing Attention on Adversarial Face images +In this subsection, we apply the Gradient-weighted Class +Activation Mapping (Grad-CAM) [50], an attention visual- +ization technique, to find the discriminative regions in the +image according to the TFV model. We use it to show the +attention of the ArcFace and ResNet-101 models for both +the original and the generated adversarial face images on the +FaceNet model. In Fig. 7, we can see that our SAA-StarGAN +focuses on the trivial regions instead of the prominent regions, +as opposed to the SemanticAdv method focuses on the most +prominent regions in the face image. Thus, SAA-StarGAN +could improve the transferability significantly. We conclude +that the prediction step for the most significant attributes plays +a key role in improving the attack transferability. +SAA-StarGAN +CS +CS-M +PS +PS-M +SemanticAdv +ArcFace +ResNet-101 +Original +Fig. 7. +Grad-CAM attention visualization for ResNet-101 and ArcFace +models being attacked by adversarial face images are generated on the FaceNet +model. This Figure compares our SAA-StarGAN based on the important facial +attributes and SemanticAdv based on indiscriminate attributes. +E. Discussion on the Similarity to the Original Image +It is evident in the literature that most methods succeed +in generating adversarial face images. Therefore, we aim to +measure the quality between the adversarial face and original +images through MSE and SSIM. MSE measures the absolute +errors, So the lowest value is the best. SSIM is used for +measuring the similarity between two images by predicting +the perceived quality of images. So, the highest value is the + +10 +best. We compare the adversarial face images crafted by our +SAA-StarGAN with the baseline methods. We select 2,000 +original images and 2,000 adversarial face images for different +attack methods for the evaluation. After that, we calculated +the MSE and SSIM for each method separately, as shown in +Table III. SAA-StarGAN-PS and SAA-StarGAN-CS for single +attribute show the largest values of SSIM compared to others. +Besides, they have the lowest values of MSE, which give +them preference and make them more suitable for generating +adversarial face images. The main reason is that our method +depends on changing a single most significant attribute. This +change leads to a slight modification in the adversarial face +image, but with a high effect in misleading the different +models. On the other hand, SemanticAdv has a high structural +similarity value. The Random Selection method randomly +applies two attributes, which is lower than the SemanticAdv +method in the structural similarity. In contrast, Sticker and +Face Mask methods on FaceNet or ArcFace model show the +highest absolute error. These methods mainly depend on pixel- +level and grid-level, which add large perturbations to the face +image, covering approximately 20 ∼ 30 % area of the face +image. We can see that any adversarial face images generated +by FGSM, BIM, PGD, and MI-FGSM are expected to be +perceptible to human eyes. Generally, SAA-StarGAN-PS and +SAA-StarGAN-CS for a single attribute are most similar to +the original images and have the lowest error. Therefore, it +is recommended to use our proposed method to craft the +adversarial face images. +TABLE III +MSE AND SSIM TO COMPARE THE ORIGINAL AND ADVERSARIAL FACE +IMAGES GENERATED ON FACENET AND ARCFACE MODELS BY OUR +SAA-STARGAN AND THE DIFFERENT BASELINES. BEST VALUES APPEAR +IN BOLD. +Attacks +FaceNet +ArcFace +MSE (↓) +SSIM (↑) +MSE (↓) +SSIM(↑) +FGSM +0.034 +0.388 +0.036 +0.374 +BIM +0.031 +0.596 +0.032 +0.585 +PGD +0.026 +0.715 +0.026 +0.700 +MI-FGSM +0.028 +0.718 +0.029 +0.705 +SemanticAdv +0.023 +0.813 +0.023 +0.813 +Random Selection +0.034 +0.727 +0.034 +0.726 +Sticker +0.958 +0.009 +0.959 +0.008 +Face Mask +0.968 +0.007 +0.969 +0.006 +SAA-StarGAN-CS +0.022 +0.822 +0.020 +0.821 +SAA-StarGAN-PS +0.015∗ +0.854 +0.016∗ +0.846 +SAA-StarGAN-CS-M +0.040 +0.729 +0.036 +0.735 +SAA-StarGAN-PS-M +0.026 +0.792 +0.026 +0.784 +VI. CONCLUSION +In this work, we present a new adversarial attack method +called the SAA-StarGAN to generate semantic adversarial face +images in both white-box and black-box settings. This method +focuses on predicting the important facial attributes for each +input image to change a few of the most important attributes +to make the model promote the trivial features in the face +images. In the white-box setting, SAA-StarGAN first predicts +the significant attributes of each input image via TFV and +uses the cosine similarity or probability score to determine the +most significant attributes. Then the most important features +are changed using a StarGAN model in the feature space. +In the black-box setting, we make a loop to change the +ordered significant attributes until the output is an adversary. +Finally, the generated adversarial face images achieve good +impersonation and dodging attacks. We observe that SAA- +StarGAN could generate high-quality and realistic images +through the experiments and achieve high attack success rates +in the white-box and black-box settings. The proposed SAA- +StarGAN exhibits significantly higher transferability than the +state-of-the-art face attack methods. 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Vedantam, D. Parikh, and +D. Batra, “Grad-CAM: Visual explanations from deep networks via +gradient-based localization,” in IEEE International Conference on Com- +puter Vision, 2017, pp. 618–626. +Yasmeen M. Khedr is currently pursuing the Ph.D. +degree with Huazhong University of Science and +Technology, Wuhan, China. She received the B.S. +and M.S. degrees in information technology from +Zagazig University, Egypt, in 2011 and 2017, re- +spectively. She is an Assistant Lecturer with the De- +partment of Information Technology, Zagazig Uni- +versity. Her research interests include adversarial +learning, artificial intelligence, multimedia security, +image processing and computer vision. +Yifeng Xiong is currently pursuing the master de- +gree with Huazhong University of Science and Tech- +nology, Wuhan, China. He received the B.S. degree +in computer science from Huazhong University of +Science and Technology, Wuhan, China, in 2020. +His research interests include adversarial learning, +machine learning, computer vision and natural lan- +guage processing. +Kun He (SM18) is currently a Professor in School +of Computer Science and Technology, Huazhong +University of Science and Technology, Wuhan, P.R. +China; and a Mary Shepard B. Upson Visiting Pro- +fessor for the 2016-2017 Academic year in Engineer- +ing, Cornell University NY, USA. She received the +Ph.D. degree in system engineering from Huazhong +University of Science & Technology, in 2006. Her +research interests include adversarial learning, ma- +chine learning, social network analysis, and combi- +natorial optimization. + +12 +APPENDIX +The supplementary material for Semantic Adversarial At- +tacks on Face Recognition through Significant Attributes. +A. Cosine Similarity (CS) +Algorithm 1 Determine the most significant attributes by CS +Input: Original image x, attributes {a}, generator G of Star- +GAN, target face verification model TFV +Output: The most significant attributes C +1: for each ai in attributes {a} do +2: +change ai using G +3: +x∗ +ai ← G(x, ai) +4: +fx∗ ← TFV (x), fx∗ai ← TFV (x∗ +ai) +5: +Sai ← CS(fx, fx∗ai ) +6: end for +7: C ← Sort {a} according to each Sai in the ascending +order +8: Output +the +most +significant +attributes +C += +(c1, c2, c3, ..., cK) +B. Probability Score (PS) +Algorithm 2 Determine the most significant attributes by PS +Input: Original image x, attributes {a}, generator G of Star- +GAN, attribute prediction model f +Output: The most significant attributes {C} +1: Use attribute prediction model to predict attributes +2: for each a′ +i in attributes {a} do +3: +Pa′ +i(x) ← f(x) +4: +Change a′ +i using G +5: +x∗ +a′ +i ← G(x, a′ +i) +6: +Pa′ +i(x∗ +a′ +i) ← f( x∗ +a′ +i) +▷ Compute the probability for +each attribute +7: +Compute ∆Pa′ +i +8: end for +9: C ← Sort {a} using ∆Pa′ +i in descending order +10: Output +the +most +significant +attributes +C += +(c1, c2, c3, ..., cK) +C. SAA-StarGAN in Black-Box Setting +To emphasize and clarify the effectiveness of the pro- +posed SAA-StarGAN method in improving the transferability +against different models. We select 1000 adversarial face +images crafted on FaceNet from each attack method against +a SphereFace model under an impersonation attack. For the +dodging attack, we choose 1000 of the generated adversarial +face images on FaceNet with the original images. We measure +the cosine similarity scores before and after the attack to see +the improvement percentage for each attack method against a +SphereFace model. In Fig. 8, we observe that most scores +between the generated face and target faces of the SAA- +StarGAN method fall above Ts @ 0.1 % FPR, demonstrating +Algorithm 3 The SAA-StarGAN for Black-Box Attack +Input: Original image x, original attribute cori, most signifi- +cant attributes {c}, encoder GE, decoder GD, target face +verification model TFV , target image xtgt, L2 distance +function dist, threshold sim th +Output: Advesarial example xadv +1: f ∗ +1 ← GE (x, cori) +2: γ ← create array in range [0,1] +3: for ci in most significant attributes C do +4: +f ∗ +ci ← GE (x, ci) +5: +for γi in γ do +6: +f ∗ +γi = γi · f ∗ +1 + (1- γi) · f ∗ +ci +7: +x∗ +γi = GD (f ∗ +γi) +8: +dγi ← dist (TFV (xtgt, x∗ +γi) +9: +score.insert(dγi, γi ) +10: +end for +11: +if impersonation attack then +12: +γopt = argmaxγ (score) +13: +else if dodging attack then +14: +γopt = argminγ (score) +15: +end if +16: +Substitute γopt in f ∗ +17: +x∗ = GD (f ∗) +18: +if sim(x, x∗) ≤ th then +19: +Return None +20: +end if +21: +if dist(TFV (x∗, xtgt) ≤ T) then +22: +xadv ← x∗ +23: +break +24: +else +25: +x ← x∗ +26: +ci ← add the next attribute from C +27: +end if +28: +if dist(TFV (x∗, x) ≥ T) then +29: +xadv ← x∗ +30: +break +31: +else +32: +x ← x∗ +33: +ci ← add the next attribute from C +34: +end if +35: end for +that the generated face images can be falsely accepted by 53, +50, 48.7, and 50.6 % in an impersonation attack. In contrast +to the baseline methods, a few scores fall above Ts. For the +dodging attack, the scores fall below Ts @ 0.1 % FPR by 77.2, +71.8, 61.4, and 66.7 %, demonstrating that the SphereFace +model can falsely reject the image pairs. We conclude that the +SAA-StarGAN significantly outperforms the baseline attacks +to improve the transferability of black-box models. + +13 +-0.2 +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +40 +50 +Frequency +FGSM + Adversarial + Original + +T@0.1% FPR +(a) Impersonation attack +-0.2 +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +40 +50 +Frequency +BIM +-0.2 +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +40 +50 +Frequency +SemanticAdv +-0.2 +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +40 +50 +Frequency +Random Selection +-0.2 +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +40 +50 +Frequency +SAA-StarGAN-CS +-0.2 +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +40 +50 +Frequency +SAA-StarGAN-PS +-0.2 +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +40 +50 +Frequency +SAA-StarGAN-CS-M +-0.2 +0.0 +0.2 +0.4 +0.6 +0 +10 +20 +30 +40 +50 +Frequency +SAA-StarGAN-PS-M +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +10 +20 +30 +40 +50 +Frequency +FGSM + Adversarial + Original + +T@0.1% FPR +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +10 +20 +30 +40 +50 +Frequency +BIM +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +10 +20 +30 +40 +50 +Frequency +SemanticAdv +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +10 +20 +30 +40 +50 +Frequency +Random Selection +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +10 +20 +30 +40 +50 +Frequency +SAA-StarGAN-CS +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +10 +20 +30 +40 +50 +Frequency +SAA-StarGAN-PS +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +10 +20 +30 +40 +50 +Frequency +SAA-StarGAN-CS-M +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +10 +20 +30 +40 +50 +Frequency +SAA-StarGAN-PS-M +(b) Dodging attack +Fig. 8. Shift in cosine similarity scores for SphereFace before and after adversarial attacks generated by SAA-StarGAN and the baselines on FaceNet. The +shift of SAA-StarGAN is much more distinct as compared with the baselines. + diff --git a/XNFLT4oBgHgl3EQfTi_1/content/tmp_files/load_file.txt b/XNFLT4oBgHgl3EQfTi_1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c642619fbcff02ddbed9f902b82f037e2226a6b --- /dev/null +++ b/XNFLT4oBgHgl3EQfTi_1/content/tmp_files/load_file.txt @@ -0,0 +1,1603 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf,len=1602 +page_content='1 Semantic Adversarial Attacks on Face Recognition through Significant Attributes Yasmeen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Khedr ˙ID , Yifeng Xiong ˙ID , and Kun He ˙ID , Senior Member, IEEE Abstract—Face recognition is known to be vulnerable to adversarial face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Existing works craft face adversarial images by indiscriminately changing a single attribute without being aware of the intrinsic attributes of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' To this end, we propose a new Semantic Adversarial Attack called SAA- StarGAN that tampers with the significant facial attributes for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We predict the most significant attributes by applying the cosine similarity or probability score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The probability score method is based on training a Face Verification model for an attribute prediction task to obtain a class probability score for each attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The prediction process will help craft adversarial face images more easily and efficiently, as well as improve the adversarial transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Then, we change the most significant facial attributes, with either one or more of the facial attributes for impersonation and dodging attacks in white-box and black- box settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Experimental results show that our method could generate diverse and realistic adversarial face images meanwhile avoid affecting human perception of the face recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' SAA- StarGAN achieves an 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='5% attack success rate against black- box models, outperforming existing methods by 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='5% under the impersonation attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Concerning the black-box setting, SAA- StarGAN achieves high attack success rates on various models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The experiments confirm that predicting the most important attributes significantly affects the success of adversarial attacks in both white-box and black-box settings and could enhance the transferability of the crafted adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Index Terms—Adversarial examples, face verification, image- to-image translation, feature fusion, adversarial transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' INTRODUCTION Face Recognition (FR) [1] is an important computer vision task widely used in solving authentication problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' FR can be categorized as Face Identification and Face Verification (FV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Over the past decades, FV, which determines whether a pair of face images belong to the same identity [2], has achieved great achievements in various applications such as mobile payment, military, finance, surveillance security, and border control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' However, Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [3] find that Deep Neural Net- works (DNNs) are susceptible to adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' These adversarial examples have tiny perturbations added to the benign images that remain imperceptible to human vision but Manuscript received October 19, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' This work is supported by International Cooperation Foundation of Hubei Province, China (2021EHB011) and National Natural Science Foundation (62076105).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Corresponding author: Kun He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Yasmeen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Khedr is with the School of Computer Science and Technol- ogy, Huazhong University of Science and Technology, Wuhan 430074, China, and also with the Faculty of Computers and Informatics, Zagazig University, Zagazig 44519, Egypt (e-mail: yasmeenkhedr@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Yifeng Xiong and Kun He are with the School of Computer Science and Technology, Huazhong University of Science and Technology, Wuhan 430074, China (e-mail: xiongyf@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' brooklet60@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' could mislead DNN models to produce incorrect predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Besides, some studies confirm the vulnerability of DNNs to input variations [2]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Also, adversarial attacks can be divided into categories with different goals and assumptions on the attacker’s knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' White-box and black-box are two main settings based on the assumption of the attacker’s knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The former supposes that the attacker can access the model’s parameter values, architecture, training method, inputs, outputs, and weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Whereas the latter assumes that the attacker only has access to the inputs and outputs of the model but knows no information about the model [1] There is a growing interest in adversarial studies for FR models [5]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Adversarial studies seek to generate adversar- ial face images to mislead facial recognition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Methods used to manipulate the facial content includes face synthesis, identity swap, face morphing, face attribute manipulation, and expression swap [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Face attributes are among the emerging soft biometrics for modern security systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Some studies use face attribute manipulation for different goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Rozsa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [10], [11] propose the Fast Flipping Attribute technique to mislead facial attribute recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Also, Mirjalili and Ross [12] use the face attribute to modify the face image for a gender classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Recently, methods based on Generative Adversarial Network (GAN) have appeared that are used to manipulate facial attribute images, such as StarGAN [13], STGAN [14], and AttGAN [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [16] use AttGAN to generate semantic attacks to deceive gender classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' These studies are limited to classification problems instead of facial recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Meanwhile, Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [6] craft adversarial examples to mislead FR by changing the attribute individually and check- ing whether the generated image is adversarial until they find an adversarial example or failed after attempting the change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' But, they craft an adversarial face image by indiscriminately distorting facial attributes without being aware of the signif- icant facial attributes on each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [17] propose Adversarial Makeup Transfer GAN (AMT-GAN) to generate adversarial face images, but it tends to produce high-quality images of females due to an imbalance of gender in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' These studies handle the attack on FR but they either suffer from the weakness of transferability to black-box models due to changing the face attribute randomly, or exhibit bias on genders due to the imbalance of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Our work aims to mislead FR models depending on chang- ing the significant facial attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' So, we propose a new attack method called the Semantic Adversarial Attack using StarGAN (SAA-StarGAN), which effectively and easily crafts semantic adversarial examples besides improving the attack arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='12046v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='CV] 28 Jan 2023 2 transferability significantly by tampering with the significant facial attributes for each input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' These attributes are sup- posed to affect the decisions of different FV models, leading to deceiving the FV models and enhancing the adversarial transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In the white-box setting, we predict the most significant attributes for each input image by using either the cosine similarity (CS) or the probability score (PS) based on the Target Face Verification (TFV) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Then, we change one or multiple via the StarGAN model in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The Attention Feature Fusion (AFF) method is used to fuse the features of inconsistent semantics to generate a realistic image and produce β different values used for interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In the black-box setting, SAA-StarGAN depends on predicting the most important attributes through the cosine similarity (CS) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' These attributes are modified on the input image according to their arrangement based on the prediction step by making an iterative loop to alter them sequentially until reaching the adversarial face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The empirical results confirm that predicting the most significant attributes (that will be changed first) plays an im- portant role in successful attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Our SAA-StarGAN method outperforms other methods significantly on the attack success rate in the black-box setting and also preserves high attack success rates in the white-box setting for both impersonation and dodging attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Our method provides perceptually real- istic images that maintain the source image identity to avoid confusing human perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We also analyze the attention map of the TFV model that is attacked by our adversarial face images using gradient-weighted class activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' As a result, our method focuses on trivial features instead of prominent features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The main contributions of this work are summarized as follows: We propose a novel attack method called Semantic Adversarial Attack using StarGAN (SAA-StarGAN) that enhances the transferabil- ity of adversarial face images by tampering with the critical facial attributes for each input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' SAA-StarGAN generates semantic adversarial face images easily and effectively in white-box and black-box settings by predicting the most significant facial attributes using two techniques, cosine similarity or probability score, for imper- sonation and dodging attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We propose modifications on SAA-StarGAN to depend only on the output of the target model in a black-box setting by applying a linear search to find the optimal value of the interpolation co- efficient that affects the generated face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The empirical results confirm that predicting the most significant attributes (which will be changed first) plays a vital role in a successful attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Our SAA-StarGAN method outperforms other methods considerably on the white-box attack success rate and black-box adversarial transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Also, it provides perceptually re- alistic images that maintain the source image identity to avoid confusing human perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' RELATED WORK In this section introduce the related work of generating adversarial examples for both image classification and FR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Adversarial Attacks on Images Many adversarial example generation methods have been proposed to mislead different image classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Most studies focus on generating restricted adversarial ex- amples by adding perturbations to the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [3] first find the existence of adversarial examples for image classification, which transforms an image by a small amount to be undetectable and thereby changes how the image is classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [4] propose a Fast Gradient Sign Method (FGSM) that uses the gradients of the neural network to generate adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' They just applied a one-step gradient update along the direction of the gradient sign at each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Kurakin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [18] propose a Basic Iterative Method (BIM), which applies FGSM perturbations of smaller magnitude for multiple iterations to improve the attack success rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' They clipped pixels in each iteration to avoid a large change Besides, Projected Gradient Descent (PGD) [19] is considered an extension of the BIM method to diversify the synthesized adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [20] and Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [21] incorporate momentum into the iterative FGSM to boost the attack transferbility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Diverse methods have been proposed, such as designing an efficient saliency adversarial map for seeking the adversarial noise [22], image denoising attack [23], and adding perturba- tion based on the attended regions and features [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Meanwhile, GAN is also used in the construction of ad- versarial examples due to its awesome ability to generate images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [25] propose AdvGAN, which constructs a generator network based on encoder-decoder to generate adversarial perturbation and then adds this perturbation to the original image to mislead the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' On the other hand, Jandial et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [26] propose AdvGAN++, showing that latent features achieve higher attack success rates than AdvGAN and craft realistic images on CIFAR and MNIST datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [27] propose a method for generating unrestricted adver- sarial examples based on the ACGAN from scratch instead of adding small perturbations on a source image for the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Adversarial Transfer on Generative Adversarial Net (AT-GAN) [28] generates non-constrained adversarial examples directly from any input noise, which aims to learn the distribution of the adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Adversarial Attacks on Face Recognition Recently, many adversarial attacks have been proposed for attacking the FR models, which can be divided into three cat- egories: (1) adding adversarial perturbations, (2) manipulating facial attributes, and (3) physical attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' One line of study focuses on changing the facial appear- ance of input images by adding small perturbations in a specific region to be imperceptible to human eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Deb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [29] propose an automated adversarial face method called 3 AdvFaces that generate minimal perturbations in the salient facial regions via GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [30] hide the attack information by the makeup effect to attack the eye regions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [5] propose Attentional Adversarial Attack Generative Network (A3GN) to deceive the FR model under impersonation attacks, where attention modules and variational autoencoder are incorporated to learn semantic information from the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Zhong and Deng [31] propose Dropout Face Attacking Network (DFANet) to improve the transferability by integrating the dropout in convolution layers in the iterative steps to generate adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The second category is based on manipulating facial attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Rozsa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [11] propose Fast Flipping Attribute (FFA) technique which found that the robustness of DNNs against adversarial attacks varies highly between facial attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Kakizaki and Yoshida [7] focus on the serious risks resulting from the ineffectiveness of conventional certified defenses against adversarial examples that are not restricted to small perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' They use image translation techniques to generate unrestricted adversarial ex- amples by translating the source image into any desired facial appearance with large perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Qiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' [6] introduce SemanticAdv, which can generate unrestricted adversarial ex- amples by altering a single facial attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The physical attacks can be generated by a variety of different tools, such as adding some adversarial face acces- sories [32], natural makeup [33] and adversarial patches [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' These methods make the attacks more dangerous in the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' FACE RECOGNITION ATTACK In this section, we first provide the problem definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Then we present a detailed description of our method in white-box and black-box settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Ultimately, we introduce a preliminary method called Random Selection as a baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Problem Definition The main purpose of the FR model is to recognize the input image by training the model on dataset D(x, y), where x is a face image sampled according to the latent distribution, and y is the corresponding ground-truth label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' According to f(x) : x → y, the model can predict the label for each input face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The main goal is to generate an adversarial face image xadv that is similar to the original image x but misleads the FV model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' FV (xadv) = y′ ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' On the other hand, adversarial attacks are divided into two types: dodging and impersonation attacks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The dodging attack (untargeted attack) is developed to fool the target model such that the output is a random identity excluding the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In contrast, the impersonation attack (targeted attack) misleads the target model by recognizing the adversarial face image as a specified target identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' For the dodging attack, we generate xadv, which is identified as not the same identity as FV (xadv) ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' For impersonation attack, we seek to make the model recognize xadv as the same identity of another given image such that FV (xadv) = ytgt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Semantic Adversarial Attack (SAA-StarGAN) We propose an efficient attack method that first predicts the most significant facial attributes for each input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Then, we generate xadv in the intermediate layers instead of the output layers because this increases internal feature distortion, leading to improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' There are two steps in the proposed framework for a white-box attack: 1) Predict the most significant attributes for each input image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 2) Generate xadv by modifying one or multiple most significant attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' These attributes are changed by using the StarGAN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' StarGAN model [13] consists of a single generator G and a discriminator D trained to learn to translate images from one domain to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 1) Significant Attribute Prediction: We propose using either the cosine similarity (CS) or the probability score (PS) to detect the significant attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The corresponding methods are denoted as SAA-StarGAN-CS and SAA-StarGAN-PS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We first apply CS or PS to predict the most significant attributes and compare their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Consequently, we retrain the G of StarGAN on all the facial attributes to use in the significant attribute prediction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' a) Cosine Similarity (CS): We utilize the TFV models [35], [36] to obtain the important attributes by computing the cosine similarity between the out- put features of the TFV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' More details of the algorithm for the predicted most significant attributes using CS are shown in Supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' For image attributes a = (a1, a2, a3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=', aK) where ai indicates the ith attribute, and K the total number of attributes, we first use StarGAN to change each ai of the input x to get the image synthesis x∗ ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Then, we extract the synthesis image features fx∗ai , and the original image features fx by the TFV model, and calculate their cosine similarity to get Sai, that refers to the degree of change in the output of TFV by changing attribute ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We then sort the attributes in a in the ascending order according to the similarity score values to get the most important attributes C = (c1, c2, c3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=', cK) where ci is the ith sorted attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The less the cosine similarity, the more the significance of the attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The cosine similarity, as illustrated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (1), is a way to measure the similarity between two vectors, ranging from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Sai = CS(fx, fx∗ai ) = fx · fx∗ai ∥fx∥ · ∥fx∗ai ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (1) b) Probability Score (PS): The basic idea of the probability score method is to use the TFV model as the Attribute Prediction model (Att-Pred) to predict the important attributes of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' For each attribute ai in input x, we use the Att-Pred model to obtain the class probability score (PS), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' If a′ i is found in the original image x, we remove it in the synthesis image x∗ and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Therefore, we use the StarGAN model to change a′ i to get the x∗ a′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Moreover, each attribute in the input face image has a different impact on the final decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Thus, we need to calculate the probability value Pa′ i of a′ i for x and x∗ to get the degree of change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' ∆Pa′ i indicates the degree of change in 4 Young Wearing_Lipstick Wavy_Hair Pointy_Nose No_Beard Heavy_Makeup Blond_Hair Attractive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Output from the attribute prediction model using FaceNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' the probability value in image x before and after changing a′ i to determine the most significant attributes for each image x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' ∆Pa′ i = Pa′ i(x) − Pa′ i(x∗ a′ i), (2) where x = a1a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='ak, x∗ a′ i = a1a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='a′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Here x∗ a′ i indicates the generated image after changing attribute a′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Finally, we sort the attributes in a in descending order using ∆Pa′ i to obtain the most significant attributes C to help us craft xadv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' These attributes represent the best attack effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The algorithm of this step is illustrated in Supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 2) Adversarial Face Image Generation: The second step focuses on generating the adversarial face images, as illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' To generate xadv, we apply two kinds of per- turbations using single or multiple attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' SAA-StarGAN- CS-M and SAA-StarGAN-PS-M indicate the methods used for multiple attributes in CS and PS techniques, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' a) For single attribute, after completing the first step of predicting C = (c1, c2, c3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=', cK), we use the generator G of StarGAN, which we have already trained on facial attributes in the significant attribute prediction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' G is composed of an encoder (GE) and a decoder (GD), as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The (GE) takes an input image x and the single significant attribute c1, then we get the output feature in intermediate layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (GD) takes the feature as input, and outputs the synthesized image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 2, we use the x as an input to the GE with significant attribute c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Then we extract the output features from the conv layer f ∗ conv and the Residual Block layer f ∗ res of the encoder, as shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' G = GE· GD, (3) f ∗ conv = GE(x, c, conv layer), (4) f ∗ res = GE(x, c, res layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (5) On the other hand, we exploit the Attention Feature Fusion (AFF) method [37] to obtain the fused feature as an input to the decoder GD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Usually, addition and concatenation methods combine features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' But recently, the attention mechanism has succeeded in many computer vision applications [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' So, we use AFF to combine the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' AFF is a framework that combines the features from different layers based on the Multi-Scale channel attention (MS) module to overcome the inconsistent semantics among the input features and generate a more realistic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' More details about the AFF framework, AFF Original image 1 1 0 0 0 0 Most significant attributes 𝒇𝒓𝒆𝒔 ∗ 𝒇𝒄𝒐𝒏𝒗 ∗ 𝒇∗ TFV Match or not Target image 𝒙𝒂𝒅𝒗 𝒙𝒂𝒅𝒗 𝒙𝒕𝒈𝒕 Adversarial loss Attribute labels Blond/Smiling/Eyeglasses/ Chubby/Bangs/Young Encoder (E) Decoder (D) MS 𝜷 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The SAA-StarGAN attack framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The original image and most significant attributes are fed into Encoder (GE) to change these attributes and then extract their features from different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The Attention Feature Fusion (AFF) framework is then used to generate β and perform fusion between features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' After that, the fused features are sent to the decoder (GD) to obtain the synthesis image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Finally, the (TFV ) model receives both the synthesis image and target identity to calculate the adversarial loss and optimize the β value at the feature level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' including the MS module, are presented in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (6) indicates the fusion weights β, where β ∈ [0, 1] is calculated from the attention weights generated by the MS module in AFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' As a result, we update the value of β until the model is misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' To get the better fused feature f ∗, we apply an interpolation in the feature space, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Finally, the decoder GD takes the fused feature f ∗ as input and gets the synthesized face image x∗ as the output, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' β = MS(f ∗ conv, f ∗ res) (6) f ∗ = β· f ∗ conv + (1 − β)· f ∗ res (7) x∗ = D(f ∗) (8) Our work obtains the xadv by modifying the significant attribute c1 for each image through feature-level interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' To achieve the impersonation attack, we use the L2 loss function to minimize the distance between the face embedding of xadv and the target image xtgt, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' xadv = Minx∗ ∥ TFV (x∗) − TFV (xtgt) ∥2 2 (9) The adversarial face image is crafted for dodging the attack by maximizing the distance between xadv and the x in the feature space, as shown in the following: xadv = Maxx∗ ∥ TFV (x∗) − TFV (x) ∥2 2 (10) b) For multiple attributes, similarly, we use the same G of StarGAN, which has been used to modify a single attribute, but in this step, to change multiple significant attributes C in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In our experiments, we change two significant attributes, c1 and c2, represented as one-hot vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' As a result, we obtain the synthesized image with a bigger change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' After that, We study the effect of this change on attacking the face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We do not change more than two attributes as we Young Wearing_Lipstick Wavy_Hair Pointy_Nose No_Beard Heavy_Makeup Blond_Hair Attractive 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='05 have tested using more significant attributes in C, but some of the resulting images are not realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 3) SAA-StarGAN in Black-Box Setting: A black-box attack differs from a white-box attack as the adversary has no access to the model’s gradient or parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Therefore, we propose modifications for our proposed method, SAA-StarGAN, to depend only on the target model’s output in the black-box setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' To generate a successful attack, we need two steps: 1) Predicting the most significant attributes for the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' This step is similar to that in the white-box setting, but we only use the CS method as it does not need to train a target model to predict the attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 2) Doing a linear search to find the largest γ value that affects the generated face image by changing the most significant attributes with γ for each face image in an iterative loop until the output misleads the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' More details of the algorithm for the black-box attack method are shown in Supplementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' To change the most significant attributes C, we use GE from the StarGAN model to get the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' But, we make an iterative loop to alter the attributes according to the most important to the least until we meet the adversarial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We use a bilinear interpolation with a variable γ value to generate the fused features f ∗ γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Therefore, we make a linear search to find the optimal γopt value according to the change in the confidence score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We create a vector γ that consists of 100 random values drawn from [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' This vector is applied to study the effect of each value γ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Then, the values are arranged according to the score change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' For an impersonation attack, we select the optimal γopt value that decreases the distance between the face embedding of the generated face image and the target face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' For the dodging attack, we select the optimal γopt value that maximizes the distance between the face embedding of the generated face image and the input face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Finally, we substitute the optimal γopt value to generate the fused feature and feed it to a decoder to get the x∗ by the following equations: f ∗ = γopt· f ∗ 1 + (1 − γopt)· f ∗ ci, (11) x∗ = GD(f ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (12) To guarantee that the generated face image will preserve semantic similarity from the original face image, we need to measure the semantic similarity sim between the generated face image and the input face image to filter out the face images that are not realistic and control their quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The adversarial face image is found when the adversarial criterion is achieved and the semantic similarity is above a threshold th (sim > th).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The above steps will repeat to add the next significant attribute in the ordered significant attributes until we find an adversarial example (Lines 18 - 34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Random Attribute Selection based Attack This subsection presents a preliminary method , the Random Selection, as a baseline method to compare with our SAA- StarGAN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We generate adversarial face images based on randomly selecting a set of attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' These attributes are changed by using the StarGAN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Therefore, we make two copies of the generator from StarGAN and re-train them separately using two different sets of attributes representing each attribute a as a one-hot vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The first generator, G1, is trained by the first set S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='The second generator, G2, is trained by the second set S2As a result, the overall Random Selection system involves three components, namely G1, G2, and the TFV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Firstly, G1 takes x with a specific attribute as1 from S1 to generate translated image with dimensions H, W, and L for height, width, and channels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The second image synthesis x∗ 2 is then obtained using the translated image as the input to G2 with another attribute as2 from S2, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Note that the two attributes are chosen randomly from the sets S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Through permutations between the attributes from the previous sets, we change two attributes using the StarGAN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Finally, interpolation is applied between the pair of images produced from G1 and G2 to generate xadv as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' According to the presented procedure, 25 adversarial face images are generated for each x due to the permutation between different attributes from the two sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' x∗ 2 = G2(G1(x, as1), as2) (13) xadv = α· x∗ 1 + (1 − α)· x∗ 2 (14) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' EXPERIMENTAL SETUP This section provide experimental setup, including dataset, models, baselines, evaluation metrics and implementation de- tails of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Dataset The proposed SAA-StarGAN method uses the CelebA dataset, which has 202,599 face images with 40 facial at- tributes and 10,177 identities [39], to generate semantic ad- versarial face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' This dataset is the most popular type used in the face recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We use 40 facial attributes to train the StarGAN model in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Also, we randomly choose 5,000 different identities as the original images and 5,000 different identities as the target images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Target Face Verification Models To evaluate the effectiveness of the proposed SAA- StarGAN, we choose ten state-of-the-art FV models, con- taining different model architectures and training loss func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We use two of them as the white-box TFV models: FaceNet [35] and ArcFace [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The two models return 512-dimensional embeddings of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Besides, we use two other publicly available models, SphereFace [40] and CosFace [41], for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Then, we select different trained models under different backbones and loss functions such as ResNet-101 [42], IResNet50 [36], [43], MobileFace [44], and ShuffleNet V2 [45] to demonstrate the effectiveness of our SAA-StarGAN on different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We compute the optimal threshold T based on the false-positive rate (FPR) for each FV model used in the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Details of each model are presented in Supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Baselines We adopt eight baseline methods to evaluate our attack method, including the Random Selection method presented in Section III-C, the typical one-step attack method of FGSM [4], BIM [18], PGD [19], and MI-FGSM [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Besides, face attack methods such as Sticker and Face mask attacks [46] are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The facial attack method of SemanticAdv [6] is the most comparable method to ours, which modifies the face attribute to generate adversarial face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' For the Random Selection method, five facial attributes are applied for G1: hair color (black-blond), heavy makeup, gender, and pale skin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' while for G2, smiling, mouth slightly open, bangs, eyeglasses, and young attributes are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' For FGSM, we set the perturbation as ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='2 for L2 attack and the number of iterations as 20 for BIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Also, we set the perturbation as ϵ = 8 for PGD and MI-FGSM with pixel values in the range [0, 255].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The number of iterations is 40, and the decay factor µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Evaluation Metrics We use several evaluation matrices to estimate our attack effectiveness over different baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We select the attack success rate to evaluate the adversarial face images crafted by SAA-StarGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' To measure the similarity between images, we use the cosine similarity and use a threshold T @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 % FPR for each TFV model to decide whether the similarity between two identity persons matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We can obtain cosine threshold Ts from the Euclidean threshold after normalizing features as Ts = 1 − (T/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The attack success rate used for impersonation attack is computed as follows: success rate = (#ImagePairs(xadv, xtgt) ≥ Ts) (#TotalImagePairs) , (15) where ImagePairs consists of an adversarial face image gen- erated by SAA-StarGAN and the matched target face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The attack success rate used for dodging attack is computed as follows: success rate = (#ImagePairs(xadv, x) < Ts) (#TotalImagePairs) , (16) where ImagePairs consists of an adversarial face image gen- erated by SAA-StarGAN and the input face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Finally, we compute the Mean Square Error (MSE) [8] and the Structural Similarity Index Measure (SSIM) [47] to evaluate the quality between the original face images x and the adversarial face images xadv for different attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Implementation Details We use the Adam optimizer [48] with a fixed learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='05 and up to 300 epochs for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' SAA-StarGAN is implemented using PyTorch v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' For the PS, we train the TFV model for the attribute prediction task to have the class probability score (PS) for each attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' All experiments are conducted on a single Titan X GPU to generate adversarial face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We perform impersonation attacks based on the original and target images with different identities and dodging attacks based on the pairs of original images with the same identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Firstly, all the selected face images pass through an MTCNN detector [49] to detect the face image and align images for the entire image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Then, we obtain the resized images in 112 × 112 × 3, but inside each model it will carry out the specific input size due to the diversity of model inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' EXPERIMENTAL RESULTS To validate the effectiveness of SAA-StarGAN, we empir- ically evaluate our adversarial face images and illustrate that SAA-StarGAN achieves higher attack success rates against different FV models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' On the other hand, SAA-StarGAN boosts the adversarial transferability with high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The black- box setting demonstrates that SAA-StarGAN achieves high attack success rates significantly and shows the importance of significant attribute prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Besides, SAA-StarGAN can generate realistic and diverse adversarial face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Table I and Table II list the attack success rates of various methods, using FaceNet and ArcFace models to generate adversarial examples respectively, for impersonation and dodging attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Comparison for White-box Attacks To validate the efficacy of our attack method, we fisrt compare SAA-StarGAN with the baseline methods in the white-box setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' For a fair comparison, we train these base- line methods using FaceNet and ArcFace models on CelebA dataset with T @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1% FPR for both impersonation and dodging attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Then we compute the attack success rate for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The first column in Table I compares our SAA- StarGAN and the baselines in the white-box setting on the FaceNet model for the impersonation attack in (a) and dodging attack in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' All variants of the proposed SAA-StarGAN method have reached a nearly 100 % attack success rate, confirming that SAA-StarGAN can mislead the TFV models successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The second column in Table II compares our SAA-StarGAN and the baselines in the white-box setting on the ArcFace model for the impersonation attack in (a) and dodging attack in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' For all the cases, it can be concluded that our SAA-StarGAN can effectively craft adversarial face images to fool TFV models in the white-box setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Comparison for Black-box Attacks Most face recognition systems do not allow access to any internal information of the neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In the black-box setting, the weights and network architectures are not included in the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' So, it is essential to evaluate the vulnerability of FRs in both the transfer-based attack setting and the attack based on score confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 1) Transferability Analysis: The transferability across dif- ferent models is one of the most important properties of adversarial examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' To demonstrate the transferability of the adversarial face images generated by SAA-StarGAN under two white-box models, we construct a dataset from successful adversaries crafted by the FaceNet and ArcFace models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Then we evaluate the attack success rate on nine different TFV models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We can observe from Table I that for impersonation 7 attacks, our SAA-StarGAN achieves a high attack success rate under different experimental conditions compared to various baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In contrast, the attack success rate of the Sticker and Face mask attacks has not exceeded 17 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' FGSM and BIM methods exhibit the weakest transferability on all TFV models after the Sticker and Face mask attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The PGD, MI- FGSM, Random Selection, and SemanticAdv methods achieve better transferability than FGSM and BIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The attack success rate of these attacks improves by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='5 % ∼ 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 %, and our SAA-StarGAN outperforms them by a clear margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Although the Random Selection and SemanticAdv methods craft the adversaries by modifying the facial attributes, they only depend on changing random attributes or a single fixed attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We can see that the transferability of SAA-StarGAN- CS for the single attribute surpasses that of other methods in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In addition, attacking the ShuffleNet V2 model using adversarial face images generated by FaceNet for the SAA-StarGAN-CS achieves the best attack success rate of 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='30 %, outperforming the baseline attacks by 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='7 % ∼ 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Finally, we conclude that our SAA-StarGAN method outperforms others on all models besides maintaining a high attack success rate on the white-box setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Similarly, our attack outperforms the other baselines significantly in dodging attack, as presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We can observe that the ad- versarial face images generated by SAA-StarGAN-CS against the SphereFace exceed 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 % ∼ 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='7 % for the baseline attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Face mask based on the grid level is effective than PGD and MI-FGSM under dodging attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In addition, this attack is 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 (b) Dodging attack SAA-StarGAN Attack Sucess Rate (%) ImSemanticAdv FaceNet ArcFace ResNet-101 CosFace SphereFace MobileFace ShuffleNet V2 IR50-Softmax IR50-CosFace IR50-SphereFace 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='2 Attack Sucess Rate (%) FaceNet ArcFace ResNet-101 CosFace SphereFace MobileFace ShuffleNet V2 IR50-Softmax IR50-CosFace IR50-SphereFace (a) Impersonation attack Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The average attack success rate (%) for black-box attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' SAA- StarGAN outperforms ImSemanticAdv by a clear margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Original Target Adversarial Single attribute (+) Receding Hairline (+) Wearing Lipstick Adversarial Multiple attributes (+) Receding Hairline (+) Chubby (+)Wearing Lipstick (+) Eyeglasses (a) SAA-StarGAN-CS FaceNet model (-) Chubby (+) Bangs (-) Chubby (+) Eyeglasses (+) Bangs (-) Mouth Slightly Open (b) SAA-StarGAN-PS ArcFace model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Adversarial face images generated by SAA-StarGAN in white-box setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' It shows the success of SAA-StarGAN in producing realistic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Adversarial face images generated by SAA-StarGAN in the black-box setting based on the score confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The figure illustrates the realistic face images, and the red borders indicate a few unrealistic adversarial face images generated by SAA-StarGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Table II illustrates that our SAA-StarGAN significantly im- proves the transferability of adversarial face images crafted by the ArcFace model against different models over the baseline attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The main reason is that SAA-StarGAN depends on manipulating the most important attributes that affect the de- cision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' As clear from Table II, SAA-StarGAN-CS outperforms the baselines of MI-FGSM, PGD, Random Selection, Seman- ticAdv, BIM, FGSM, Face Mask, and Sticker by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='8, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='8, 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='5, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0, 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='5, 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0, and 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 % respectively under the impersonation attack, against IR50-CosFace model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' To dodge an attack, The results are illustrated in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We conclude that the main advantage of SAA-StarGAN-CS is that it does not need to train the TFV models for the attribute prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We can directly apply different TFV models easily and efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Besides, we observe that the models within the same backbone have good transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' So, the generated adversarial face images crafted on the ArcFace model against IR50-Softmax, IR50-CosFace, and IR50-SphereFace have high transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' And ShuffleNet V2 and MobileFace have light weights which are easily attacked by the adversarial face images generated on ArcFace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In contrast, the FaceNet model is challenging to transfer to other models, where it is trained on IncepetionResNetV2 based on the softmax loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 8 TABLE I TRANSFERABILITY OF THE ADVERSARIAL EXAMPLES GENERATED BY SAA-STARGAN AND THE BASELINES AGAINST BLACK-BOX MODELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' WE USE ADVERSARIAL EXAMPLES GENERATED ON FACENET MODEL TO ATTACK NINE DIFFERENT TFV MODELS FOR IMPERSONATION & DODGING ATTACKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' VALUES REPRESENT THE ATTACK SUCCESS RATE (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' ∗ INDICATES WHITE-BOX ATTACKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' THE NUMBERS MARKED IN BOLD REPRESENT THE BEST ATTACK SUCCESS RATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (a) Impersonation attack Attack FaceNet ArcFace ResNet-101 CosFace SphereFace MobileFace ShuffleNet V2 IR50-Softmax IR50-CosFace IR50-SphereFace FGSM 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='7∗ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='3 BIM 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0∗ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 19.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='9 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='7 MI-FGSM 99.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 TABLE II TRANSFERABILITY OF THE ADVERSARIAL EXAMPLES GENERATED BY SAA-STARGAN AND THE BASELINES AGAINST BLACK-BOX MODELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' WE USE ADVERSARIAL EXAMPLES GENERATED ON ARCFACE MODEL TO ATTACK NINE DIFFERENT TFV MODELS FOR IMPERSONATION & DODGING ATTACKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' VALUES REPRESENT THE ATTACK SUCCESS RATE (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' ∗ INDICATES WHITE-BOX ATTACKS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' THE NUMBERS MARKED IN BOLD REPRESENT THE BEST ATTACK SUCCESS RATE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' (a) Impersonation attack Attack FaceNet ArcFace ResNet-101 CosFace SphereFace MobileFace ShuffleNet V2 IR50-Softmax IR50-CosFace IR50-SphereFace FGSM 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='5∗ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 23.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='2 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='7 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='8 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 Face Mask 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0∗ 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='5 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='7 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 SAA-StarGAN-CS 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='2 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0∗ 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='3 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='9 SAA-StarGAN-PS 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0∗ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='7 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='8 SAA-StarGAN-CS-M 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='2 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0∗ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='9 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='9 SAA-StarGAN-PS-M 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='9 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0∗ 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 9 Original Target FGSM SemanticAdv SAA-StarGAN-CS SAA-StarGAN-PS (+) Eyeglasses (+) Heavy Makeup (+) Heavy Makeup (+) Bangs BIM MI-FGSM PGD Sticker Face Mask (+) Blond Hair Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The adversarial face images generated on FaceNet by different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' SAA-StarGAN can generate high quality adversarial face images as compared with the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We provide an additional experiment described in the Sup- plementary to confirm the effectiveness of the SAA-StarGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 2) Comparison of Attacks based on Score Confidence: We also evaluate the performance of SAA-StarGAN of black-box attacks based on score confidence to empirically demonstrate that the proposed method in the black-box setting can generate adversarial images that mislead different FV models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We have improved the SemanticAdv method (denoted as ImSemanti- cAdv) to add multiple attributes in black-box setting, based on the random attribute selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We compare SAA-StarGAN with ImSemanticAdv that follows our method’s procedure when changing attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' This method is performed by mak- ing a loop to change the random attributes until reaching the adversary randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The main results of black-box attacks based on score confidence for the impersonation and dodging are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We can observe that SAA-StarGAN outperforms ImSemanticAdv across all different models by a large margin for the impersonation and dodging black- box attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Compared with ImSemanticAdv, SAA-StarGAN improves the average attack success rate by 13 % ∼ 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='8 %, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='9 % ∼ 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='7 % under impersonation and dodging attacks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Therefore, predicting the most important attributes of the target model affects the attack’s performance significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Visualization for SAA-StarGAN The goal of adversarial face images is to mislead the FV models Without fooling humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Consequently, we illus- trate samples of adversarial face images generated by SAA- StarGAN-CS and SAA-StarGAN-PS on FaceNet and ArcFace respectively in the white-box setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 4, the label of images, sign (+) indicates the addition of attribute while, sign (-) denotes removing the attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We can see that the proposed method achieves the goal of producing realistic images and the change on the images is slight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 5 presents a set of adversarial face images that are generated on FaceNet in the black-box setting based on the score confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We observe that a few images are not realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' But in general, the proposed method can generate realistic and diverse images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The comparison between SAA- StarGAN and baselines on FaceNet model is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Although two attributes are applied to the proposed SAA- StarGAN-CS and SAA-StarGAN-PS methods, the realistic of the generated adversarial face images are close enough to those produced by SemanticAdv with single attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' On the other hand, the adversarial face images generated by FGSM, BIM, PGD, and MI-FGSM are not clear compared to our SAA- StarGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The face images generated by Sticker and Face Mask attacks are unrealistic and misleading to the human eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We can conclude that the goal has been successfully achieved in the current study, as the presented faces are realistic and can be easily identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Visualizing Attention on Adversarial Face images In this subsection, we apply the Gradient-weighted Class Activation Mapping (Grad-CAM) [50], an attention visual- ization technique, to find the discriminative regions in the image according to the TFV model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We use it to show the attention of the ArcFace and ResNet-101 models for both the original and the generated adversarial face images on the FaceNet model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 7, we can see that our SAA-StarGAN focuses on the trivial regions instead of the prominent regions, as opposed to the SemanticAdv method focuses on the most prominent regions in the face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Thus, SAA-StarGAN could improve the transferability significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We conclude that the prediction step for the most significant attributes plays a key role in improving the attack transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' SAA-StarGAN CS CS-M PS PS-M SemanticAdv ArcFace ResNet-101 Original Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Grad-CAM attention visualization for ResNet-101 and ArcFace models being attacked by adversarial face images are generated on the FaceNet model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' This Figure compares our SAA-StarGAN based on the important facial attributes and SemanticAdv based on indiscriminate attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Discussion on the Similarity to the Original Image It is evident in the literature that most methods succeed in generating adversarial face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Therefore, we aim to measure the quality between the adversarial face and original images through MSE and SSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' MSE measures the absolute errors, So the lowest value is the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' SSIM is used for measuring the similarity between two images by predicting the perceived quality of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' So, the highest value is the 10 best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We compare the adversarial face images crafted by our SAA-StarGAN with the baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We select 2,000 original images and 2,000 adversarial face images for different attack methods for the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' After that, we calculated the MSE and SSIM for each method separately, as shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' SAA-StarGAN-PS and SAA-StarGAN-CS for single attribute show the largest values of SSIM compared to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Besides, they have the lowest values of MSE, which give them preference and make them more suitable for generating adversarial face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The main reason is that our method depends on changing a single most significant attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' This change leads to a slight modification in the adversarial face image, but with a high effect in misleading the different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' On the other hand, SemanticAdv has a high structural similarity value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The Random Selection method randomly applies two attributes, which is lower than the SemanticAdv method in the structural similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In contrast, Sticker and Face Mask methods on FaceNet or ArcFace model show the highest absolute error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' These methods mainly depend on pixel- level and grid-level, which add large perturbations to the face image, covering approximately 20 ∼ 30 % area of the face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We can see that any adversarial face images generated by FGSM, BIM, PGD, and MI-FGSM are expected to be perceptible to human eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Generally, SAA-StarGAN-PS and SAA-StarGAN-CS for a single attribute are most similar to the original images and have the lowest error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Therefore, it is recommended to use our proposed method to craft the adversarial face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' TABLE III MSE AND SSIM TO COMPARE THE ORIGINAL AND ADVERSARIAL FACE IMAGES GENERATED ON FACENET AND ARCFACE MODELS BY OUR SAA-STARGAN AND THE DIFFERENT BASELINES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' BEST VALUES APPEAR IN BOLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Attacks FaceNet ArcFace MSE (↓) SSIM (↑) MSE (↓) SSIM(↑) FGSM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='374 BIM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='585 PGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='700 MI-FGSM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='705 SemanticAdv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='813 Random Selection 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='726 Sticker 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='959 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='008 Face Mask 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='006 SAA-StarGAN-CS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='822 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='821 SAA-StarGAN-PS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='015∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='016∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='846 SAA-StarGAN-CS-M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='729 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='735 SAA-StarGAN-PS-M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='784 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' CONCLUSION In this work, we present a new adversarial attack method called the SAA-StarGAN to generate semantic adversarial face images in both white-box and black-box settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' This method focuses on predicting the important facial attributes for each input image to change a few of the most important attributes to make the model promote the trivial features in the face images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In the white-box setting, SAA-StarGAN first predicts the significant attributes of each input image via TFV and uses the cosine similarity or probability score to determine the most significant attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Then the most important features are changed using a StarGAN model in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In the black-box setting, we make a loop to change the ordered significant attributes until the output is an adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Finally, the generated adversarial face images achieve good impersonation and dodging attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We observe that SAA- StarGAN could generate high-quality and realistic images through the experiments and achieve high attack success rates in the white-box and black-box settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The proposed SAA- StarGAN exhibits significantly higher transferability than the state-of-the-art face attack methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The results demonstrate the significant improvements in attacking various face recog- nition models due to the step of important attributes prediction for each input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' REFERENCES [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Akhtar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Mian, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Kardan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Das, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Vedantam, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Parikh, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Batra, “Grad-CAM: Visual explanations from deep networks via gradient-based localization,” in IEEE International Conference on Com- puter Vision, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 618–626.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Yasmeen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Khedr is currently pursuing the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' degree with Huazhong University of Science and Technology, Wuhan, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' She received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' degrees in information technology from Zagazig University, Egypt, in 2011 and 2017, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' She is an Assistant Lecturer with the De- partment of Information Technology, Zagazig Uni- versity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Her research interests include adversarial learning, artificial intelligence, multimedia security, image processing and computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Yifeng Xiong is currently pursuing the master de- gree with Huazhong University of Science and Tech- nology, Wuhan, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' He received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' degree in computer science from Huazhong University of Science and Technology, Wuhan, China, in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' His research interests include adversarial learning, machine learning, computer vision and natural lan- guage processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Kun He (SM18) is currently a Professor in School of Computer Science and Technology, Huazhong University of Science and Technology, Wuhan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' and a Mary Shepard B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Upson Visiting Pro- fessor for the 2016-2017 Academic year in Engineer- ing, Cornell University NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' She received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' degree in system engineering from Huazhong University of Science & Technology, in 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Her research interests include adversarial learning, ma- chine learning, social network analysis, and combi- natorial optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 12 APPENDIX The supplementary material for Semantic Adversarial At- tacks on Face Recognition through Significant Attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Cosine Similarity (CS) Algorithm 1 Determine the most significant attributes by CS Input: Original image x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' attributes {a},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' generator G of Star- GAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' target face verification model TFV Output: The most significant attributes C 1: for each ai in attributes {a} do 2: change ai using G 3: x∗ ai ← G(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' ai) 4: fx∗ ← TFV (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' fx∗ai ← TFV (x∗ ai) 5: Sai ← CS(fx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' fx∗ai ) 6: end for 7: C ← Sort {a} according to each Sai in the ascending order 8: Output the most significant attributes C = (c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' c3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=', cK) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Probability Score (PS) Algorithm 2 Determine the most significant attributes by PS Input: Original image x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' attributes {a},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' generator G of Star- GAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' attribute prediction model f Output: The most significant attributes {C} 1: Use attribute prediction model to predict attributes 2: for each a′ i in attributes {a} do 3: Pa′ i(x) ← f(x) 4: Change a′ i using G 5: x∗ a′ i ← G(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' a′ i) 6: Pa′ i(x∗ a′ i) ← f( x∗ a′ i) ▷ Compute the probability for each attribute 7: Compute ∆Pa′ i 8: end for 9: C ← Sort {a} using ∆Pa′ i in descending order 10: Output the most significant attributes C = (c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' c3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=', cK) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' SAA-StarGAN in Black-Box Setting To emphasize and clarify the effectiveness of the pro- posed SAA-StarGAN method in improving the transferability against different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We select 1000 adversarial face images crafted on FaceNet from each attack method against a SphereFace model under an impersonation attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' For the dodging attack, we choose 1000 of the generated adversarial face images on FaceNet with the original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We measure the cosine similarity scores before and after the attack to see the improvement percentage for each attack method against a SphereFace model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 8, we observe that most scores between the generated face and target faces of the SAA- StarGAN method fall above Ts @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 % FPR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' demonstrating Algorithm 3 The SAA-StarGAN for Black-Box Attack Input: Original image x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' original attribute cori,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' most signifi- cant attributes {c},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' encoder GE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' decoder GD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' target face verification model TFV ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' target image xtgt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' L2 distance function dist,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' threshold sim th Output: Advesarial example xadv 1: f ∗ 1 ← GE (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' cori) 2: γ ← create array in range [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1] 3: for ci in most significant attributes C do 4: f ∗ ci ← GE (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' ci) 5: for γi in γ do 6: f ∗ γi = γi · f ∗ 1 + (1- γi) · f ∗ ci 7: x∗ γi = GD (f ∗ γi) 8: dγi ← dist (TFV (xtgt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' x∗ γi) 9: score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='insert(dγi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' γi ) 10: end for 11: if impersonation attack then 12: γopt = argmaxγ (score) 13: else if dodging attack then 14: γopt = argminγ (score) 15: end if 16: Substitute γopt in f ∗ 17: x∗ = GD (f ∗) 18: if sim(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' x∗) ≤ th then 19: Return None 20: end if 21: if dist(TFV (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' xtgt) ≤ T) then 22: xadv ← x∗ 23: break 24: else 25: x ← x∗ 26: ci ← add the next attribute from C 27: end if 28: if dist(TFV (x∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' x) ≥ T) then 29: xadv ← x∗ 30: break 31: else 32: x ← x∗ 33: ci ← add the next attribute from C 34: end if 35: end for that the generated face images can be falsely accepted by 53,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='7, and 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 % in an impersonation attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' In contrast to the baseline methods, a few scores fall above Ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' For the dodging attack, the scores fall below Ts @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1 % FPR by 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='2, 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='8, 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4, and 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='7 %, demonstrating that the SphereFace model can falsely reject the image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' We conclude that the SAA-StarGAN significantly outperforms the baseline attacks to improve the transferability of black-box models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='6 0 10 20 30 40 50 Frequency FGSM Adversarial Original T@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='1% FPR (a) Impersonation attack 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='2 0.' metadata={'source': 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+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content='0 0 10 20 30 40 50 Frequency SAA-StarGAN-PS-M (b) Dodging attack Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' Shift in cosine similarity scores for SphereFace before and after adversarial attacks generated by SAA-StarGAN and the baselines on FaceNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} +page_content=' The shift of SAA-StarGAN is much more distinct as compared with the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFLT4oBgHgl3EQfTi_1/content/2301.12046v1.pdf'} diff --git a/Z9FPT4oBgHgl3EQfvDVz/content/tmp_files/2301.13158v1.pdf.txt b/Z9FPT4oBgHgl3EQfvDVz/content/tmp_files/2301.13158v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b934beba3a54b006abdda7063c9d3170f7efd8a5 --- /dev/null +++ b/Z9FPT4oBgHgl3EQfvDVz/content/tmp_files/2301.13158v1.pdf.txt @@ -0,0 +1,894 @@ +arXiv:2301.13158v1 [math.GN] 30 Jan 2023 +SELECTION PRINCIPLES AND PROOFS FROM THE BOOK +BOAZ TSABAN +Abstract. I provide simplified proofs for each of the following fundamental theorems re- +garding selection principles: +(1) The Quasinormal Convergence Theorem, due to the author and Zdomskyy, asserting +that a certain, important property of the space of continuous functions on a space is +actually preserved by Borel images of that space. +(2) The Scheepers Diagram Last Theorem, due to Peng, completing all provable implica- +tions in the diagram. +(3) The Menger Game Theorem, due to Telg´arsky, determining when Bob has a winning +strategy in the game version of Menger’s covering property. +(4) A lower bound on the additivity of Rothberger’s covering property, due to Carlson. +The simplified proofs lead to several new results. +To Adina +1. Introduction +The study of selection principles unifies notions and studies originating from dimension +theory (Menger and Hurewicz), measure theory (Borel), convergence properties (Cs´asz´ar– +Laczkovicz), and function spaces (Gerlits–Nagy and Arhangel’ski˘ı), notions analyzed and +developed in numerous studies of later mathematicians, especially since the 1996 paper of +Just, Miller, Scheepers and Szeptycki [9]. The unified notions include, among others, many +classic types of special sets of real numbers, local properties in function spaces, and more +recent types of convergence properties. +Selective topological covering properties form the kernel of selection principles. +These +covering properties are related via the Scheepers Diagram (Figure 1). This is a diagram +Ufin(O, Γ) +Hurewicz +� Ufin(O, Ω) +� Sfin(O, O) +Menger +Sfin(Γ, Ω) +�♠ +♠ +♠ +♠ +♠ +♠ +♠ +S1(Γ, Γ) +� +�♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +S1(Γ, Ω) +� +�♥ +♥ +♥ +♥ +♥ +♥ +♥ +S1(Γ, O) +�t +t +t +t +t +t +t +t +t +t +t +t +t +t +t +t +t +Sfin(Ω, Ω) +� +S1(Ω, Γ) +Gerlits–Nagy +� +� +S1(Ω, Ω) +� +� +�♥ +♥ +♥ +♥ +♥ +♥ +♥ +S1(O, O) +Rothberger +� +Figure 1. The Scheepers Diagram +2020 Mathematics Subject Classification. Primary: 37F20; Secondary 26A03, 03E75 . +1 + +2 +BOAZ TSABAN +of covering properties and implications among them. The properties in this diagram are +obtained as follows. +For families A and B of sets, let S1(A, B) be the statement: For each sequence of elements +of the family A, we can pick one element from each sequence member, and obtain an element +of the family B. When A = B = O(X), the family of open covers of a topological space +X, we obtain Rothberger’s property (1941), the topological version of Borel’s strong measure +zero. We say that a space X satisfies S1(O, O) if the assertion S1(O(X), O(X)) holds, and +similarly for the other selective properties. +The hypothesis Sfin(A, B) is obtained by replacing one by finitely many in the above +definition. The property Sfin(O, O) is, by an observation of Hurewicz (1925), equivalent to +Menger’s basis property, a dimension-type property. The property Ufin(A, B) is obtained by +further allowing us to take the unions of the selected finite subsets—this matters for some +types of covers. For technical reasons, this property does not consider all covers of type A, +but only those that have no finite subcover. +A cover of a space is an ω-cover if no member of the cover covers the entire space, but +every finite subset of the space is covered by some member of the cover. For a space X, Ω(X) +is the family of open ω-covers of the space. A point-cofinite cover is an infinite cover where +every point of the space belongs to all but finitely many members of the cover. Γ(X) is the +family of open point-cofinite covers of the space X. +Applying the mentioned selection principles to the cover types O, Ω and Γ, we obtain ad- +ditional important properties, such as Hurewicz’s property Ufin(O, Γ) (1925). We also obtain +the Gerlits–Nagy γ-property S1(Ω, Γ) (1982), characterizing the Fr´echet–Urysohn property of +the function space Cp(X) of continuous real-valued functions, with the topology of pointwise +convergence: A topological space is Fr´echet–Urysohn if every point in the closure of a set +is actually a limit of a sequence in the set. This duality between the spaces X and Cp(X) +also translates various tightness and convergence properties of the space Cp(X)—discovered +earlier by Arhangel’ski˘ı, Bukovsk´y, Sakai, and others—to the selective covering properties +Sfin(Ω, Ω), S1(Γ, Γ), and S1(Ω, Ω). In Section 2, we provide a surprisingly simple proof of one +of the most important results of this type. While the result itself does not involve selective +covering properties explicitly, its proof does that extensively. +A topological space is Lindel¨of if every open cover has a countable subcover. For exam- +ple, all sets of real numbers are Lindel¨of. Since all selection principles concern countable +sequences, the theory mainly deals with Lindel¨of spaces. For Lindel¨of spaces, the Scheepers +Diagram is the result of a classification of all properties thus introduced; each property is +equivalent to one in the diagram [9]. It was long open whether any additional implication +could be established among the properties in the diagram. In Section 3 we deal with the +recent, surprising solution of this problem. +Menger’s covering property Sfin(O, O) is the oldest, most general, and most applied prop- +erty in the Scheepers Diagram. Initially, Menger conjectured his property to coincide with +σ-compactness. +While this turned out false [20], the game version of this property does +provide a characterization of σ-compactness. A very transparent proof of this deep result is +presented in Section 4. +In Section 5 we consider a connection to combinatorial set theory. We show that a nontrivial +lower bound on the additivity of Rothberger’s property follows easily from basic knowledge +on selection principles. + +PROOFS FROM THE BOOK +3 +The Book is a popular myth by Paul Erd˝os: A transfinite book containing the most simple +proofs for all theorems. I would like to believe that the proofs presented here are similar to +ones from the Book . . . or from some of its preliminary drafts, at any rate. +2. The Quasinormal Convergence Theorem +By real set we mean a topological space where every open set is a countable union of clopen +sets. Such are, for example, totally disconnected subsets of the real line and, in particular, +subsets of the Cantor space {0, 1}N. In general, every perfectly normal space with any of the +properties considered in this section is a real set. +Let X be a real set. A sequence of real-valued functions f1, f2, . . . on X converges quasi- +normally to a real-valued function f if there are positive real numbers ǫ1, ǫ2, . . . converging +to 0 such that for each point x ∈ X, we have +|fn(x) − f(x)| ≤ ǫn +for all but finitely many n. Quasinormal convergence generalizes uniform convergence. +A real set X is a QN space if every sequence of continuous real-valued functions on X +that converges to 0 pointwise, converges to 0 quasinormally. Equivalently, convergence in the +space Cp(X) is quasinormal. +QN spaces were studied intensively, e.g., by Bukovsk´y, Rec�law, Repick´y, Scheepers, Nowik, +Sakai, and Haleˇs [21, and references therein]. This and other properties of similar type are +preserved by continuous images, and all experience prior to the paper of the author and +Zdomskyy [21] suggested that they are not preserved by Borel images. Thus, the following +theorem [21, Theorem 9] came as a surprise. +The Baire space NN is quasiordered by the relation ≤∗ of eventual dominance: f ≤∗ g if +f(n) ≤ g(n) for all but finitely many n. A subset Y of the Baire space NN is bounded if it is +bounded with respect to evntual dominance. +Theorem 2.1 (Quasinormal Convergence Theorem). The following assertions are equivalent +for real sets X: +(1) The set X is a QN space. +(2) Every Borel image of the set X in the Baire space NN is bounded. +The second property in the theorem is well known and straightforward to apply: The most +natural transformations needed in proofs regarding these notions are always easily seen to +be Borel. Consequently, the theorem had a dramatic impact on the study of QN spaces: +First, many of the previous sophisticated arguments could be replaced by straightforward +ones. Second, many properties that were hitherto considered separately turned out provably +equivalent. Consequently, this theorem settled all problems concerning these properties [21]. +The original proof of the Quasinormal Convergence Theorem is long and involved, and some +of its parts are difficult to follow. A more natural proof was later published by Bukovsk´y and +ˇSupina [5, §4]. Inspired by a paper of Gerlits and Nagy [7], I have discovered the following +surprisingly simple proof. All needed proof ingredients were already available at the time the +Quasinormal Convergence Theorem was established. The following lemma provides the key +to the proof. +For a space X, let PF(X) be the collection of countably infinite point-finite families of +open sets in X. +Lemma 2.2. Let X be a topological space. The following assertions are equivalent: + +4 +BOAZ TSABAN +(1) Every Borel image of the space X in the Baire space NN is bounded. +(2) The space X satisfies S1(PF, PF). +Proof. Let F(X) (respectively, B(X)) be the family of countable closed (respectively, Borel) +covers of the set X, and FΓ(X) (respectively, BΓ(X)) be the family of infinite closed (re- +spectively, Borel) point-cofinite covers of the set X. The properties (1), Ufin(B, BΓ), and +S1(BΓ, BΓ) are equivalent [17, Theorem 1]. For a family U of open sets, we have U ∈ PF(X) +if and only if +{ Uc : U ∈ U } ∈ FΓ(X). +It follows that S1(PF, PF) = S1(FΓ, FΓ). +(1) ⇒ (2): Clearly, S1(BΓ, BΓ) implies S1(FΓ, FΓ). +(2) ⇒ (1): A theorem of Bukovsk´y–Rec�law–Repick´y [4, Corollary 5.3] asserts that +Ufin(F, FΓ) = Ufin(B, BΓ). +The usual argument [16, Proposition 11] shows that S1(FΓ, FΓ) implies Ufin(F, FΓ): If { Cn : +n ∈ N} ∈ F(X) and there is no finite subcover, then { �n +k=1 Ck : n ∈ N} ∈ FΓ(X). +□ +A topological space Y has Arhangel’ski˘ı’s property α1 if for every sequence s1, s2, . . . of +sequences converging to the same point, there is a sequence s such that the sets im(sn)\im(s) +are finite for all natural numbers n. This property is defined by properties of sets (images of +sequences) rather than sequences. Fix a bijection ϕ: N × N → N. For sequences s1, s2, . . . , +with sn = (s(n,1), s(n,2), . . . ) for each n, define +∞ +� +n=1 +sn := (sϕ(1), sϕ(2), . . . ). +Since convergence of a sequence does not depend on the order of its elements, it does not +matter, for our purposes, which bijection ϕ is used. A sequence �∞ +n=1 sn converges to a point +p if and only if each sequence sn converges to p, and for each neighborhood U of p, we have +im(sn) ⊆ U for all but finitely many n. +Lemma 2.3. Let Y be an α1 space. For every sequence s1, s2, . . . of sequences in the space +Y converging to the same point p, there are tails tn of sn, for n ∈ N, such that the sequence +�∞ +n=1 tn converges to p. +Proof. There is a sequence s such that the sets im(sn)\im(s) are finite for all natural numbers +n. By moving to a subsequence, we may assume that im(s) ⊆ �∞ +n=1 im(sn). Suppose that +s = (a1, a2, . . . ). For each natural number n, since the sequence sn coverges to the point p, +every element other than p may appear in the sequence sn only finitely often. Thus, there is +a tail tn of the sequence sn such that +im(tn) ⊆ { ak : k ≥ n } ∪ {p}. +Let U be a neighborhood of p. There is a natural number N such that +im(tn) ⊆ { ak : k ≥ n } ∪ {p} ⊆ U +for all natural numbers n ≥ N. Thus, the direct sum t := �∞ +n=1 tn converges to the point +p. +□ +Sakai [13, Theorem 3.7] and Bukovsk´y–Haleˇs [3, Theorem 11] proved that a real set X is a +QN space if, and only if, the space Cp(X) is an α1 space. Thus, the Quasinormal Convergence +Theorem can be stated, and proved, as follows. + +PROOFS FROM THE BOOK +5 +Theorem 2.4. The following assertions are equivalent for real sets X: +(1) The space Cp(X) is an α1 space. +(2) Every Borel image of the set X in the Baire space NN is bounded. +Proof. (2) ⇒ (1): This is the straightforward implication. For completeness, we reproduce +its proof [21, Theorem 9]. +Let s1, s2, . . . be sequences in the space Cp(X) that converge to a function f ∈ Cp(X). +For each natural number n, suppose that +sn = (f n +1 , f n +2 , f n +3 , . . . ). +Define a Borel function Ψ: X → NN by +Ψ(x)(n) := min +� +k : (∀m ≥ k) |f n +m(x) − f(x)| ≤ 1 +n +� +. +Let g ∈ NN be a ≤∗-bound for the image Ψ[X]. Then the sequence +∞ +� +n=1 +(f n +g(n), f n +g(n)+1, f n +g(n)+2, . . . ) +converges to the function f. +(1) ⇒ (2): This is the main implication. By Lemma 2.2, it suffices to prove that the set +X satisfies S1(PF, PF). Let U1, U2, . . . ∈ PF(X). By thinning out the point-finite covers, we +may assume that they are pairwise disjoint [16, Lemma 4]. For each set U ∈ �∞ +n=1 Un, let CU +be a countable family of clopen sets with � CU = U. For each natural number n, let +Vn := +� +U∈Un +CU. +Every set C ∈ Vn is contained in at most finitely many sets U ∈ Un. Thus, the family Vn is +infinite and point-finite. Let sn be a bijective enumeration of the family +{ χV : V ∈ Vn }. +The sequence sn is in Cp(X), and it converges to the constant function 0. +As the space Cp(X) is α1, there are for each n a tail tn of the sequence sn such that the +sequence s := �∞ +n=1 tn converges to 0. For each natural number n, pick a set Un ∈ Un with +CUn ⊆ im(tn). Then the family { Un : n ∈ N} is infinite and point-finite. +□ +For a set X ⊆ {0, 1}N, Gerlits and Nagy (and, independently, Nyikoˇs) define a space T(X) +as follows. Let {0, 1}∗ denote the set of finite sequences of elements of the set {0, 1}. Let +X ⊆ {0, 1}N. For each point x ∈ X, let Ax ⊆ {0, 1}∗ be the set of initial segments of the +point x. Let X ∪ {0, 1}∗ be the topological space where the points of the set {0, 1}∗ are +isolated, and for each point x ∈ X, a neighborhood base of x is given by the sets {x} ∪ B, +where B is a cofinite subset of the set Ax. Let T(X) be the one-point compactification of +this space, and ∞ be the compactifying point. +Gerlits and Nagy prove that if a set X ⊆ {0, 1}N is a Siepi´nski set, then the space T(X) +is α1, and that if the space T(X) is α1, then the set X is a σ-set [7, Theorem 4] . The +following theorem unifies these results and improves upon them. Indeed, every Borel image +of a Siepi´nski set in the Baire space is bounded, and every set with bounded Borel images in +the Baire space is a σ-set [17, and references therein]. +Theorem 2.5. Let X ⊆ {0, 1}N. The following assertions are equivalent: + +6 +BOAZ TSABAN +(1) The space T(X) is an α1 space. +(2) Every Borel image of the set X in the Baire space NN is bounded. +Proof. For a finite sequence s ∈ {0, 1}∗, let [s] be the basic clopen subset of the Cantor +space {0, 1}N consisting of all functions extending s. Every open set in the space {0, 1}N is a +disjoint union of basic clopen sets. A sequence a1, a2, . . . in the set {0, 1}∗ converges to ∞ in +the space T(X) if, and only if, the set { [an] : n ∈ N} is point-finite in the space X [7]. The +argument in the proof of Theorem 2.4 applies. +□ +3. The Scheepers Diagram Last Theorem +The implications in the Scheepers Diagram 1 were all rather straightforward to establish, +and almost all other potential implications were ruled out by counterexamples [9]. Only +two problems remained open: Does Ufin(O, Ω) imply Sfin(Γ, Ω)? And if not, does Ufin(O, Γ) +imply Sfin(Γ, Ω)? [9, Problems 1 and 2]. For nearly three decades it was expected that the +remaining two potential implications were refutable. +Only when Peng came up with an +entirely new method for refuting implications among selective covering properties [12], were +these problems resolved. But not in the expected way: Having proved that Ufin(O, Ω) does +not imply Sfin(Γ, Ω), Peng tried to refute the last remaining potential implication. And he +failed. His close examination of the failure suggested a path for proving the last potential +implication [12, Theorem 23]. Peng’s results establish the final form of the Scheepers Diagram +(Figure 2). +Ufin(O, Γ) +Hurewicz +� Sfin(Γ, Ω) +� Ufin(O, Ω) +� Sfin(O, O) +Menger +S1(Γ, Γ) +� +�♥ +♥ +♥ +♥ +♥ +♥ +♥ +S1(Γ, Ω) +� +�♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +S1(Γ, O) +�♠ +♠ +♠ +♠ +♠ +♠ +♠ +Sfin(Ω, Ω) +� +S1(Ω, Γ) +Gerlits–Nagy +� +� +S1(Ω, Ω) +� +� +�♥ +♥ +♥ +♥ +♥ +♥ +♥ +S1(O, O) +Rothberger +� +Figure 2. The Final Scheepers Diagram +Peng’s proof of the last implication is somewhat involved. The proof given below identifies +the heart of Peng’s argument, and replaces the other parts with simple, quotable observations +about selective covering properties. +Let k be a natural number. A cover of a space is a k-cover if no member of the cover +covers the entire space, but every k-element subset of the space is covered by some member +of the cover. Thus, a cover is an ω-cover if and only if it is a k-cover for all natural numbers +k. For a space X and a natural number k, let Ok(X) be the family of open k-covers of the +space. +Lemma 3.1. Let Π be a selection principle, and A a type of open covers. The following +assertions are equivalent: +(1) Π(A, Ω). +(2) For each natural number k we have Π(A, Ok). + +PROOFS FROM THE BOOK +7 +Proof. (1) ⇒ (2): Obvious. +(2) ⇒ (1): Let U1, U2, . . . be a sequence in A. Split the sequence into infinitely many +disjoint sequences. +For each natural number k, apply Π(A, Ok) to the kth sequence, to +obtain a k-cover Vk. Then �∞ +k=1 Vk is an ω-cover, in accordance to the required property +Π(A, Ω). +□ +Theorem 3.2 (Peng [12, Theorem 23]). The Hurewicz property Ufin(O, Γ) implies Sfin(Γ, Ω). +Proof. Let X be a Huewicz space. By Lemma 3.1, it suffices to prove that Sfin(Γ, Ok) holds +for all natural numbers k. Fix a natural number k. +Let U1, U2, . . . be a sequence in Γ(X). By moving to countably infinite subcovers, we may +enumerate +Un = { Un +m : m ∈ N } +for each n. For each n and m, we may replace the set Un +m with the smaller set +U1 +m ∩ U2 +m ∩ · · · ∩ Un +m, +so that we may assume that +U1 +m ⊇ U2 +m ⊇ U3 +m ⊇ · · · +for all natural numbers m. The refined covers Un remain in Γ(X). +Let g0(m) := m for all m. We define, by induction, increasing functions g1, . . . , gk ∈ NN. +Let l < k and assume that the function gl is defined. For natural numbers n, m and i, let +V l,n +i +:= +gl(i) +� +m=i +Un +m; +W l,n +m +:= +� +{ V l,n +i +: n ≤ i, gl(i) ≤ m }. +For each l and n, the sets W l,n +m +are increasing with m, and cover the space X. +By the +Hurewicz property, there is an increasing function gl+1 ∈ NN such that +{ W l,n +gl+1(n) : n ∈ N} ∈ Γ(X). +This completes the iductive construction. +We will show that +{ Un +m : n ∈ N, m ≤ gk(n) } ∈ Ok(X). +Let x1, . . . , xk ∈ X. Since { W l,n +gl+1(n) : n ∈ N} ∈ Γ(X) for all l = 0, . . . , k − 1, there is a +natural number N with +x1, . . . , xk ∈ W l,n +gl+1(n) +for all l = 1, . . . , k and all n ≥ N. Fix a number n0 ≥ N. +Since x1 ∈ W k−1,n0 +gk(n0) , there is n1 with n0 ≤ n1, gk−1(n1) ≤ gk(n0) and +x1 ∈ V k−1,n0 +n1 += +gk−1(n1) +� +m=n1 +Un0 +m . +Since x2 ∈ W k−2,n1 +gk−1(n1), there is n2 with n1 ≤ n2, gk−2(n2) ≤ gk−1(n1) and +x2 ∈ V k−2,n1 +n2 += +gk−2(n2) +� +m=n2 +Un1 +m ⊆ +gk−2(n2) +� +m=n2 +Un0 +m . + +8 +BOAZ TSABAN +Since x3 ∈ W k−3,n2 +gk−2(n2), there is n3 with n2 ≤ n3, gk−3(n3) ≤ gk−2(n2) and +x3 ∈ V k−3,n2 +n3 += +gk−3(n3) +� +m=n3 +Un2 +m ⊆ +gk−3(n3) +� +m=n3 +Un0 +m . +... +Since xk ∈ W 0,nk−1 +g1(nk−1), there is nk with nk−1 ≤ nk = g0(nk) ≤ g1(nk−1) and +xk ∈ V 0,nk−1 +nk += Unk−1 +nk +⊆ Un0 +nk . +It follows that x1, . . . , xk ∈ Un0 +nk , and nk ≤ gk(n0). +□ +The proof of Theorem 3.2 establishes a stronger result. +To this end, we need the fol- +lowing definitions and lemma. An infinite cover of a space X is ω-groupable (respectively, +k-groupable, for a natural number k) if there is a partition of the cover into finite parts such +that for each finite (respectively, k-element) set F ⊆ X and all but finitely many parts P of +the partition, there is a set U ∈ P with F ⊆ U [10]. Let Ωgp(X) (respectively, Ogp +k (X)) be +the family of open ω-groupable (respectively, k-groupable) covers of the space X. +Lemma 3.3. Let Π be a selection principle, and A a type of open covers. The following +assertions are equivalent: +(1) Π(A, Ωgp); +(2) For each natural number k we have Π(A, Ogp +k ). +Proof. The proof is similar to that of Lemma 3.1, once we observe that if { Un : n ∈ N} is a +k-grouable cover for all k, then it is ω-groupable. This follows easily from the fact that for +each countable family { Pk : k ∈ N } of partitions of N into finite sets, there is a partition P +of N into finite sets that is eventually coarser than all of the given partitions, that is, such +that for each k, all but finitely many members of the partition P contain a member of the +partition Pk. +□ +Koˇcinac and Scheepers proved that if all finite powers of a space X are Hurewicz, then +every open ω-cover of the space is ω-groupable. Together with Peng’s Theorem 3.2, we have +that if all finite powers of a space X are Hurewicz, then the space satisfies Sfin(Γ, Ωgp). The +following theorem shows that the assumption on the finite powers is not needed. +In the theorem, we also mention Sfin(Γ, Λgp). An open cover is large if each point is in +infinitely many members of the cover. Let Λ(X) be the family of large open covers of the +space X. An open cover U is in Λgp(X) [10] (also denotedג(Γ), depending on the context [14]) +if there is a partition of the cover into finite parts such that for each point x ∈ X and all but +finitely many parts P of the partition, we have x ∈ � P. +Theorem 3.4. The following assertions are equivalent: +(1) Ufin(O, Γ), +(2) Sfin(Γ, Ωgp); and +(3) Sfin(Γ, Λgp). +Proof. (1) ⇒ (2): The proof of Peng’s Theorem 3.2, as written above, shows, for a prescribed +number k, that for each k-element set F there is a natural number N such that for each +n ≥ N there is a member of the finite set Fn = { Un +m : m ≤ gk(n) } that contains the set F. +By thinning out the point-cofinite covers, we may assume that they are pairwise disjoint [16, + +PROOFS FROM THE BOOK +9 +Lemma 4], and consequently so are the finite sets Fn. Thus, �∞ +n=1 Fn ∈ Ogp +k . This proves +Sfin(Γ, Ogp +k ) for all k. Apply Lemma 3.3. +(2) ⇒ (3): Ωgp ⊆ Λgp. +(3) ⇒ (1): This implication is standard and should be known. +For completeness, we +provide a proof. Assume that the space X satisfies Sfin(Γ, Λgp). It suffices to prove that +it satisfies Ufin(Γ, Γ). +Given a sequence U1, U2, . . . in Γ(X), we may (as in the proof of +Theorem 3.2) assume that the covers get finer with n. Apply Sfin(Γ, Λgp) to obtain a cover +U ∈ Λgp, with parts Pn (for n ∈ N) witnessing that. +Let F1 ⊆ U1 be a finite set refined by P1. Let n2 be minimal with Pn2 ⊆ �∞ +n=2 Un, and +F2 ⊆ U2 be a finite set refined by Pn2. Let n3 be minimal with Pn3 ⊆ �∞ +n=3 Un, and F3 ⊆ U3 +be a finite set refined by Pn3. Continuing in this manner, we obtain finite sets Fn ⊆ Un for +n ∈ N, with { � Fn : n ∈ N} ∈ Γ(X). +□ +Koˇcinac and Scheepers proved that Ufin(O, Γ) = Sfin(Ω, Λgp) = Sfin(Λ, Λgp) [10, Theo- +rem 14]. However, Ufin(O, Γ) ̸= Sfin(Ω, Ωgp): The latter property is equivalent to satisfying +Ufin(O, Γ) in all finite powers [10, Theorem 16], a property strictly stronger than Ufin(O, Γ) [9, +Theorem 2.12]. +4. When Bob has a winning strategy in the Menger game +Menger [11] conjectured that his property Sfin(O, O) implies σ-compactness. While his +conjecture turned out false [20, and references therein], a closely related assertion is true. +The Menger game [8], Gfin(O, O), is the game associated to Menger’s property Sfin(O, O). It +is played on a topological space X, and has an inning per each natural number n. In each +inning, Alice picks an open cover Un of the space, and Bob chooses a finite set Fn ⊆ Un. Bob +wins if �∞ +n=1 Fn is a cover of the space, and otherwise Alice wins. Telg´arsky [18] proved that +if Bob has a winning strategy in the Menger game played on a metric space, then the space +is σ-compact. +Scheepers [15, Theorem 1] provided a direct proof of Telg´arsky’s Theorem, using the notion +of H-closed sets. We will eliminate the notion of H-closed sets and the closure operations +from Scheepers’s proof, and obtain a more transparent proof. As a bonus, the separation +hypotheses on the space are eliminated. +A subset K of a topological space X is relatively compact if every open cover U of the +entire space X has a finite subcover of the set K. A set K is relatively compact if and only +if its closure is compact. +Lemma 4.1. Let κ be a cardinal number. If a space X is a union of at most κ relatively +compact sets, then it is the union of at most κ compact sets. +Proof. If X = � +α<κ Kα, then X = � +α<κ Kα. +□ +For a basis B for the topology of a space X, let OB(X) be the family of subsets of B that +cover the space X. +Lemma 4.2. Let X be a topological space with a basis B, and σ be a function on the family +OB(X) such that for each cover U ∈ OB(X), σ(U) is a finite subset of U. Then the set +K := +� +U∈OB(X) +� +σ(U) +is relatively compact. + +10 +BOAZ TSABAN +Proof. Let U be an open cover of X. Let V ∈ OB(X) be a cover that refines the cover U. +Then K ⊆ � σ(V), and there is a finite set F ⊆ U with � σ(V) ⊆ � F. +□ +Scheepers [15, Theorem 1] proves the following theorem for metric spaces. If Bob has a +winning strategy in the Menger game played on X, then the space X is Menger and, in +particular, Lindel¨of. If X is, in addition, metric, then the space is second countable. +Theorem 4.3. Let X be a second countable topological space. If Bob has a winning strategy +in the Menger game Gfin(O, O) played on X, then the space X is σ-compact. +Proof. We follow steps of Scheepers’s proof, removing what is not necessary. Let σ be a +winning stratery for Bob. Fix a countable base B for the topology of the space X. Let N∗ be +the set of finite sequences of natural numbers. We consider all possible games where Alice +chooses her covers from the family OB(X). +Since the base B is countable, the family { σ(U) : U ∈ OB } (the possible first responds of +Bob) is countable, too. Choose elements U1, U2, . . . ∈ OB with +{ σ(Un) : n ∈ N} = { σ(U) : U ∈ OB }. +By induction, for a given natural number n and each sequence s ∈ Nn, the family +{ σ(Us1, Us1,s2, . . . , Us, U) : U ∈ OB } +is countable. Choose elements Us,1, Us,2, . . . ∈ OB with +{ σ(Us1, . . . , Us, Us,n) : n ∈ N} = { σ(Us1, . . . , Us, U) : U ∈ OB }. +This completes our inductive construction. +By Lemma 4.2, for each sequence s ∈ N∗, the set +Ks := +∞ +� +n=1 +� +σ(Us1, . . . , Us, Us,n) +is relatively compact. By Lemma 4.1, it remains to see that X = � +s∈N∗ Ks. +Assume that some element x ∈ X is not in � +s∈N∗ Ks. +(1) Since x /∈ K(), there is m1 with x /∈ � σ(Um1). +(2) Since x /∈ Km1, there is m2 with x /∈ � σ(Um1,m2). +(3) Since x /∈ Km1,m2, there is m3 with x /∈ � σ(Um1,m2,m3). +(4) Etc. +Then the play +Um1, σ(Um1), Um1,m2, σ(Um1,m2), Um1,m2,m3, σ(Um1,m2,m3), . . . +is lost by Bob; a contradiction. +□ +Let α be an ordinal number. The transfinite Menger game Gα +fin(O, O) is defined as the +ordinary Menger game, with the only difference that now there is an inning per each ordinal +number β < α. Clearly, if α1 < α2 and Bob has a winning strategy in the α1-Menger game, +then Bob has a winning strategy in the α2-Menger game: He can use a winning startegy in +the first α1 innings, and then play arbitrarily. Thus, the following theorem is stronger than +Theorem 4.3, and has no assumption on the topological space X. +The weight of a topological space is the minimal cardinality of a base for its topology. +Theorem 4.4. Let X be a topological space of weight κ. Bob has a winning strategy in the +game Gκ +fin(O, O) if and only if the space X is a union of at most κ compact sets. + +PROOFS FROM THE BOOK +11 +Proof. (⇒) The proof is identical to that of Theorem 4.3, only that here we begin with a +base of cardinality κ. +(⇐) In the α-th inning, Bob covers the α-th compact set. +□ +Corollary 4.5. Let X be a topological space of weight κ. If Bob has a winning strategy in +the Menger game, then the space X is a union of at most κ compact sets. +□ +The converse of Corollary 4.5 is false: The discrete space of cardinality κ has weight κ, +and it is a union of κ compact sets (singletons). This space is not Lindel¨of, and thus not +Menger, so Bob has no winning strategy in the Menger game played on this space. +For a space X, let O(X) be the families U of open sets with � +U∈U U = X. A space X is +almost Menger if it satisfies Sfin(O, O). For regular spaces, almost Menger is equivalent to +Menger [1]. And similarly for the other notions considered below. Thus, the remainder of +this section is mainly relevant for nonregular spaces. The almost Menger game is the game +associated to the property Sfin(O, O). The corresponding notion of almost Lindel¨of is classic, +and so is the notion of almost compact space: A space K is almost compact if every open +cover of K has a finite subset F with dense union. This notion appears in the literature +under various names. For Hausdorff spaces, it is known to be equivalent to Hausdorff closed, +that is, being closed in all Hausdorff superspaces. +A set K in a space X is relatively almost compact if every open cover of the space X has +a finite subset F with K ⊆ � +U∈F U. The standard proof that every almost compact space +is H-closed shows that every relatively almost compact set in a Hausdorff space is closed in +that space. However, no separation hypothesis is needed in the following theorem. Since +the proof of the following theorem is provided by the first part of Scheepers’s argument, we +attribute the theorem to Scheepers. +Theorem 4.6 (Scheepers). Let X be a second countable topological space. The following +assertions are equivalent: +(1) Bob has a winning strategy in the game Gfin(O, O) played on X. +(2) The space X is a countable union of relatively almost compact sets. +Proof. (2) ⇒ (1): This is easy. +(1) ⇒ (2): This is a part of the argument of Scheepers [15, Theorem 1]. We provide it, for +completion and verification. +Repeat the inductive construction of Theorem 4.3 verbatim. Having completed it, define +for each sequence s ∈ N∗: +Ks := +∞ +� +n=1 +� +U∈σ(Us1,...,Us,Us,n) +U. +The set Ks is relatively almost compact. It remains to see that X = � +s∈N∗ Ks. +Assume that some element x ∈ X is not in � +s∈N∗ Ks. +(1) Since x /∈ K(), there is m1 with x /∈ � +U∈σ(Um1) U. +(2) Since x /∈ Km1, there is m2 with x /∈ � +U∈σ(Um1,m2) U. +(3) Since x /∈ Km1,m2, there is m3 with x /∈ � +U∈σ(Um1,m2,m3) U. +(4) Etc. +Then the play +Um1, σ(Um1), Um1,m2, σ(Um1,m2), Um1,m2,m3, σ(Um1,m2,m3), . . . + +12 +BOAZ TSABAN +is lost by Bob; a contradiction. +□ +The assertions analogous to the more general Theorem 4.4 and Corollary 4.5 also hold. +Theorem 4.6 answers a question of Babinkostova, Pansera and Scheepers [1, Question 26(2)] +in the case of second countable spaces. +5. The additivity of Rothberger’s property +Let add(N ) be the minimal cardinality of a family F ⊆ NN such that there is no function +S : N → [N]<∞ with |S(n)| ≤ n for all n, such that for each function f ∈ F we have +f(n) ∈ S(n) for all but finitely many n. +The notation add(N ) is explained by a result of Bartoszy´nski and Judah [2, Theorem 2.11]: +The cardinal number add(N ) is the minimal cardinality of a family of Lebesgue null sets of +real numbers whose union is not Lebesgue null. In general, the additivity of a property is the +minimum cardinality of a family of sets satisfying the property, whose union does not. The +following theorem is attributed to Carlson by Bartoszy´nski and Judah [2, Theorem 2.9]. +Theorem 5.1 (Carlson). Let κ < add(N ). If a Lindel¨of space is a union of at most κ +spaces satisfying S1(O, O), then the space X satisfies S1(O, O). That is, for Lindel¨of spaces, +add(N ) ≤ add(S1(O, O)). +This theorem is an easy consequence of a simple, basic fact concerning selection principles. +We need the following lemmata. +Let A and B be types of open covers. A topological space X satisfies Sn(A, B) if for all +U1, U2, · · · ∈ A(X), there are finite sets F1 ⊆ U1, F2 ⊆ U2, . . . such that |Fn| ≤ n for all n, +and �∞ +n=1 Fn ∈ B(X). +Garcia-Ferreira and Tamariz-Mascarua [6, Lemma 3.12] established the following observa- +tion in the case A = O. +Lemma 5.2 ([20, Theorem A.1]). Let A be a type of countable covers such that every pair +of covers of type A has a joint refinement of type A. Then Sn(A, O) = S1(A, O). +Lemma 5.3 (Folklore). If a space X satisfies S1(A, O), then for each sequence U1, U2, . . . in +O(X) there are elements Um1 ∈ Um1, Um2 ∈ Um2, . . . such that for each point x ∈ X, we have +x ∈ Umn for infinitely many n. +Proof. As usual, we split the sequence of open covers to infinitely many disjoint sequences, +and apply the property S1(A, O) to each subsequence separately. +□ +Theorem 5.4. Let A be a type of countable covers such that every pair of covers of type A +has a joint refinement of type A. Then add(N ) ≤ add(S1(A, O)). +Proof. Let κ < add(N ) and X = � +α<κ Xα, where each space Xα satisfies S1(A, O). By +Lemma 5.2, it suffices to show that the space X satisfies Sn(A, O). +Let Un = {Un +m : m ∈ N} ∈ A(X), for n ∈ N. For each ordinal number α < κ, as the space +Xα satisfies S1(A, O), there is a function fα ∈ NN such that for each point x ∈ Xα we have +x ∈ Un +fα(n) +for infinitely many n. +There is a function S : N → [N]<∞ with |S(n)| ≤ n for all n, such that for each α < κ we +have +fα(n) ∈ S(n) +for all but finitely many n. Then �∞ +n=1{ Un +m : m ∈ S(n) } ∈ O(X). +□ + +PROOFS FROM THE BOOK +13 +Judging by an extensive survey on the topic [19], the result in the second item below seems +to be new. +Corollary 5.5. +(1) For Lindel¨of spaces, add(N ) ≤ add(S1(O, O)). +(2) add(N ) ≤ add(S1(Γ, O)). +Proof. (1) Since the spaces are Lindel¨of, we may restrict attention to countable covers, and +the assumptions of Theorem 5.4 hold. +(2) A countably infinite subset of a point-cofinite cover is also a point-cofinite cover. Thus, +we may restrict attention to countable point-cofinite covers. +It is well-known that every +pair of point-cofinite covers has a joint refinement that is a point-cofinite cover. Indeed, +let U and V be countable point-cofinite covers. Enumerate them U = { Un : n ∈ N} and +V = { Vn : n ∈ N}. Then { Un ∩ Vn : n ∈ N} ∈ Γ(X). Theorem 5.4 applies. +□ +We can extract additional information from this proof method. A topological space X +satisfies Un(Γ, Γ) [20] if for all U1, U2, · · · ∈ Γ(X), there are finite sets F1 ⊆ U1, F2 ⊆ U2, . . . +such that |Fn| ≤ n for all n, and { � Fn : n ∈ N} ∈ Γ(X). This property is strictly inbetween +S1(Γ, Γ) and Ufin(O, Γ) [20, Theorems 3.3 and 3.8]. +Theorem 5.6. add(N ) ≤ add(Un(Γ, Γ)). +Proof. Let κ < add(N ) and X = � +α<κ Xα, where each space Xα satisfies Un(Γ, Γ). It suffices +to show that the space X satisfies Un2(Γ, Γ), where the cardinality of the n-th selected finite +set is at most n2 [20, Lemma 3.2]. +Let Un = {Un +m : m ∈ N} ∈ Γ(X), for n ∈ N. For each α < κ, as the space Xα satisfies +Un(Γ, Γ), there is a function Sα: N → � +n[N]≤n such that for each point x ∈ Xα we have +x ∈ +� +m∈Sα(n) +Un +m +for infinitely many n. +There is a function S : N → � +n[N]≤n with |S(n)| ≤ n for all n, such that for each α < κ +we have +Sα(n) ∈ S(n) +for all but finitely many n. For each natural number n, let Fn := � S(n). Then |Fn| ≤ n2 +for all n, and { � +m∈Fn Un +m : n ∈ N} ∈ Γ(X). +□ +Acknowledgments. This is the first paper that I write after a long, callenging period. I +thank all those who helped and encouraged me throughout, and supported my return to +normal track afterwards. Above all, I thank my wife, Adina, for her faith, support, and +patience. +References +[1] L. Babinkostova, B. Pansera, M. Scheepers, Weak covering properties and infinite games, Topology and +its Applications 159 (2012), 3644–3657. +[2] T. Bartoszy´nski, H. Judah, On cofinality of the smallest covering of the real line by meager sets II, +Proceedings of the American Mathematical Society 123 (1995), 1879–1885. +[3] L. Bukovsk´y, J. Haleˇs, QN-space, wQN-space and covering properties, Topology and its Applications +154 (2007), 848–858. + +14 +BOAZ TSABAN +[4] L. Bukovsk´y, I. Rec�law, M. Repick´y, Spaces not distinguishing pointwise and quasinormal convergence +of real functions, Topology and its Applications 41 (1991), 25–41. +[5] L. Bukovsk´y, J. ˇSupina, Sequence selection principles for quasi-normal convergence, Topology and its +Applications 159 (2012), 283–289. +[6] S. Garcia-Ferreira, A. Tamariz-Mascarua, Some generalizations of rapid ultrafilters and Id-fan tightness, +Tsukuba Journal of Mathematics 19 (1995), 173–185. +[7] J. Gerlits, Zs. Nagy, On Frechet spaces, Rendiconti del Circolo Matematico di Palermo, serie ii, supple- +mento 18 (1988), 51–71. +[8] W. Hurewicz, ¨Uber eine Verallgemeinerung des Borelschen Theorems, Mathematische Zeitschrift 24 +(1925), 401–421. +[9] W. Just, A. Miller, M. Scheepers, and P. Szeptycki, The combinatorics of open covers II, Topology and +its Applications 73 (1996), 241–266. +[10] L. Koˇcinac, M. Scheepers, Combinatorics of open covers (VII): Groupability, Fundamenta Mathematicae +179 (2003), 131–155. +[11] K. Menger, Einige ¨Uberdeckungss¨atze der Punktmengenlehre, Sitzungsberichte der Wiener Akademie +133 (1924), 421–444. +[12] Y. Peng, Scheepers’ conjecture and the Scheepers Diagram, Transactions of the American Mathematical +Society, to apper. +[13] M. Sakai, The sequence selection properties of Cp(X), Topology and its Applications 154 (2007), 552– +560. +[14] N. Samet, M. Scheepers, B. Tsaban, Partition relations for Hurewicz-type selection hypotheses, Topology +and its Applications 1562009, 616–623. +[15] M. Scheepers, A direct proof of a theorem of Telg´arsky, Proceedings of the American Mathematical +Society 123 (1995), 3483–3485. +[16] M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology and its Applications 69 (1996), +31–62. +[17] M. Scheepers, B. Tsaban, The combinatorics of Borel covers, Topology and its Applications 121 (2002), +357–382. +[18] R. Telg´arsky, On games of Topsoe, Mathematica Scandinavica 54 (1984), 170–176. +[19] B. Tsaban, Additivity numbers of covering properties, in: Selection Principles and Covering Prop- +erties in Topology (L. Koˇcinac, ed.), Quaderni di Matematica 18, Seconda Universita di Napoli, +Caserta 2006, 245–282. +[20] B. Tsaban, Menger’s and Hurewicz’s Problems: Solutions from “The Book” and refinements, Contem- +porary Mathematics 533 (2011), 211–226. +[21] B. Tsaban, L. Zdomskyy, Hereditarily Hurewicz spaces and Arhangel’ski˘ı sheaf amalgamations, Journal +of the European Mathematical Society 12 (2012), 353–372. +Boaz Tsaban, Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel +Email address: tsaban@math.biu.ac.il + diff --git a/Z9FPT4oBgHgl3EQfvDVz/content/tmp_files/load_file.txt b/Z9FPT4oBgHgl3EQfvDVz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..04626dce32fdaf81f53ab5facaa619652c250fd1 --- /dev/null +++ b/Z9FPT4oBgHgl3EQfvDVz/content/tmp_files/load_file.txt @@ -0,0 +1,638 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf,len=637 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='13158v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='GN] 30 Jan 2023 SELECTION PRINCIPLES AND PROOFS FROM THE BOOK BOAZ TSABAN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' I provide simplified proofs for each of the following fundamental theorems re- garding selection principles: (1) The Quasinormal Convergence Theorem, due to the author and Zdomskyy, asserting that a certain, important property of the space of continuous functions on a space is actually preserved by Borel images of that space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) The Scheepers Diagram Last Theorem, due to Peng, completing all provable implica- tions in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (3) The Menger Game Theorem, due to Telg´arsky, determining when Bob has a winning strategy in the game version of Menger’s covering property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (4) A lower bound on the additivity of Rothberger’s covering property, due to Carlson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The simplified proofs lead to several new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' To Adina 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Introduction The study of selection principles unifies notions and studies originating from dimension theory (Menger and Hurewicz), measure theory (Borel), convergence properties (Cs´asz´ar– Laczkovicz), and function spaces (Gerlits–Nagy and Arhangel’ski˘ı), notions analyzed and developed in numerous studies of later mathematicians, especially since the 1996 paper of Just, Miller, Scheepers and Szeptycki [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The unified notions include, among others, many classic types of special sets of real numbers, local properties in function spaces, and more recent types of convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Selective topological covering properties form the kernel of selection principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' These covering properties are related via the Scheepers Diagram (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' This is a diagram Ufin(O, Γ) Hurewicz � Ufin(O, Ω) � Sfin(O, O) Menger Sfin(Γ, Ω) �♠ ♠ ♠ ♠ ♠ ♠ ♠ S1(Γ, Γ) � �♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ S1(Γ, Ω) � �♥ ♥ ♥ ♥ ♥ ♥ ♥ S1(Γ, O) �t t t t t t t t t t t t t t t t t Sfin(Ω, Ω) � S1(Ω, Γ) Gerlits–Nagy � � S1(Ω, Ω) � � �♥ ♥ ♥ ♥ ♥ ♥ ♥ S1(O, O) Rothberger � Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The Scheepers Diagram 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Primary: 37F20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Secondary 26A03, 03E75 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' 1 2 BOAZ TSABAN of covering properties and implications among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The properties in this diagram are obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For families A and B of sets, let S1(A, B) be the statement: For each sequence of elements of the family A, we can pick one element from each sequence member, and obtain an element of the family B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' When A = B = O(X), the family of open covers of a topological space X, we obtain Rothberger’s property (1941), the topological version of Borel’s strong measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' We say that a space X satisfies S1(O, O) if the assertion S1(O(X), O(X)) holds, and similarly for the other selective properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The hypothesis Sfin(A, B) is obtained by replacing one by finitely many in the above definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The property Sfin(O, O) is, by an observation of Hurewicz (1925), equivalent to Menger’s basis property, a dimension-type property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The property Ufin(A, B) is obtained by further allowing us to take the unions of the selected finite subsets—this matters for some types of covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For technical reasons, this property does not consider all covers of type A, but only those that have no finite subcover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A cover of a space is an ω-cover if no member of the cover covers the entire space, but every finite subset of the space is covered by some member of the cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For a space X, Ω(X) is the family of open ω-covers of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A point-cofinite cover is an infinite cover where every point of the space belongs to all but finitely many members of the cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Γ(X) is the family of open point-cofinite covers of the space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Applying the mentioned selection principles to the cover types O, Ω and Γ, we obtain ad- ditional important properties, such as Hurewicz’s property Ufin(O, Γ) (1925).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' We also obtain the Gerlits–Nagy γ-property S1(Ω, Γ) (1982), characterizing the Fr´echet–Urysohn property of the function space Cp(X) of continuous real-valued functions, with the topology of pointwise convergence: A topological space is Fr´echet–Urysohn if every point in the closure of a set is actually a limit of a sequence in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' This duality between the spaces X and Cp(X) also translates various tightness and convergence properties of the space Cp(X)—discovered earlier by Arhangel’ski˘ı, Bukovsk´y, Sakai, and others—to the selective covering properties Sfin(Ω, Ω), S1(Γ, Γ), and S1(Ω, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' In Section 2, we provide a surprisingly simple proof of one of the most important results of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' While the result itself does not involve selective covering properties explicitly, its proof does that extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A topological space is Lindel¨of if every open cover has a countable subcover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For exam- ple, all sets of real numbers are Lindel¨of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Since all selection principles concern countable sequences, the theory mainly deals with Lindel¨of spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For Lindel¨of spaces, the Scheepers Diagram is the result of a classification of all properties thus introduced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' each property is equivalent to one in the diagram [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' It was long open whether any additional implication could be established among the properties in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' In Section 3 we deal with the recent, surprising solution of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Menger’s covering property Sfin(O, O) is the oldest, most general, and most applied prop- erty in the Scheepers Diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Initially, Menger conjectured his property to coincide with σ-compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' While this turned out false [20], the game version of this property does provide a characterization of σ-compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A very transparent proof of this deep result is presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' In Section 5 we consider a connection to combinatorial set theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' We show that a nontrivial lower bound on the additivity of Rothberger’s property follows easily from basic knowledge on selection principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' PROOFS FROM THE BOOK 3 The Book is a popular myth by Paul Erd˝os: A transfinite book containing the most simple proofs for all theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' I would like to believe that the proofs presented here are similar to ones from the Book .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' or from some of its preliminary drafts, at any rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The Quasinormal Convergence Theorem By real set we mean a topological space where every open set is a countable union of clopen sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Such are, for example, totally disconnected subsets of the real line and, in particular, subsets of the Cantor space {0, 1}N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' In general, every perfectly normal space with any of the properties considered in this section is a real set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let X be a real set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A sequence of real-valued functions f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' on X converges quasi- normally to a real-valued function f if there are positive real numbers ǫ1, ǫ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' converging to 0 such that for each point x ∈ X, we have |fn(x) − f(x)| ≤ ǫn for all but finitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Quasinormal convergence generalizes uniform convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A real set X is a QN space if every sequence of continuous real-valued functions on X that converges to 0 pointwise, converges to 0 quasinormally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Equivalently, convergence in the space Cp(X) is quasinormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' QN spaces were studied intensively, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=', by Bukovsk´y, Rec�law, Repick´y, Scheepers, Nowik, Sakai, and Haleˇs [21, and references therein].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' This and other properties of similar type are preserved by continuous images, and all experience prior to the paper of the author and Zdomskyy [21] suggested that they are not preserved by Borel images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Thus, the following theorem [21, Theorem 9] came as a surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The Baire space NN is quasiordered by the relation ≤∗ of eventual dominance: f ≤∗ g if f(n) ≤ g(n) for all but finitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A subset Y of the Baire space NN is bounded if it is bounded with respect to evntual dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='1 (Quasinormal Convergence Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The following assertions are equivalent for real sets X: (1) The set X is a QN space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) Every Borel image of the set X in the Baire space NN is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The second property in the theorem is well known and straightforward to apply: The most natural transformations needed in proofs regarding these notions are always easily seen to be Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Consequently, the theorem had a dramatic impact on the study of QN spaces: First, many of the previous sophisticated arguments could be replaced by straightforward ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Second, many properties that were hitherto considered separately turned out provably equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Consequently, this theorem settled all problems concerning these properties [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The original proof of the Quasinormal Convergence Theorem is long and involved, and some of its parts are difficult to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A more natural proof was later published by Bukovsk´y and ˇSupina [5, §4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Inspired by a paper of Gerlits and Nagy [7], I have discovered the following surprisingly simple proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' All needed proof ingredients were already available at the time the Quasinormal Convergence Theorem was established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The following lemma provides the key to the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For a space X, let PF(X) be the collection of countably infinite point-finite families of open sets in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let X be a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The following assertions are equivalent: 4 BOAZ TSABAN (1) Every Borel image of the space X in the Baire space NN is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) The space X satisfies S1(PF, PF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let F(X) (respectively, B(X)) be the family of countable closed (respectively, Borel) covers of the set X, and FΓ(X) (respectively, BΓ(X)) be the family of infinite closed (re- spectively, Borel) point-cofinite covers of the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The properties (1), Ufin(B, BΓ), and S1(BΓ, BΓ) are equivalent [17, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For a family U of open sets, we have U ∈ PF(X) if and only if { Uc : U ∈ U } ∈ FΓ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' It follows that S1(PF, PF) = S1(FΓ, FΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (1) ⇒ (2): Clearly, S1(BΓ, BΓ) implies S1(FΓ, FΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) ⇒ (1): A theorem of Bukovsk´y–Rec�law–Repick´y [4, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='3] asserts that Ufin(F, FΓ) = Ufin(B, BΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The usual argument [16, Proposition 11] shows that S1(FΓ, FΓ) implies Ufin(F, FΓ): If { Cn : n ∈ N} ∈ F(X) and there is no finite subcover, then { �n k=1 Ck : n ∈ N} ∈ FΓ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ A topological space Y has Arhangel’ski˘ı’s property α1 if for every sequence s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' of sequences converging to the same point, there is a sequence s such that the sets im(sn)\\im(s) are finite for all natural numbers n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' This property is defined by properties of sets (images of sequences) rather than sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Fix a bijection ϕ: N × N → N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For sequences s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' , with sn = (s(n,1), s(n,2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' ) for each n, define ∞ � n=1 sn := (sϕ(1), sϕ(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Since convergence of a sequence does not depend on the order of its elements, it does not matter, for our purposes, which bijection ϕ is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A sequence �∞ n=1 sn converges to a point p if and only if each sequence sn converges to p, and for each neighborhood U of p, we have im(sn) ⊆ U for all but finitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let Y be an α1 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For every sequence s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' of sequences in the space Y converging to the same point p, there are tails tn of sn, for n ∈ N, such that the sequence �∞ n=1 tn converges to p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' There is a sequence s such that the sets im(sn)\\im(s) are finite for all natural numbers n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' By moving to a subsequence, we may assume that im(s) ⊆ �∞ n=1 im(sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Suppose that s = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For each natural number n, since the sequence sn coverges to the point p, every element other than p may appear in the sequence sn only finitely often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Thus, there is a tail tn of the sequence sn such that im(tn) ⊆ { ak : k ≥ n } ∪ {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let U be a neighborhood of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' There is a natural number N such that im(tn) ⊆ { ak : k ≥ n } ∪ {p} ⊆ U for all natural numbers n ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Thus, the direct sum t := �∞ n=1 tn converges to the point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ Sakai [13, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='7] and Bukovsk´y–Haleˇs [3, Theorem 11] proved that a real set X is a QN space if, and only if, the space Cp(X) is an α1 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Thus, the Quasinormal Convergence Theorem can be stated, and proved, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' PROOFS FROM THE BOOK 5 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The following assertions are equivalent for real sets X: (1) The space Cp(X) is an α1 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) Every Borel image of the set X in the Baire space NN is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) ⇒ (1): This is the straightforward implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For completeness, we reproduce its proof [21, Theorem 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' be sequences in the space Cp(X) that converge to a function f ∈ Cp(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For each natural number n, suppose that sn = (f n 1 , f n 2 , f n 3 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Define a Borel function Ψ: X → NN by Ψ(x)(n) := min � k : (∀m ≥ k) |f n m(x) − f(x)| ≤ 1 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let g ∈ NN be a ≤∗-bound for the image Ψ[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Then the sequence ∞ � n=1 (f n g(n), f n g(n)+1, f n g(n)+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' ) converges to the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (1) ⇒ (2): This is the main implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='2, it suffices to prove that the set X satisfies S1(PF, PF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' ∈ PF(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' By thinning out the point-finite covers, we may assume that they are pairwise disjoint [16, Lemma 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For each set U ∈ �∞ n=1 Un, let CU be a countable family of clopen sets with � CU = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For each natural number n, let Vn := � U∈Un CU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Every set C ∈ Vn is contained in at most finitely many sets U ∈ Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Thus, the family Vn is infinite and point-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let sn be a bijective enumeration of the family { χV : V ∈ Vn }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The sequence sn is in Cp(X), and it converges to the constant function 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' As the space Cp(X) is α1, there are for each n a tail tn of the sequence sn such that the sequence s := �∞ n=1 tn converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For each natural number n, pick a set Un ∈ Un with CUn ⊆ im(tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Then the family { Un : n ∈ N} is infinite and point-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ For a set X ⊆ {0, 1}N, Gerlits and Nagy (and, independently, Nyikoˇs) define a space T(X) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let {0, 1}∗ denote the set of finite sequences of elements of the set {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let X ⊆ {0, 1}N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For each point x ∈ X, let Ax ⊆ {0, 1}∗ be the set of initial segments of the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let X ∪ {0, 1}∗ be the topological space where the points of the set {0, 1}∗ are isolated, and for each point x ∈ X, a neighborhood base of x is given by the sets {x} ∪ B, where B is a cofinite subset of the set Ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let T(X) be the one-point compactification of this space, and ∞ be the compactifying point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Gerlits and Nagy prove that if a set X ⊆ {0, 1}N is a Siepi´nski set, then the space T(X) is α1, and that if the space T(X) is α1, then the set X is a σ-set [7, Theorem 4] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The following theorem unifies these results and improves upon them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Indeed, every Borel image of a Siepi´nski set in the Baire space is bounded, and every set with bounded Borel images in the Baire space is a σ-set [17, and references therein].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let X ⊆ {0, 1}N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The following assertions are equivalent: 6 BOAZ TSABAN (1) The space T(X) is an α1 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) Every Borel image of the set X in the Baire space NN is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For a finite sequence s ∈ {0, 1}∗, let [s] be the basic clopen subset of the Cantor space {0, 1}N consisting of all functions extending s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Every open set in the space {0, 1}N is a disjoint union of basic clopen sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A sequence a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' in the set {0, 1}∗ converges to ∞ in the space T(X) if, and only if, the set { [an] : n ∈ N} is point-finite in the space X [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The argument in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='4 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The Scheepers Diagram Last Theorem The implications in the Scheepers Diagram 1 were all rather straightforward to establish, and almost all other potential implications were ruled out by counterexamples [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Only two problems remained open: Does Ufin(O, Ω) imply Sfin(Γ, Ω)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' And if not, does Ufin(O, Γ) imply Sfin(Γ, Ω)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' [9, Problems 1 and 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For nearly three decades it was expected that the remaining two potential implications were refutable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Only when Peng came up with an entirely new method for refuting implications among selective covering properties [12], were these problems resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' But not in the expected way: Having proved that Ufin(O, Ω) does not imply Sfin(Γ, Ω), Peng tried to refute the last remaining potential implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' And he failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' His close examination of the failure suggested a path for proving the last potential implication [12, Theorem 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Peng’s results establish the final form of the Scheepers Diagram (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Ufin(O, Γ) Hurewicz � Sfin(Γ, Ω) � Ufin(O, Ω) � Sfin(O, O) Menger S1(Γ, Γ) � �♥ ♥ ♥ ♥ ♥ ♥ ♥ S1(Γ, Ω) � �♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ S1(Γ, O) �♠ ♠ ♠ ♠ ♠ ♠ ♠ Sfin(Ω, Ω) � S1(Ω, Γ) Gerlits–Nagy � � S1(Ω, Ω) � � �♥ ♥ ♥ ♥ ♥ ♥ ♥ S1(O, O) Rothberger � Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The Final Scheepers Diagram Peng’s proof of the last implication is somewhat involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The proof given below identifies the heart of Peng’s argument, and replaces the other parts with simple, quotable observations about selective covering properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let k be a natural number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A cover of a space is a k-cover if no member of the cover covers the entire space, but every k-element subset of the space is covered by some member of the cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Thus, a cover is an ω-cover if and only if it is a k-cover for all natural numbers k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For a space X and a natural number k, let Ok(X) be the family of open k-covers of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let Π be a selection principle, and A a type of open covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The following assertions are equivalent: (1) Π(A, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) For each natural number k we have Π(A, Ok).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' PROOFS FROM THE BOOK 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (1) ⇒ (2): Obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) ⇒ (1): Let U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' be a sequence in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Split the sequence into infinitely many disjoint sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For each natural number k, apply Π(A, Ok) to the kth sequence, to obtain a k-cover Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Then �∞ k=1 Vk is an ω-cover, in accordance to the required property Π(A, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='2 (Peng [12, Theorem 23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The Hurewicz property Ufin(O, Γ) implies Sfin(Γ, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let X be a Huewicz space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='1, it suffices to prove that Sfin(Γ, Ok) holds for all natural numbers k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Fix a natural number k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' be a sequence in Γ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' By moving to countably infinite subcovers, we may enumerate Un = { Un m : m ∈ N } for each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For each n and m, we may replace the set Un m with the smaller set U1 m ∩ U2 m ∩ · · · ∩ Un m, so that we may assume that U1 m ⊇ U2 m ⊇ U3 m ⊇ · · · for all natural numbers m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The refined covers Un remain in Γ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let g0(m) := m for all m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' We define, by induction, increasing functions g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' , gk ∈ NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let l < k and assume that the function gl is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For natural numbers n, m and i, let V l,n i := gl(i) � m=i Un m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' W l,n m := � { V l,n i : n ≤ i, gl(i) ≤ m }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For each l and n, the sets W l,n m are increasing with m, and cover the space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' By the Hurewicz property, there is an increasing function gl+1 ∈ NN such that { W l,n gl+1(n) : n ∈ N} ∈ Γ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' This completes the iductive construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' We will show that { Un m : n ∈ N, m ≤ gk(n) } ∈ Ok(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' , xk ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Since { W l,n gl+1(n) : n ∈ N} ∈ Γ(X) for all l = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' , k − 1, there is a natural number N with x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' , xk ∈ W l,n gl+1(n) for all l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' , k and all n ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Fix a number n0 ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Since x1 ∈ W k−1,n0 gk(n0) , there is n1 with n0 ≤ n1, gk−1(n1) ≤ gk(n0) and x1 ∈ V k−1,n0 n1 = gk−1(n1) � m=n1 Un0 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Since x2 ∈ W k−2,n1 gk−1(n1), there is n2 with n1 ≤ n2, gk−2(n2) ≤ gk−1(n1) and x2 ∈ V k−2,n1 n2 = gk−2(n2) � m=n2 Un1 m ⊆ gk−2(n2) � m=n2 Un0 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' 8 BOAZ TSABAN Since x3 ∈ W k−3,n2 gk−2(n2), there is n3 with n2 ≤ n3, gk−3(n3) ≤ gk−2(n2) and x3 ∈ V k−3,n2 n3 = gk−3(n3) � m=n3 Un2 m ⊆ gk−3(n3) � m=n3 Un0 m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Since xk ∈ W 0,nk−1 g1(nk−1), there is nk with nk−1 ≤ nk = g0(nk) ≤ g1(nk−1) and xk ∈ V 0,nk−1 nk = Unk−1 nk ⊆ Un0 nk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' It follows that x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' , xk ∈ Un0 nk , and nk ≤ gk(n0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='2 establishes a stronger result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' To this end, we need the fol- lowing definitions and lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' An infinite cover of a space X is ω-groupable (respectively, k-groupable, for a natural number k) if there is a partition of the cover into finite parts such that for each finite (respectively, k-element) set F ⊆ X and all but finitely many parts P of the partition, there is a set U ∈ P with F ⊆ U [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let Ωgp(X) (respectively, Ogp k (X)) be the family of open ω-groupable (respectively, k-groupable) covers of the space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let Π be a selection principle, and A a type of open covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The following assertions are equivalent: (1) Π(A, Ωgp);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) For each natural number k we have Π(A, Ogp k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The proof is similar to that of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='1, once we observe that if { Un : n ∈ N} is a k-grouable cover for all k, then it is ω-groupable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' This follows easily from the fact that for each countable family { Pk : k ∈ N } of partitions of N into finite sets, there is a partition P of N into finite sets that is eventually coarser than all of the given partitions, that is, such that for each k, all but finitely many members of the partition P contain a member of the partition Pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ Koˇcinac and Scheepers proved that if all finite powers of a space X are Hurewicz, then every open ω-cover of the space is ω-groupable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Together with Peng’s Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='2, we have that if all finite powers of a space X are Hurewicz, then the space satisfies Sfin(Γ, Ωgp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The following theorem shows that the assumption on the finite powers is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' In the theorem, we also mention Sfin(Γ, Λgp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' An open cover is large if each point is in infinitely many members of the cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let Λ(X) be the family of large open covers of the space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' An open cover U is in Λgp(X) [10] (also denotedג(Γ), depending on the context [14]) if there is a partition of the cover into finite parts such that for each point x ∈ X and all but finitely many parts P of the partition, we have x ∈ � P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The following assertions are equivalent: (1) Ufin(O, Γ), (2) Sfin(Γ, Ωgp);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' and (3) Sfin(Γ, Λgp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (1) ⇒ (2): The proof of Peng’s Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='2, as written above, shows, for a prescribed number k, that for each k-element set F there is a natural number N such that for each n ≥ N there is a member of the finite set Fn = { Un m : m ≤ gk(n) } that contains the set F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' By thinning out the point-cofinite covers, we may assume that they are pairwise disjoint [16, PROOFS FROM THE BOOK 9 Lemma 4], and consequently so are the finite sets Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Thus, �∞ n=1 Fn ∈ Ogp k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' This proves Sfin(Γ, Ogp k ) for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) ⇒ (3): Ωgp ⊆ Λgp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (3) ⇒ (1): This implication is standard and should be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For completeness, we provide a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Assume that the space X satisfies Sfin(Γ, Λgp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' It suffices to prove that it satisfies Ufin(Γ, Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Given a sequence U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' in Γ(X), we may (as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='2) assume that the covers get finer with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Apply Sfin(Γ, Λgp) to obtain a cover U ∈ Λgp, with parts Pn (for n ∈ N) witnessing that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let F1 ⊆ U1 be a finite set refined by P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let n2 be minimal with Pn2 ⊆ �∞ n=2 Un, and F2 ⊆ U2 be a finite set refined by Pn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let n3 be minimal with Pn3 ⊆ �∞ n=3 Un, and F3 ⊆ U3 be a finite set refined by Pn3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Continuing in this manner, we obtain finite sets Fn ⊆ Un for n ∈ N, with { � Fn : n ∈ N} ∈ Γ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ Koˇcinac and Scheepers proved that Ufin(O, Γ) = Sfin(Ω, Λgp) = Sfin(Λ, Λgp) [10, Theo- rem 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' However, Ufin(O, Γ) ̸= Sfin(Ω, Ωgp): The latter property is equivalent to satisfying Ufin(O, Γ) in all finite powers [10, Theorem 16], a property strictly stronger than Ufin(O, Γ) [9, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' When Bob has a winning strategy in the Menger game Menger [11] conjectured that his property Sfin(O, O) implies σ-compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' While his conjecture turned out false [20, and references therein], a closely related assertion is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The Menger game [8], Gfin(O, O), is the game associated to Menger’s property Sfin(O, O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' It is played on a topological space X, and has an inning per each natural number n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' In each inning, Alice picks an open cover Un of the space, and Bob chooses a finite set Fn ⊆ Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Bob wins if �∞ n=1 Fn is a cover of the space, and otherwise Alice wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Telg´arsky [18] proved that if Bob has a winning strategy in the Menger game played on a metric space, then the space is σ-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Scheepers [15, Theorem 1] provided a direct proof of Telg´arsky’s Theorem, using the notion of H-closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' We will eliminate the notion of H-closed sets and the closure operations from Scheepers’s proof, and obtain a more transparent proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' As a bonus, the separation hypotheses on the space are eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A subset K of a topological space X is relatively compact if every open cover U of the entire space X has a finite subcover of the set K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A set K is relatively compact if and only if its closure is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let κ be a cardinal number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' If a space X is a union of at most κ relatively compact sets, then it is the union of at most κ compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' If X = � α<κ Kα, then X = � α<κ Kα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ For a basis B for the topology of a space X, let OB(X) be the family of subsets of B that cover the space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let X be a topological space with a basis B, and σ be a function on the family OB(X) such that for each cover U ∈ OB(X), σ(U) is a finite subset of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Then the set K := � U∈OB(X) � σ(U) is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' 10 BOAZ TSABAN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let U be an open cover of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let V ∈ OB(X) be a cover that refines the cover U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Then K ⊆ � σ(V), and there is a finite set F ⊆ U with � σ(V) ⊆ � F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ Scheepers [15, Theorem 1] proves the following theorem for metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' If Bob has a winning strategy in the Menger game played on X, then the space X is Menger and, in particular, Lindel¨of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' If X is, in addition, metric, then the space is second countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let X be a second countable topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' If Bob has a winning strategy in the Menger game Gfin(O, O) played on X, then the space X is σ-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' We follow steps of Scheepers’s proof, removing what is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let σ be a winning stratery for Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Fix a countable base B for the topology of the space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let N∗ be the set of finite sequences of natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' We consider all possible games where Alice chooses her covers from the family OB(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Since the base B is countable, the family { σ(U) : U ∈ OB } (the possible first responds of Bob) is countable, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Choose elements U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' ∈ OB with { σ(Un) : n ∈ N} = { σ(U) : U ∈ OB }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' By induction, for a given natural number n and each sequence s ∈ Nn, the family { σ(Us1, Us1,s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' , Us, U) : U ∈ OB } is countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Choose elements Us,1, Us,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' ∈ OB with { σ(Us1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' , Us, Us,n) : n ∈ N} = { σ(Us1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' , Us, U) : U ∈ OB }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' This completes our inductive construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='2, for each sequence s ∈ N∗, the set Ks := ∞ � n=1 � σ(Us1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' , Us, Us,n) is relatively compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='1, it remains to see that X = � s∈N∗ Ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Assume that some element x ∈ X is not in � s∈N∗ Ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (1) Since x /∈ K(), there is m1 with x /∈ � σ(Um1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) Since x /∈ Km1, there is m2 with x /∈ � σ(Um1,m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (3) Since x /∈ Km1,m2, there is m3 with x /∈ � σ(Um1,m2,m3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (4) Etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Then the play Um1, σ(Um1), Um1,m2, σ(Um1,m2), Um1,m2,m3, σ(Um1,m2,m3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' is lost by Bob;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ Let α be an ordinal number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The transfinite Menger game Gα fin(O, O) is defined as the ordinary Menger game, with the only difference that now there is an inning per each ordinal number β < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Clearly, if α1 < α2 and Bob has a winning strategy in the α1-Menger game, then Bob has a winning strategy in the α2-Menger game: He can use a winning startegy in the first α1 innings, and then play arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Thus, the following theorem is stronger than Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='3, and has no assumption on the topological space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The weight of a topological space is the minimal cardinality of a base for its topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let X be a topological space of weight κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Bob has a winning strategy in the game Gκ fin(O, O) if and only if the space X is a union of at most κ compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' PROOFS FROM THE BOOK 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (⇒) The proof is identical to that of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='3, only that here we begin with a base of cardinality κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (⇐) In the α-th inning, Bob covers the α-th compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let X be a topological space of weight κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' If Bob has a winning strategy in the Menger game, then the space X is a union of at most κ compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ The converse of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='5 is false: The discrete space of cardinality κ has weight κ, and it is a union of κ compact sets (singletons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' This space is not Lindel¨of, and thus not Menger, so Bob has no winning strategy in the Menger game played on this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For a space X, let O(X) be the families U of open sets with � U∈U U = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A space X is almost Menger if it satisfies Sfin(O, O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For regular spaces, almost Menger is equivalent to Menger [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' And similarly for the other notions considered below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Thus, the remainder of this section is mainly relevant for nonregular spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The almost Menger game is the game associated to the property Sfin(O, O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The corresponding notion of almost Lindel¨of is classic, and so is the notion of almost compact space: A space K is almost compact if every open cover of K has a finite subset F with dense union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' This notion appears in the literature under various names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For Hausdorff spaces, it is known to be equivalent to Hausdorff closed, that is, being closed in all Hausdorff superspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A set K in a space X is relatively almost compact if every open cover of the space X has a finite subset F with K ⊆ � U∈F U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The standard proof that every almost compact space is H-closed shows that every relatively almost compact set in a Hausdorff space is closed in that space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' However, no separation hypothesis is needed in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Since the proof of the following theorem is provided by the first part of Scheepers’s argument, we attribute the theorem to Scheepers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='6 (Scheepers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let X be a second countable topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The following assertions are equivalent: (1) Bob has a winning strategy in the game Gfin(O, O) played on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) The space X is a countable union of relatively almost compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) ⇒ (1): This is easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (1) ⇒ (2): This is a part of the argument of Scheepers [15, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' We provide it, for completion and verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Repeat the inductive construction of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='3 verbatim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Having completed it, define for each sequence s ∈ N∗: Ks := ∞ � n=1 � U∈σ(Us1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=',Us,Us,n) U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The set Ks is relatively almost compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' It remains to see that X = � s∈N∗ Ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Assume that some element x ∈ X is not in � s∈N∗ Ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (1) Since x /∈ K(), there is m1 with x /∈ � U∈σ(Um1) U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) Since x /∈ Km1, there is m2 with x /∈ � U∈σ(Um1,m2) U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (3) Since x /∈ Km1,m2, there is m3 with x /∈ � U∈σ(Um1,m2,m3) U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (4) Etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Then the play Um1, σ(Um1), Um1,m2, σ(Um1,m2), Um1,m2,m3, σ(Um1,m2,m3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' 12 BOAZ TSABAN is lost by Bob;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ The assertions analogous to the more general Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='4 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='5 also hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='6 answers a question of Babinkostova, Pansera and Scheepers [1, Question 26(2)] in the case of second countable spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The additivity of Rothberger’s property Let add(N ) be the minimal cardinality of a family F ⊆ NN such that there is no function S : N → [N]<∞ with |S(n)| ≤ n for all n, such that for each function f ∈ F we have f(n) ∈ S(n) for all but finitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The notation add(N ) is explained by a result of Bartoszy´nski and Judah [2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='11]: The cardinal number add(N ) is the minimal cardinality of a family of Lebesgue null sets of real numbers whose union is not Lebesgue null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' In general, the additivity of a property is the minimum cardinality of a family of sets satisfying the property, whose union does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' The following theorem is attributed to Carlson by Bartoszy´nski and Judah [2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='1 (Carlson).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let κ < add(N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' If a Lindel¨of space is a union of at most κ spaces satisfying S1(O, O), then the space X satisfies S1(O, O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' That is, for Lindel¨of spaces, add(N ) ≤ add(S1(O, O)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' This theorem is an easy consequence of a simple, basic fact concerning selection principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' We need the following lemmata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let A and B be types of open covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A topological space X satisfies Sn(A, B) if for all U1, U2, · · · ∈ A(X), there are finite sets F1 ⊆ U1, F2 ⊆ U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' such that |Fn| ≤ n for all n, and �∞ n=1 Fn ∈ B(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Garcia-Ferreira and Tamariz-Mascarua [6, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='12] established the following observa- tion in the case A = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='2 ([20, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let A be a type of countable covers such that every pair of covers of type A has a joint refinement of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Then Sn(A, O) = S1(A, O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='3 (Folklore).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' If a space X satisfies S1(A, O), then for each sequence U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' in O(X) there are elements Um1 ∈ Um1, Um2 ∈ Um2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' such that for each point x ∈ X, we have x ∈ Umn for infinitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' As usual, we split the sequence of open covers to infinitely many disjoint sequences, and apply the property S1(A, O) to each subsequence separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let A be a type of countable covers such that every pair of covers of type A has a joint refinement of type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Then add(N ) ≤ add(S1(A, O)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let κ < add(N ) and X = � α<κ Xα, where each space Xα satisfies S1(A, O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='2, it suffices to show that the space X satisfies Sn(A, O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let Un = {Un m : m ∈ N} ∈ A(X), for n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For each ordinal number α < κ, as the space Xα satisfies S1(A, O), there is a function fα ∈ NN such that for each point x ∈ Xα we have x ∈ Un fα(n) for infinitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' There is a function S : N → [N]<∞ with |S(n)| ≤ n for all n, such that for each α < κ we have fα(n) ∈ S(n) for all but finitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Then �∞ n=1{ Un m : m ∈ S(n) } ∈ O(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ PROOFS FROM THE BOOK 13 Judging by an extensive survey on the topic [19], the result in the second item below seems to be new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (1) For Lindel¨of spaces, add(N ) ≤ add(S1(O, O)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) add(N ) ≤ add(S1(Γ, O)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (1) Since the spaces are Lindel¨of, we may restrict attention to countable covers, and the assumptions of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='4 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' (2) A countably infinite subset of a point-cofinite cover is also a point-cofinite cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Thus, we may restrict attention to countable point-cofinite covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' It is well-known that every pair of point-cofinite covers has a joint refinement that is a point-cofinite cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Indeed, let U and V be countable point-cofinite covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Enumerate them U = { Un : n ∈ N} and V = { Vn : n ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Then { Un ∩ Vn : n ∈ N} ∈ Γ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='4 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ We can extract additional information from this proof method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' A topological space X satisfies Un(Γ, Γ) [20] if for all U1, U2, · · · ∈ Γ(X), there are finite sets F1 ⊆ U1, F2 ⊆ U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' such that |Fn| ≤ n for all n, and { � Fn : n ∈ N} ∈ Γ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' This property is strictly inbetween S1(Γ, Γ) and Ufin(O, Γ) [20, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' add(N ) ≤ add(Un(Γ, Γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let κ < add(N ) and X = � α<κ Xα, where each space Xα satisfies Un(Γ, Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' It suffices to show that the space X satisfies Un2(Γ, Γ), where the cardinality of the n-th selected finite set is at most n2 [20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Let Un = {Un m : m ∈ N} ∈ Γ(X), for n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For each α < κ, as the space Xα satisfies Un(Γ, Γ), there is a function Sα: N → � n[N]≤n such that for each point x ∈ Xα we have x ∈ � m∈Sα(n) Un m for infinitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' There is a function S : N → � n[N]≤n with |S(n)| ≤ n for all n, such that for each α < κ we have Sα(n) ∈ S(n) for all but finitely many n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' For each natural number n, let Fn := � S(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Then |Fn| ≤ n2 for all n, and { � m∈Fn Un m : n ∈ N} ∈ Γ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' □ Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' This is the first paper that I write after a long, callenging period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' I thank all those who helped and encouraged me throughout, and supported my return to normal track afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Above all, I thank my wife, Adina, for her faith, support, and patience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' References [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Babinkostova, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Pansera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Scheepers, Weak covering properties and infinite games, Topology and its Applications 159 (2012), 3644–3657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} 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+page_content=' Tsaban, Additivity numbers of covering properties, in: Selection Principles and Covering Prop- erties in Topology (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Koˇcinac, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' ), Quaderni di Matematica 18, Seconda Universita di Napoli, Caserta 2006, 245–282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' [20] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Tsaban, Menger’s and Hurewicz’s Problems: Solutions from “The Book” and refinements, Contem- porary Mathematics 533 (2011), 211–226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Tsaban, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Zdomskyy, Hereditarily Hurewicz spaces and Arhangel’ski˘ı sheaf amalgamations, Journal of the European Mathematical Society 12 (2012), 353–372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content=' Boaz Tsaban, Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel Email address: tsaban@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='biu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} +page_content='il' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FPT4oBgHgl3EQfvDVz/content/2301.13158v1.pdf'} diff --git a/aNE5T4oBgHgl3EQfDg77/content/tmp_files/2301.05407v1.pdf.txt b/aNE5T4oBgHgl3EQfDg77/content/tmp_files/2301.05407v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..097f77f602990b68f2ac64417ff2292eabe071ff --- /dev/null +++ b/aNE5T4oBgHgl3EQfDg77/content/tmp_files/2301.05407v1.pdf.txt @@ -0,0 +1,4950 @@ +One-loop contributions to decays eb → eaγ and (g − 2)ea anomalies, +and Ward identity +L.T. Hue,1, 2, ∗ H. N. Long,1, 2, † V. H. Binh,3 H. L. T. Mai,4 and T. Phong Nguyen ‡5, § +1Subatomic Physics Research Group, +Science and Technology Advanced Institute, +Van Lang University, Ho Chi Minh City 70000, Vietnam +2Faculty of Applied Technology, School of Technology, +Van Lang University, Ho Chi Minh City 70000, Vietnam +3Institute of Physics, Vietnam Academy of Science and Technology, +10 Dao Tan, Ba Dinh, Hanoi 10000, Vietnam +4Faculty of Physics Science, Can Tho Medical College, +Nguyen Van Cu Street, Can Tho 94000, Vietnam +5Department of Physics, Can Tho University, +3/2 Street, Can Tho 94000, Vietnam +Abstract +In this paper, we will present analytic formulas to express one-loop contributions to lepton flavor +violating decays eb → eaγ, which are also relevant to the anomalous dipole magnetic moments of +charged leptons ea. These formulas were computed in the unitary gauge, using the well-known +Passarino-Veltman notations. We also show that our results are consistent with those calculated +previously in the ’t Hooft-Veltman gauge, or in the limit of zero lepton masses. At the one-loop +level, we show that the appearance of fermion-scalar-vector type diagrams in the unitary gauge will +violate the Ward Identity relating to an external photon. As a result, the validation of the Ward +Identity guarantees that the photon always couples with two identical particles in an arbitrary +triple coupling vertex containing a photon. +PACS numbers: +‡ corresponding author +∗Electronic address: lethohue@vlu.edu.vn +†Electronic address: hoangngoclong@vlu.edu.vn +§Electronic address: thanhphong@ctu.edu.vn +1 +arXiv:2301.05407v1 [hep-ph] 13 Jan 2023 + +I. +INTRODUCTION +The lepton sector is one of the most interesting object for experiments to search for new +physics (NP) beyond the prediction of the standard model (SM). For example, the evidence +of neutrino oscillation confirms that the SM must be extended. Recently, the experimental +data of anomalous magnetic moments (AMM) of charged leptons (g −2)ea/2 ≡ aea has been +updated, where the deviation between SM prediction and the lasted experiment data for +muon is [1] +∆aNP +µ +≡ aexp +µ +− aSM +µ += (251 ± 59) × 10−11, +(1) +corresponding to the 4.2σ deviation from standard model (SM) prediction [2] combined +from various contributions [3–23]. For the electron anomaly, the deviation between SM and +experiment is 1.6σ discrepancy [24]. +On the other hand, ∆ae,µ are strongly constrained by the experimental data obtained +from searching for the charged lepton flavor violating (cLFV) decays eb → eaγ are [25, 26]: +Br(τ → µγ) < 4.4 × 10−8, Br(τ → eγ) < 3.3 × 10−8, Br(µ → eγ) < 4.2 × 10−13. +(2) +This important property was discussed previously, for example see discussions for a general +estimation in Ref. [27], and many particular models beyond the standard model (BSM) +[28–33]. General formulas expressing simultaneously both one-loop contributions to AMM +and cLFV amplitudes were introduced in the limits of new heavy scalar and/or gauge boson +exchanges m2 +B ≫ m2 +a with ma being the mass of a charged lepton ea = e, µ, τ [27]. Other +calculations in the unitary gauge were discussed [34, 35] for the one-loop contributions to +aea with ma ̸= 0, without the relations with the cLFV amplitudes. The analytic one-loop +formulas for cLFV amplitudes calculated in the ’t Hooft Feynman (HF) gauge were also +shown in Ref. [36], using the notations of the Passarino-Veltman (PV) functions [37, 38] +with ma ̸= mb. The approximate formulas with ma = mb = 0 were introduced and consistent +with those given in Ref. [27], as shown particularly in Ref. [39] for 3-3-1 models. The general +analytic formulas of these PV functions were introduced for numerical investigations. They +are consistent with the results generated by LoopTools [40], which can be transformed +into other PV notations implemented in the Fortran numerical package Collier [41], used +to investigate cLFV decays in a two Higgs doublet model (2HDM) [42]. Many particular +expressions to compute the AMM and/or cLFV decay amplitudes predicted by different +2 + +particular BSM were constructed [28]. The relations among them can be checked by using +suitable transformations, starting from the set of particular PV notations in this work. On +the other hand, in a discussion on analytic formulas for one-loop contributions to AMM, +a class of fermion-scalar-vector (FSV ) diagrams consisting of a photon coupling with two +different physical particles, namely one scalar and one gauge boson, were considered even +in the unitary gauge [34]. It leads us to a question whether the Ward identity (WI) for +the external photon is still valid with the presence of this diagram type. We emphasize +that the general results for one-loop contributions to decays eb → eaγ and AMM of leptons +introduced in many previous works do not include this FSV diagrams. Moreover, they +imply the existence of the triple photon coupling with two distinguishable physical particles +that has never been mentioned previously. In particular, many works introducing general +one-loop contributions for AMM of charged leptons [27, 28, 35], or decays relating with +photon such as cLFV decays eb → eaγ [27, 28, 36], loop-induced Higgs decays h → γγ +[43, 44], h → Zγ, f ¯fγ [44–47], quark decays q → q′γ, . . . . Excluding the FSV vertex type +will reduce a huge number of related one- and two-loop diagrams as well as confirm the +validation of general one-loop calculation introduced previously. +In this work, we will show precisely the important steps to derive the one-loop contri- +butions to both AMM and cLFV decays. The calculation is performed by hand, which is +consistent with another cross-checking using FORM package [48]. The final formulas are +expressed exactly in terms of the PV functions defined by LoopTools. The results are then +easily to change into all of the other available forms using suitable transformations. The +convention of the PV-functions are very convenient to derive the exact formulas before solv- +ing particular pure mathematical problems. We also determine contributions arising from +a new form of photon coupling with vector bosons such as leptoquarks and confirm the +consistence between our results and those introduced in Refs. [44, 49, 50]. +Our paper is organized as follows. Section I explains our aim of this work. Section II +introduces notations and important formulas to establish the relations between AMM and +cLFV amplitudes. Section III shows discussions to confirm the consistence of our results +and previous works, and the validation of the WI for the relevant analyitic formulas. Section +IV summarizes main features of our work. Finally, we provide many appendices showing +precisely many intermediate steps and notations to derive the final results mentioned in this +work, including the analytic formulas of the PV functions consistent with LoopTools given +3 + +in appendix A. +II. +GENERAL AMPLITUDES AND NOTATIONS +It is well-known that analytic formulas of one-loop contributions to the cLFV amplitudes +eb(p2) → ea(p1)γ(q) and AMM of SM charged leptons ea can be presented in the same +expressions, see for example Ref. [27] corresponding to the presence of new heavy particles in +BSM. Possible one-loop Feynman diagrams contributing to aea and cLFV decay amplitudes +eb → eaγ in BSM are shown in Fig. 1, where F is a fermion coupling with the SM charged +lepton ea = e, µ, τ; and the boson B = h, V is a scalar or gauge boson, respectively. We note +(1) +h+ +F +eb +ea +γ +p1 +q +p2 +(2) +F +h+ +eb +ea +ea +F +eb +eb +(3) +(4) +ea +ea +F +eb +(5) +eb +ea +γ +V + +F +(6) +F +V + +eb +ea +(7) +(8) +FIG. 1: Feynman diagrams for one-loop contribution to aea and cLFV amplitudes eb → eaγ in the +unitary gauge. +here that Ref. [34] argues another type of FSV one-loop diagrams giving new contributions +to the AMM. They will be discussed in details in this work. +Firstly, we adopt the Lagrangian generating one-loop diagrams in Fig. 1, namely [27] +Lh = F(gL +a,FhPL + gR +a,FhPR)eah + h.c., +(3) +LV = Fγµ(gL +a,FV PL + gR +a,FV PR)eaVµ + h.c., +(4) +where the fermion F and the boson B = Vµ, h have electric charges QF and QB, and +masses mF and mB, respectively. These Lagrangians (3) and (4) are consistent with those +in Ref. [36]. Moreover, the photon couplings with all physical particles should be mentioned +clearly, as given in Ref. [36], i.e., we will adopt the Feynman rules that the photon alway +couples with two identical physical particles, as given in table I, where Γµνλ(p0, p+, p−) = +4 + +Vertex +Coupling +Vertex +Couplings +Vertex Couplings +Aµ(p0)V ν(p+)V ∗λ(p−) −ieQV Γµνλ(p0, p+, p−) Aµh(p+)h∗(p−) ieQh(p+ − p−)µ AµFF +ieQF γµ +TABLE I: Feynman rules for cubic couplings of photon Aµ, where p0,± are incoming momenta into +the relevant vertex. +gµν(p0 − p+)λ + gνλ(p+ − p−)µ + gλµ(p− − p0)ν is the standard form. The more general form +of Γµνλ(p0, p+, p−) introduced in Refs. [44, 49, 50] will be discussed in details later. +All couplings listed in Lagrangians (3), (4), and table I result in the following form factors +relevant with one-loop contributions: +cab +RB = +e +16π2gL∗ +a,FBgR +b,FBmF × fB(xB) + QFgB(xB) +m2 +B ++ +e +16π2 +� +mbgL∗ +a,FBgL +b,FB + magR∗ +a,FBgR +b,FB +� +× +˜fB(xB) + QF ˜gB(xB) +m2 +B +, +(5) +where xB ≡ m2 +F/m2 +B. The four scalar functions fB(x), gB(x), ˜fB(x), and ˜gB(x) are listed +in Eq. (A22) of appendix A, as the approximate formulas in the limit ma, mb ≪ mB. The +formula in Eq. (5) does not contain contributions from the FSV diagrams mentioned in +Ref. [34], because of the absence of photon coupling AV h. The corresponding formulas of +AMM and cLFV decay rates are: +aea ≡ −2ma +e +(caa +R + caa∗ +R ) = −4ma +e Re[caa +R ], +(6) +Br(eb → eaγ) = m3 +b +4πΓb +���cab +R +��2 + +��cba +R +��2� +, +(7) +where ma, mb, and Γb are the masses and total decay width of the leptons ea, eb, and +cab +R ≡ +� +B,F +cab +RB. +(8) +The amplitude for a vertex ¯eaeaAµ in Ref. [51] is consistent with the following form +presenting both AMM and cLFV amplitudes [52, 53] +iM = −ieua(p1) +� +γµF1 − σµνqν +2ma +� +iF2 + γ5F3 +�� +ub(p2)ε∗ +µ, +(9) +where σµν ≡ i +2 [γµγν − γνγµ]; F1,2,3 are scalar form factors; ε∗ +µ and qν is the polarized vector +of the external photon. The form factor F2,3 gets contribution only from loop corrections. +They relate with the well-known experimental quantities called the anomalous magnetic +5 + +moment aea and electric dipole moment dea for b = a, respectively. Specifically we have +F1 = 1; +aea = F2; +dea = − e +2ma +F3. +(10) +Regarding to the LFV decay eb → eaγ the amplitude can also be written in the same form +[36, 54], suggesting that F2 can be calculated based on the one-loop corrections to LFV +decays. In particular, the second term of the amplitude (9) can be expanded as follows [39] +M = (2p1.ε∗)ua +� +C(ab)LPL + C(ab)RPR +� +ub + ua +� +D(ab)L/ε∗PL + D(ab)R/ε∗PR +� +ub, +(11) +where ma = mb and we can prove that C(ab)LPL + C(ab)RPR = +e +2ma(F2 − iγ5F3). The WI for +the external photon gives +D(ab)L = −(mbC(ab)R + maC(ab)L), D(ab)R = −(mbC(ab)L + maC(ab)R). +(12) +The hermiticity that C(aa)R = C∗ +(aa)L [53] gives +aea = ma(C(aa)L + C(aa)R) +e += 2maRe[C(aa)L,R] +e +, +dea = i(C(aa)R − C(aa)L) = Im[C(aa)L] = −Im[C(aa)R]. +(13) +Hence, the following relations between two different notations must be satisfied: +cab +R = −1 +2C(ab)R and cba +R = −1 +2C(ab)L. +(14) +From the above discussion, we see that one-loop contributions to the aea and dea can be +written in terms of well-known PV functions, see detailed discussions in Ref. [39] or general +formula introduced for calculations the LFV decay rates of charged leptons [36], with the +identification that σL,R ≡ −C(ab)L,R. In the limit of 0 ≃ ma, mb ≪ mB, the numerical values +of aea can be evaluated using the numerical packages such as LoopTools [40] or Collier +[41]. Although the exact analytic formulas of one-loop three point functions presented in +Ref. [39] can not be applied to calculate aea, but the limit of mb → ma can be used to solve +this problem. The analytic formulas of aea were introduced completely in Ref. [34]. +Because of the relations in Eq. +(12), only C(ab)L,R is needed to determine aea and +Br(eb → eaγ). Because all two-point diagrams give contributions to just D(ab)L,R, C(ab)L,R +are calculated by considering only three-point diagrams. In this work, the analytic formulas +of D(ab)L,R will be determined directly from all diagrams in Fig. 1 to check the validation of +the WI in the presence of the FSV . +6 + +The analytic formulas for one-loop contributions to the cLFV decay amplitudes presented +in this work are more general than the results introduced in Ref. [39] for general 3-3-1 +models. Many important steps in our calculations were shown in appendix C. Using this +unitary gauge, the assumption for a particular form of the Goldstone boson couplings given +in Ref. [36] is unnecessary. In contrast, we use the same photon couplings to other physical +particles in an arbitrary BSM, as given in table I. Namely, a tree-level photon coupling +always contains two identical physical particles. This implies that the contributions from +the FSV diagrams are not included. +Using the notations of PV-functions defined in appendix A, the Fhh contributions from +diagram (1) in Fig. 1 are: +CFhh +(ab)L =−eQh +16π2 +� +magL∗ +a,FhgL +b,FhXf +1 + mbgR∗ +a,FhgR +b,FhXf +2 − mFgR∗ +a,FhgL +b,FhXf +0 +� +, +CFhh +(ab)R =−eQh +16π2 +� +magR∗ +a,FhgbR +FhXf +1 + mbgL∗ +a,FhgL +b,FhXf +2 − mFgL∗ +a,FhgR +b,FhXf +0 +� +, +(15) +where Xf +0 , Xf +1 , . . . are linear combinations of the PV-functions C0,00,i,ij defined precisely in +appendix A. +The diagram (2) in Fig. 1 gives hFF contributions as follows: +ChFF +(ab)L =−eQF +16π2 +� +magL∗ +a,FhgbL +b,FhXh +1 + mbgR∗ +a,FhgR +b,FhXh +2 + mFgR∗ +a,FhgL +b,FhXh +3 +� +, +ChFF +(ab)R =−eQF +16π2 +� +magR∗ +a,FhgR +b,FhXh +1 + mbgL∗ +a,FhgL +b,FhXh +2 + mFgL∗ +a,FhgR +b,FhXh +3 +� +. +(16) +where Xh +1,2,3 are linear combinations of C0,i,ij(m2 +h, m2 +F, m2 +F). The above result are completely +consistent with the results introduced in Ref. [36], except an overall sign and the signs before +the PV-functions ¯c1,2, arising from the different definitions of the external momenta pi in +the denominators of the one-loop integrals. We also give the analytic formulas of DFhh +(ab)L,R +and DhFF +(ab)L,R, used to confirm the WI given in Eq. (12) for the only-scalar contributions. +The PV-functions derived from the diagram (2) defined as Xh +i are different from Xf +i defined +for three diagrams (1), (3) and (4). In contrast, the equal functions are denoted as follows: +B(i) +0 +≡ B(i)f +0 += B(i)h +0 +, X0 ≡ Xf +0 = Xh +0 , i = 1, 2. +The form factors D(ab)L,R originated from scalar contributions are: +DFhh +(ab)L =−eQH +16π2 +� +gL∗ +a,FhgL +b,Fh × 2Cf +00 +� +7 + ++ +−eQe +16π2(m2 +a − m2 +b) +�� +mbgL∗ +a,FhgR +b,Fh + magR∗ +a,FhgL +b,Fh +� +mF +� +B(1) +0 +− B(2) +0 +� +− gL∗ +a,FhgL +b,Fh +� +m2 +aB(1)f +1 +− m2 +bB(2)f +1 +� +− mambgR∗ +a,FhgR +b,Fh +� +B(1)f +1 +− B(2)f +1 +�� +, +DFhh +(ab)R =DFHH +(ab)L +� +gL +a,Fh ↔ gR +a,Fh, gL +b,Fh ↔ gR +b,Fh +� +, +DhFF +(ab)L = − eQF +16π2 +� +gL∗ +a,FhgL +b,Fh +� +m2 +FCh +0 + (2 − d)Ch +00 − m2 +aXh +1 − m2 +bXh +2 +� ++gR∗ +a,FhgR +b,FhmambX0 + +� +gR∗ +a,FhgL +b,Fhma + gL∗ +a,FhgR +b,Fhmb +� +mFCh +0 +� +, +DhFF +(ab)R =DhFF +(ab)L +� +gL +a,Fh ↔ gR +a,Fh, gL +b,Fh ↔ gR +b,Fh +� +. +(17) +It is noted that the Fhh contributions are the sum of three diagrams (1), (3), and (4), +while the hFF contributions are from the only diagram (2). We emphasize that the electric +charge conversation QF = Qh + Qe is one of the necessary requirements to guarantee the +WI given in Eq. (12), see a detailed proof in appendix C. We can see this crudely from +the necessary condition that div[DhFF +(ab)L] + div[DFhh +(ab)L] ∼ gL∗ +a gL +b (Qe + Qh − QF) = 0 and +div[DhFF +(ab)R] + div[DFhh +(ab)R] ∼ gR∗ +a gR +b (Qe + Qh − QF) = 0. This conclusion supports completely +the only case of electric conversation among the remaining ones mentioned in Ref. [36]. +Regarding Lagrangian (4), which results in four diagrams in the second line of Fig. 1, +diagram (5) gives the following FV V contributions: +CFV V +(ab)L = − eQV +16π2 +� +gR∗ +a,FV gL +b,FV mF +� +3Xf +3 + +1 +2m2 +V +� +− gL∗ +a,FV gR +b,FV mF × mamb +m2 +V +Xf +012 ++ gL∗ +a,FV gL +b,FV ma +� +2(Xf +1 − Xf +3 ) + m2 +FXf +01 + m2 +bXf +2 +m2 +V +� ++gR∗ +a,FV gR +b,FV mb +� +2(Xf +2 − Xf +3 ) + m2 +FXf +02 + m2 +aXf +1 +m2 +V +�� +, +(18) +where Xf +i = Xi(m2 +F, m2 +V , m2 +V ), and +CFV V +(ab)R = − eQV +16π2 +� +gL∗ +a,FV gR +b,FV mF +� +3Xf +3 + +1 +2m2 +V +� +− gR∗ +a,FV gL +b,FV mF × mamb +m2 +V +Xf +012 ++ gR∗ +a,FV gR +b,FV ma +� +2(Xf +1 − Xf +3 ) + m2 +FXf +01 + m2 +bXf +2 +m2 +V +� ++gL∗ +a,FV gL +b,FV mb +� +2(Xf +2 − Xf +3 ) + m2 +FXf +02 + m2 +aXf +1 +m2 +V +�� +. +(19) +Diagram (6) gives V FF contributions: +CV FF +(ab)L = − eQF +16π2 +� +magL∗ +a,FV gL +b,FV +� +2Xv +01 + m2 +F (Xv +1 − Xv +3) + m2 +bXv +2 +m2 +V +� +8 + ++ mbgR∗ +a,FV gR +b,FV +� +2Xv +02 + m2 +F (Xv +2 − Xv +3) + m2 +aXv +1 +m2 +V +� +− gR∗ +a,FV gL +b,FV mF +� +4X0 + m2 +aXv +1 + m2 +bXv +2 − m2 +FXv +3 +m2 +V +� +−gL∗ +a,FV gR +b,FV +mamb +m2 +V +× mF(Xv +12 − Xv +3) +� +, +(20) +where all Xv +i are expressed in terms of PV functions CV FF +0,i,ij = C0,i,ij(m2 +V , m2 +F, m2 +F), and +CV FF +(ab)R = − eQF +16π2 +� +magR∗ +a,FV gR +b,FV +� +2Xv +01 + m2 +F (Xv +1 − Xv +3) + m2 +bXv +2 +m2 +V +� ++ mbgL∗ +a,FV gL +b,FV +� +2Xv +02 + m2 +F (Xv +2 − Xv +3) + m2 +aXv +1 +m2 +V +� +− gL∗ +a,FV gR +b,FV mF +� +4Xv +0 + m2 +aXv +1 + m2 +bXv +2 − m2 +FXv +3 +m2 +V +� +−gR∗ +a,FV gL +b,FV +mbma +m2 +V +× mF(Xv +12 − Xv +3) +� +. +(21) +Finally, using the simple notations gL,R +a +≡ gL,R +a,FV , the formulas of D(ab)L and D(ab)R are +D(78) +(ab)L =D(7) +(ab)L + D(8) +(ab)L += +eQe +16π2(m2 +a − m2 +b) +� � +gL∗ +a gR +b mb + gR∗ +a gL +b ma +� +3mF +� +B(1) +0 +− B(2) +0 +� +− mb +� +magR∗ +a gR +b + mbgL∗ +a gL +b +� +�� +2 + m2 +F + m2 +b +m2 +V +� +B(2)v +1 ++ A0(m2 +V ) + 2m2 +FB(1) +0 +m2 +V ++ 1 +� ++ ma +� +mbgR∗ +a gR +b + magL∗ +a gL +b +� +�� +2 + m2 +F + m2 +a +m2 +V +� +B(1)v +1 ++ A0(m2 +V ) + 2m2 +FB(2) +0 +m2 +V ++ 1 +�� +, +(22) +D(78) +(ab)R =D(78) +(ab)L +� +gL +a ↔ gR +a , gL +b ↔ gR +b +� +. +DFV V +(ab)L = − eQV +16π2 +� +gL∗ +a gL +b +� +2(d − 2)Cf +00 + 2(m2 +a + m2 +b)Xf +3 +− 1 +m2 +V +� +m2 +F(B(1) +0 ++ B(2) +0 +− 2Cf +00) + A0(m2 +V ) + m2 +aB(1)f +1 ++ m2 +bB(2)f +1 +�� ++ gR∗ +a gR +b mamb +� +4Xf +3 + 2Cf +00 +m2 +V +� ++ gR∗ +a gL +b × mamF +� +3Cf +0 − +1 +2m2 +V ++ m2 +bXf +012 +m2 +V +� ++gL∗ +a gR +b × mbmF +� +3Cf +0 − +1 +2m2 +V ++ m2 +aXf +012 +m2 +V +�� +, +DFV V +(ab)R =CFV V +(ab)L +� +gL +a ↔ gR +a , gL +b ↔ gR +a +� +. +(23) +9 + +The remaining formulas of D(ab)L,R from diagram (6) of Fig. 1 are +DV FF +(ab)L = eQF +16π2 +� +gL∗ +a gL +b +� +− 2m2 +FC0 + (d − 2)2Cv +00 + 2m2 +aXv +01 + 2m2 +bXv +02 +− 1 +m2 +V +� +(2 − d)m2 +FCv +00 + A0(m2 +V ) + m2 +F +� +B(1) +0 ++ B(2) +0 +� +−m2 +a +� +B(1)v +0 ++ B(1) +1 +� +− m2 +b +� +B(2)v +0 ++ B(2)v +1 +� ++ m2 +am2 +bX0 +−m2 +F +� +(m2 +a + m2 +b − m2 +F)C0 + m2 +aXv +1 + m2 +bXv +2 +� �� ++gR∗ +a gR +b mamb +� +2X0 − +1 +m2 +V +� +(2 − d)Cv +00 + m2 +FXv +3 − m2 +aXv +1 − m2 +bXv +2 +�� ++gR∗ +a gL +b mamF +m2 +V +� +−B(1)v +1 ++ (2 − d)C00 − m2 +aXv +1 + m2 +b(Xv +3 − Xv +2) +� ++gL∗ +a gR +b mbmF +m2 +V +� +−B(2)v +1 ++ (2 − d)Cv +00 − m2 +bXv +2 + m2 +a(Xv +3 − Xv +1) +�� +, +DV FF +(ab)R =DV FF +(ab)L +� +gL +a ↔ gR +a , gL +b ↔ gR +b +� +. +(24) +We note that all results presented here are crosschecked by FORM package [48], +using intermediate steps given in appendix C. There is a property that CX +(ab)R += +CX +(ab)L +� +gL +a ↔ gR +a , gL +b ↔ gR +b +� +for all X = Fhh, hFF, FV V, V FF. The above results of one- +loop contribution to C(ab)L,R are totally consistent with those introduced in Ref. [36], after +some transformations of notations presented in appendix B. In the limit of m2 +h, m2 +V ≫ +m2 +a, m2 +b, i.e., m2 +a/m2 +B, m2 +b/m2 +h ≃ 0 with B = h, V , we get consistent results with those given +in Refs. [27, 55, 56]. To derive the above results for gauge boson exchanges, we start with +many important features different from those mentioned in Ref. [36], namely: i) we do not +use the typical form of couplings relating with Goldstone bosons going along with the pres- +ence of new gauge bosons, ii) we have to use the massless property of the on-shell photon +q2 = 0, iii) to confirm the WI for all diagrams given in Fig. 1, we need the charge conver- +sation law corresponding to the Lagrangian (1): QF = QV + Qe. Therefore, our calculation +is another independent approach to confirm the result given in Ref. [36]. The details of the +calculation to confirm the WI for all one-loop contributions are given in appendix C. We +remind that our results are derived from the photon couplings listed in the table I, and do +not contain the contributions from the FSV diagrams. In the following, we pay attention to +the possibility of adding the FSV diagrams or the new forms of the photon couplings. +10 + +III. +DISCUSSION ON WI AND PREVIOUS RESULTS +A. +WI to constrain the form of photon couplings +Now we focus on the feature that the WI of the on-shell photon will constrain strongly the +forms of the cubic photon couplings with two physical particles in a renormalized Lagrangian. +Now we consider the existence of the photon couplings type at tree level: +LγXX =eQFAµ � +F1γµF2 + h.c. +� ++ eQhAµ [(h∗ +1∂µh2 − h2∂µh∗ +1) + h.c.] +− +� +eQV AµV ν +1 V λ∗ +2 Γµνλ(p0, p+p−) + h.c. +� ++ +� +gγhV gµνh−QAµV Qν + h.c. +� +, +(25) +where all couplings are more general than those well-known as the standard forms given in +Table I. The last term corresponds to the photon couplings with a scalar h and a gauge +boson V . The above Lagrangian results in the following decays from the heavy particle to +lighter one: i) F2 → F1γ, ii) h2 → h1γ, iii) V2 → V1γ, and iv) V → hγ. The WI for these +decay amplitudes at tree level is Mµ(X1 → X2γ)p0µ = 0 with p0µ being the external photon +momentum. It can be derived that: +• Using the same convention of external momenta given in Fig. 1, we have Mµ(F2 → +F1γ)qµ ∼ (mF2 − mF1)uF2(p2)uF1(p1) = 0, where p0 ≡ −q. Therefore, mF2 = mF1. +This case is automatically satisfied for the tree level AMM amplitude. +• Mµ(h2 → h1γ)p0µ ∼ (p2−p1).(p2+p1) = (m2 +h2 −m2 +h1) = 0, where all on-shell momenta +are incoming the vertex Aµh∗ +1h2, implying that p0 = −(p1 + p2) and p2 +1,2 = m2 +h1,2. The +consequence is mh1 = mh2. +• Mµ(V → hγ)p0µ ∼ εv.p0 = 0, where εv and p0 are the polarization of gauge boson V +and the external momentum of the photon Aµ. Hence the presence of a AhV vertex +does not automatically satisfy the WI. One-loop contributions for all diagrams arising +from this vertex must be checked for the validation of WI. +• Mµ(V1 → V2γ)pµ +0 ∼ εν +1ελ∗ +2 pµ +0Γµνλ(p0, p1, p2) = 0, where ε1,2, and p1,2,0 are the polariza- +tion of the gauge boson V1,2 and the external momentum of the gauge bosons V1,2 and +photon Aµ, respectively. We will use the following properties of the external gauge +bosons Vi(i = 1, 2) and photon: εi.pi = 0, p2 +0 = 0, p2 +i = m2 +Vi, and the momentum +11 + +conversation p0 + p1 + p2 = 0 following notations in table I. After some intermediate +steps of calculation, we have: +Mµ(V1 → V2γ)pµ +0 ∼(p0.ε1) [(p0 − p1).ε∗ +2] + (ε1.ε∗ +2) [(p1 − p2).p0] + (p0.ε∗ +2) [(p2 − p0).ε1] +=(ε1.ε∗ +2) +� +m2 +V2 − m2 +V1 +� += 0. +(26) +Hence, mV1 = mV2 is necessary. +From this, we consider the more general photon +coupling with a gauge boson [49] describing the couplings of a leptoquark field [50] +Γ′ +µνλ(p0, p1, p2) =gµν(kvp0 − p1)λ + gνλ(p1 − p2)µ + gλµ(p2 − kvp0)ν +=Γµνλ(p0, p1, p2) + δkv (gµνp0λ − gλµp0ν) , +(27) +with δkv = kv − 1 showing the deviation from the standard vertex listed in table +I. This may change the one-loop contributions of the diagram (5) in Fig. 1, hence +change the formulas of CFV V +(ab)L,R given in Eqs. (18) and (19), respectively. One can +prove immediately that the vertex deviation +δΓµνλ(p0, p1, p2) ≡ Γ′ +µνλ(p0, p1, p2) − Γµνλ(p0, p1, p2) = δkv (gµνp0λ − gλµp0ν) +(28) +guarantees the WI. The new one-loop contributions arising from δΓ are also satisfied +the WI, see analytic formulas given in Eq. (C36). +Now we start from the point that all results of one loop contributions given from Eq. +(15) to Eq. (24) based on the standard forms of photon couplings given in table I, where +a photon always couples with two identical physical fields. On the other hand, a recent +work [34] assumed the existence of a new photon coupling kind ASV , which may appear in +some BSM, in which the photon couples with one gauge boson V and one scalar S. The +appearance of a boson V or S will generate by itself the one-loop contributions that always +guarantee the WI by the respective set of four diagrams given in Fig. 1. Hence, the two +FSV diagrams must give contributions satisfying the WI themselves, namely +DFSV +(ab)L + maCFSV +(ab)L + mbCFSV +(ab)R = DFSV +(ab)R + maCFSV +(ab)R + mbCFSV +(ab)L = 0. +(29) +As a result, the divergent parts for both L and R parts give: +0 =gγSV +� +2gL∗ +ahgL +bV mF − gL∗ +ahgR +bV mb − gR∗ +ah gL +bV ma +� +12 + +=gγSV +� +2gR∗ +ah gR +bV mF − gR∗ +ah gL +bV mb − gL∗ +ahgR +bV ma +� +. +(30) +Considering the case of gγSV ̸= 0. Then, all quantities gL +ah, gR +bh, gL +bV , and gR +bV are zero if at +least one of them is zero. More strictly, we require that the two Eqs. (29) must be hold +for both divergent and finite parts arising from D(ab)L,R and C(ab)L,R given in appendix D. +Consequently, gγSV = 0, i.e., the FSV diagram type does not satisfy the WI. +Regarding to the vertex deviation of the AV V couplings defined in Eq. (28), the new +one-loop contributions relating to CFV V +(ab)L,R and DFV V +(ab)L,R are shown in Eq. (C36) of appendix +C. Our results are consistent with previous works [49, 50]. Although they satisfy the WI, +they contain divergent parts, for example +div +� +−δCFV V +L +� += δkveQV +32π2m2 +V +� +gL∗ +a gL +b ma + gR∗ +a gR +b mb − 2gR∗ +a gL +b mF +� +. +(31) +Hence, δkv = 0 is equivalent to the renormalizable condition of the theory, see a more +detailed explanation in Ref. [49]. This confirms that the AV V coupling listed in table I +is still valid for a general UV-complete model. Consequently, δCFV V +L += 0, implying that +the results of CFV V +(ab)L,R given in Eqs. (18) and (19) are unchanged for many renormalizable +theories. +B. +Discussions on previous results +It is easy to derive that C(ab)L,R = σL,R corresponding to the notations given in Ref. +[36], see a detailed explanation in appendix B. This confirms a perfect consistence of the +two results obtaining from different original assumptions that we have indicted above. In +addition, these results are also consistent with those given in Ref. [27] in the limit of heavy +boson masses in the loops, which are very useful for studying the correlations of AMM and +cLFV decays. +In some BSM, SM light quark may play role of the light fermions u, d ≡ F in the Yukawa +couplings [29], hence the condition m2 +F ≫ m2 +a, m2 +b is not held. But numerical illustrations +[39] to investigate cLFV decays eb → eaγ with very light neutrinos show that the case +of m2 +F ≪ m2 +a are also valid for approximation formulas with m2 +a = m2 +b = 0, provided +m2 +a, m2 +b ≪ m2 +h, m2 +V . +An analytic approximation to explain this result was given in, for +example Ref. [57]. +13 + +For analytic formulas of cLFV and aea introduced in Ref. [28], They can be changed into +the form of PV-functions consistent with our results. An exceptional case mentioned there +is the couplings of a double charged boson with two identical leptons. For example, the +Lagrangian containing couplings of a doubly charged Higgs boson is [28]: +Lint = gij +s3φ++ℓC +i ℓj + gij +p3φ++γ5ℓC +i ℓj + h.c., +(32) +where we can identify that gR +a,Fh = gij +s3 +gij +p3 and gR +a,Fh = gij +s3 −gij +p3. But the Feynman rules for +the a vertex ℓC +i ℓjφ++ containing two identical leptons gives an extra factor 2, implying that +C(ab)L,R given in Eqs. (15) and (16) must be added a factor 4. Instead of many particular +formulas to calculate one-loop contributions relating to different charged particles, the one- +loop results for (g − 2)ea and eb → eaγ decays can be generalized for aea with an arbitrary +electric charge QF of a new fermion and the boson with QB = QF − Qe with B = h, V . +Namely, the aea formulas are +aea(h) =Qhma +16π2 +� 1 +0 +dx × x(x − 1) +� +2Re[gRL]mF + (gLL + gRR)max +� +(1 − x)m2 +F + x [m2 +h + m2 +a(x − 1)] ++ QFma +16π2 +� 1 +0 +dx × x2 � +−2gRL[gRL]mF + (gLL + gRR)ma(x − 1) +� +(1 − x)m2 +h + x [m2 +F + m2 +a(x − 1)] +, +(33) +aea(V ) = − QV ma +16π2m2 +V +� 1 +0 +dx × +�Re[gRL]mF [m2 +F(x − 1) + m2 +V x(6x − 1) + m2 +ax(3 − 5x + 2x2)] +(1 − x)m2 +F + x [m2 +V + m2 +a(x − 1)] +−ma(gLL + gRR) [m2 +F(2 − 3x + x2) + m2 +V 2x(x + 1) + m2 +ax(x − 1)] +(1 − x)m2 +F + x [m2 +V + m2 +a(x − 1)] +� ++ QFma +16π2m2 +V +� 1 +0 +dx +�2gRL[gRL]mFx [m2 +Fx − 4m2 +V (1 − x) + m2 +ax(2x − 1)] +(1 − x)m2 +V + x [m2 +F + m2 +a(x − 1)] ++(gLL + gRR)max [m2 +Fx(1 + x) + 2m2 +V (2 − 3x + x2) + m2 +ax(x − 1)] +(1 − x)m2 +V + x [m2 +F + m2 +a(x − 1)] +� +, +(34) +where gRL = gR∗ +a,FBgL +a,FB, gLL = gL∗ +a,FBgL +a,FB, and gRR = gR∗ +a,FBgR +a,FB with B = h, V . The +coupling identifications are gR +a,Fh = gaa +sk + gaa +pk and gR +a,Fh = gaa +sk − gaa +pk for k = 1, 2, 3 relating to +neutral, singly, and doubly charged Higgs bosons. Similarly for the gauge bosons, gR +a,FV = +gaa +vk + gaa +ak and gR +a,FV = gaa +vk − gaa +ak for QV = 1, 0, −1, 2 corresponding k = 1, 2, 3, 4. The two +formulas (33) and (34) are derived by inserting the PV functions given in appendix A in the +limit p2 +1 = p2 +2 = m2 +a into C(ab)L,R. We have checked that our results are consistent with all +HFF, FHH, and V FF contributions relating to the diagrams (1), (2), and (6) in Fig. 1, +respectively. For the one-loop FV V contributions arising from the diagram (5), there is a +14 + +difference between our result and that in Ref. [28], namely +δ(aea)(FV V ) = QV mamF +16π2m2 +V +(|gaa +vk|2 − |gaa +ak|2) +� 1 +0 +dx(2x + 1) = QV mamF +8π2m2 +V +(|gaa +vk|2 − |gaa +ak|2). +It shows that the two results are consistent if gaa +vk = ±gaa +ak,i.e., gL +a,FBgR +a,FB = 0, which appears +in many BSM such as the SM, 3-3-1 models,... We also see that the FV V contribution to +aea of the doubly gauge boson given in Ref. [28] has an opposite sign with our result. +We note that our results are also valid as the exact solutions for studying the AMM and +eb → eaγ decay in BSM consisting of very light bosons mB ≪ m2 +a, m2 +b such as an axion-like +particle (ALP) [58, 59], or a new scalar singlet [60]. +IV. +CONCLUSION +Using the unitary gauge, we confirm the exact results of analytic formulas in terms of PV +functions for one-loop contributions to the cLFV decay rates eb → eaγ given in Ref. [36], +which are also applicable to compute the AMM of charged leptons. These results are con- +sistent with those given in Ref. [27] in the limit of heavy bosons mB ≫ ma, mb. The general +expressions in terms of PV-functions are very convenient to change into available forms. Our +calculations here are in many new feature as follows. Our calculation is independent with +the Goldstone boson couplings of the new gauge bosons. The Ward Identity of the external +photon constrains allows only the couplings of photon with two identical physical particles, +as given in table I. At tree-level, the ASV couplings do not satisfy the WI if εv.p0 ̸= 0, +where εv and p0 are the polarization of gauge boson V and the external momentum of the +photon, respectively. The one-loop FSV contributions arising from this vertex type to cLFV +amplitudes and AMM do violate the WI. Therefore, the results given in Refs. [27, 36] are +valid in all renormalizable BSM respecting the WI. They are still applied for other similar +decays of quarks q → q′γ. The photon-scalar-vector ASV vertex does not appear in BSM +satisfying the WI. Our conclusion is very useful for constructing loop calculations relating +to photon couplings, where only the vertex types listed in Table I are valid. +Acknowledgments +This research is funded by the Vietnam National Foundation for Science and Technology +Development (NAFOSTED) under the grant number 103.01-2019.387. L. T. Hue is thankful +15 + +to Van Lang University. +Appendix A: PV functions for one loop contributions defined by LoopTools +1. +General notations +The PV-functions used here were listed in Ref. [39], namely +A0(m2) = (2πµ)4−d +iπ2 +� +ddk +k2 − m2 + iδ, +B{0,µ,µν}(p2 +i , M 2 +1, M 2 +2) = (2πµ)4−d +iπ2 +� ddk × {1, kµ, kµkν} +D0Di +, i = 1, 2, +C0,µ,µν = (2πµ)4−d +iπ2 +� ddk{1, kµ, kµkν} +D0D1D2 +, +Cµ = (−p1µ) C1 + (−p2µ) C2, +Cµν = gµνC00 + p1µp1νC11 + p2µp2νC22 + (p1µp2ν + p2µp1ν)C12, +(A1) +where D0 ≡ k2 − M 2 +1 + iδ, Di ≡ (k − pi)2 − M 2 +2 + iδ, C0,µ,µν = C0,µ,µν(p2 +1, 0, p2 +2; M 2 +1, M 2 +2, M 2 +2), +µ is an arbitrary mass parameter introduced via dimensional regularization [61]. In this +work, we discuss only the case of external photon q2 = (p2 − p1)2 = 0. The scalar functions +A0, B0, C0, C00, Ci, Cij (i, j = 1, 2) are well-known PV functions, which are consistent with +those defined by LoopTools [40]. The well-known relations are: +B(i) +0 +≡ B(i) +0 (p2 +i ; M 2 +1, M 2 +2) = B(i) +0 (p2 +i ; M 2 +2, M 2 +1), +B(i) +1 +≡ B(i) +1 (p2 +i ; M 2 +1, M 2 +2) = − 1 +2p2 +i +� +A0(M 2 +2) − A0(M 2 +1) + fiB(i) +0 +� +, +(A2) +where fi = p2 +i + M 2 +2 − M 2 +1. The scalar functions A0, B0, C0 can be calculated using the +techniques of [38]. Other PV functions needed in this work are +B0,µ,µν(M2) = (2πµ)4−d +iπ2 +� ddk {1, kµ, kµkν} +D1D2 +. +(A3) +For simplicity, we define the following notations appearing in many important formulas: +X0 ≡ C0 + C1 + C2, +X1 ≡ C11 + C12 + C1, +X2 ≡ C12 + C22 + C2, +16 + +X3 ≡ C1 + C2 = X0 − C0, +X012 ≡ X0 + X1 + X2, Xij = Xi + Xj. +(A4) +Depending on the form of the PV-functions, we have Xf +i += Xi(m2 +F, m2 +B, m2 +B), Xh +i +∼ +Xi(m2 +B, m2 +F, m2 +F), and Xv +i ∼ Xi(m2 +V , m2 +F, m2 +F), corresponds to the diagrams FBB, HFF, +and V FF with B = h, V . +2. +p2 +1 ̸= p2 +2 ̸= 0 and p2 +1, p2 +2 ̸= 0. +From the definitions of PV-functions given in Eq. (A1), it can be proved that: +B(0) +0 +≡ B(0) +0 (M2) ≡ B0(0; M2, M2) = CUV − ln(M 2 +2) + O(ϵ), +A0(M) = M 2 � +B(0) +0 (M) + 1 +� +, +(A5) +Bµ(M2) = 1 +2B(0) +0 (pµ +1 + pµ +2), +(A6) +Bµν(M2) = gµν +2 M 2 +2 +� +B(0) +0 ++ 1 +� ++ 1 +6B(0) +0 +(2pµ +1pν +1 + pµ +1pν +2 + pµ +2pν +1 + 2pµ +2pν +2) , +C00 = 1 +4 +� +2M 2 +2C0 + (M 2 +2 − M 2 +1 + m2 +a) B(1) +0 +− (M 2 +2 − M 2 +1 + m2 +b) B(2) +0 +m2 +a − m2 +b ++ 1 +� +, +(A7) +where CUV is defined as the divergent part of the PV functions when D → 4, CUV = +1/ϵ − γE + log(4πµ2) with γE being Euler’s constant and D = 4 − 2ϵ. It is well-known that +the PV-functions having non-zero divergent parts are: +div +� +B(0) +0 +� += div +� +B(1) +0 +� += div +� +B(2) +0 +� += −2div +� +B(1) +1 +� += −2div +� +B(2) +1 +� += 4div [C00] = CUV , +div [A0(M)] = M 2CUV . +(A8) +As mentioned in Ref. [39], we can derive all formulas of Ci, and Cij as functions of A0, B(i) +0 , +and C0 consistent with Ref. [39], using the following relations: +2m2 +aC1 + (m2 +a + m2 +b)C2 = −faC0 − B(0) +0 ++ B(2) +0 , +(m2 +a + m2 +b)C1 + 2m2 +bC2 = −fbC0 − B(0) +0 ++ B(1) +0 , +2C00 + 2m2 +aC11 + (m2 +a + m2 +b)C12 = 1 +2B(0) +0 +− faC1, +2m2 +aC12 + (m2 +a + m2 +b)C22 = 1 +2B(0) +0 ++ B(2) +1 +− faC2 +2C00 + (m2 +a + m2 +b)C12 + 2m2 +bC22 = 1 +2B(0) +0 +− fbC2, +17 + +(m2 +a + m2 +b)C11 + 2m2 +bC12 = 1 +2B(0) +0 ++ B(1) +1 +− fbC1, +4C00 − 1 +2 + m2 +aC11 + (m2 +a + m2 +b)C12 + m2 +bC22 = B(0) +0 ++ M 2 +1C0, +(A9) +where fa,b = M 2 +2 −M 2 +1 +m2 +a,b, and C12 = C21 is used. In this work, we need just combinations +of these PV-functions for our immediate steps. In particular, we can prove that: +X0 = −B(1) +0 +− B(2) +0 +m2 +a − m2 +b +, +X12 = −B(1) +1 +− B(2) +1 +m2 +a − m2 +b += A0(M 2 +1) − A0(M 2 +2) +2m2 +am2 +b ++ (M 2 +1 − M 2 +2) +2(m2 +a − m2 +b) +� +B(1) +0 +m2 +a +− B(2) +0 +m2 +b +� +− 1 +2X0, +m2 +aB(1) +1 +− m2 +bB(2) +1 += −1 +2 +�� +m2 +a + M 2 +2 − M 2 +1 +� +B(1) +0 +− +� +m2 +b + M 2 +2 − M 2 +1 +� +B(2) +0 +� +, +b1 ≡ m2 +aB(1) +1 +− m2 +bB(2) +1 +(m2 +a − m2 +b) += −(2C00 + m2 +aX1 + m2 +bX2), +(2 − d)C00 + M 2 +2C0 = −2C00 + 1 +2 + M 2 +2C0 += −(m2 +a + M 2 +1 − M 2 +2) B(1) +0 +− (m2 +b + M 2 +1 − M 2 +2) B(2) +0 +2(m2 +a − m2 +b) += b1 + (M 2 +2 − M 2 +1)X0, +(A10) +where A0(M 2 +2) = M 2 +2(B(0) +0 ++ 1) and A0(M 2 +1) = M 2 +1(B(0) +0 ++ 1 + ln(M 2 +2/M 2 +1)). +It was proved previously, for example [39], that +B0(p2; M 2 +1, M 2 +2) = B0(p2; M 2 +2, M 2 +1) = CUV − ln(M 2 +2) + 2 − +� +σ=± +(1 − 1 +xσ +) ln (1 − xσ) , +C0(m2 +a, 0, m2 +b; M 2 +1, M 2 +2, M 2 +2) = − +1 +m2 +a − m2 +b +� +σ=± +[Li2(yaσ) − Li2(ybσ)] , +(A11) +where p = pa, pb; and +x± = +1 +2M 2 +2 +� +(M 2 +2 − M 2 +1 + p2) ± +� +(M 2 +2 − M 2 +1 + p2)2 − 4M 2 +2p2 +� +, +ya± = +1 +2M 2 +2 +� +(M 2 +2 − M 2 +1 + m2 +a) ± Λ +� +, +yb± = xa± [b → a] +(A12) +with Λ = (M 4 +1 + M 4 +2 + m4 +a − 2M 2 +1M 2 +2 − 2M 2 +1m2 +a − 2M 2 +2m2 +a)1/2. The above formula of C0 is +also consistent with that introduced in loop-induced decay amplitude of h → Zγ [62]. +18 + +3. +m2 +a = p2 +a = p2 +b ̸= 0 +Formulas for AMM in Ref. +[34] require that analytic formulas of PV functions with +mb = ma. It seems that the results of PV-functions listed in Ref. [39] are not valid. But +the limit mb = ma can be derived mathematically. For example, the result of C0 given in +Eq. (A11) leads to a consequence that +C0(m2 +a, 0, m2 +a; M 2 +1, M 2 +2, M 2 +2) = +lim +mb→ma C0(m2 +a, 0, m2 +b; M 2 +1, M 2 +2, M 2 +2) += − +∂ +∂(m2 +a) +� +σ=± +Li2(yaσ) = +� +σ=± +y′ +aσ ln(1 − yaσ) +yaσ += +� +σ=± +ln(1 − yaσ) +2M 2 +2yaσ +× +� +1 − σ × M 2 +1 + M 2 +2 − m2 +a +Λ +� +, +(A13) +where f ′ ≡ ∂f/(∂m2 +a) denotes a well-known derivative notation. In addition, B(1) +0 += B(2) +0 +and B(1) +1 += B(2) +1 +is automatically satisfied. Many formulas containing (m2 +a − m2 +b) in the +denominators corresponding a derivative in the limit ma → mb: +X0 = −B(1)′ +0 += +� +σ=± +y′ +aσ [yaσ + ln(1 − yaσ)] +y2 +aσ +, +X12 = −B(1)′ +1 +. . . . , +(A14) +In this way, we can confirm all results introduced in Ref. [34]. There is another way to +calculate form factors, using the Feynman trick: +1 +D0D1D2 += Γ(3) +� 1 +0 +dx dy dz δ(1 − x − y − z) +D3 +, +(A15) +where +D = [k − (yp1 + zp2)]2 − M 2 + iδ, +M 2 = y(y + z − 1)p2 +1 + z(y + z − 1)p2 +2 + xM 2 +1 + (1 − x)M 2 +2. +(A16) +With M 2 +0 = (p2 +2 − p2 +1)xy − x(1 − x)p2 +2 + xM 2 +1 + (1 − x)M 2 +2, the PV functions are: +C{0,1,2,11,22,12} = − +� 1 +0 +dx +� 1−x +0 +dy {1, −y, −(1 − x − y), y2, (1 − x − y)y, (1 − x − y)2} +M 2 +0 +, +X0,1,2,3 = − +� 1 +0 +dx +� 1−x +0 +dy × {x, −xy, −x(1 − x − y), −(1 − x)} +M 2 +0 +. +(A17) +19 + +The expressions of Xi in Eq. (A17) are very convenient for the case of (g − 2) anomaly, +where p2 +1 = p2 +2 = m2 +a results in M 2 +0 = −x(1−x)m2 +a +xM 2 +1 +(1−x)M 2 +2, which is independent +with y. Consequently, the +X0,1,2,3 = − +� 1 +0 +dx{x(1 − x), −x(1 − x)2/2, −x(1 − x)2/2, −(1 − x)2} +M 2 +0 += − +� 1 +0 +dx{x(1 − x), −(1 − x)x2/2, −(1−)x2/2, −x2} +M 2 +0 +, +(A18) +Formulas of Eq. (A18) are enough to check the consistence between our results with those +of (g − 2) anomalies and cLFV amplitudes mentioned in ref. [28]. Using the second line of +Eq. (A18), we can write the general formulas of aµ as shown in Eqs. (33) and (34). +In deed, all integrals in Eqs. (33) and (34) can be solved analytically. Starting from the +general formulas of M 2 +0 as a functions of x: M 2 +0(x) = m2 +a(x − x+)(x − x−) corresponding +to the two solutions x±. All numerators in Eqs. (33) and (34) are always written in the +following forms: +ax2 + bx2 + c = a1M 2 +0 + b1 +dM 2 +0 +dx + c1. +(A19) +The consequence is +� 1 +0 +dx × ax2 + bx2 + c +M 2 +0 += a1 + b1 ln M 2 +1 +M 2 +2 ++ c1 +√ +Λ +ln +�(1 − x−)x+ +(1 − x+)x− +� +. +(A20) +The result in this way must be consistent with those discussed in Ref. [34], hence we do not +show precisely here. +4. +p2 +a = p2 +b = 0 +Results for the case of p2 +a = p2 +b = 0 were provided in Ref. [36], namely +C0 = a = +M 2 +1 − M 2 +2 + M 2 +1 ln +� +M2 +2 +M2 +1 +� +(M 2 +1 − M 2 +2)2 +, +C1 = C2 = c = − +3M 4 +1 − 4M 2 +1M 2 +2 + M 4 +2 + 2M 4 +1 ln +� +M2 +2 +M2 +1 +� +4(M 2 +1 − M 2 +2)3 +, +C11 = C22 = 2C12 = d = +11M 6 +1 − 18M 4 +1M 2 +2 + 9M 2 +1M 4 +2 − 2M 6 +2 + 6M 6 +1 ln +� +M2 +2 +M2 +1 +� +18(M 2 +1 − M 2 +2)4 +. +(A21) +20 + +This approximate formulas of PV functions give results consistent with those given in Ref. +[27], namely +fh(x) = 2˜gh(x) = x2 − 1 − 2x log x +4(x − 1)3 +, +gh(x) = x − 1 − log x +2(x − 1)2 +, +˜fh(x) = 2x3 + 3x2 − 6x + 1 − 6x2 log x +24(x − 1)4 +, +(A22) +fV (x) = x3 − 12x2 + 15x − 4 + 6x2 log x +4(x − 1)3 +, +gV (x) = x2 − 5x + 4 + 3x log x +2(x − 1)2 +, +˜fV (x) = −4x4 + 49x3 − 78x2 + 43x − 10 − 18x3 log x +24(x − 1)4 +, +˜gV (x) = −3(x3 − 6x2 + 7x − 2 + 2x2 log x) +8(x − 1)3 +, +where x ≡ m2 +F/m2 +B. The diagrams FBB and BFF corresponds to different identifications +that {M1, M2} = {mF, mB} or and {M1, M2} = {mB, mF}. +Appendix B: Notations in Ref. [36] +Corresponding to the two one-loop diagram classes FV V and V FF, we have the following +equivalence between two classes of notations +{a, c1, c2, d1, d2, f, g} ≡ {C0, C2, C1, C22, C11, C12, C00}B , +� +¯a, −¯c1, −¯c2, ¯d1, ¯d2, ¯f, ¯g +� +≡ {C0, C2, C1, C11, C22, C12, C00}f , +where B = h, V are gauge bosons in the loop. In addtion, the diferent notations in the +definitions of the one-loop integrals given in Eq. (A1), we have {m1, m2} ≡ {mb, ma} while +{p1, p2} ≡ {−p2, −p1} and {p1, p2} ≡ {p2, p1} for the diagrams V FF and FV V respectively. +The couplings in the Yukawa Lagrangians of physical bosons are L1 ≡ gL +b , R1 ≡ gR +b , L2 ≡ gL +a , +and R2 ≡ ga +R, which result in the following equvalences: λ ≡ gL∗ +a gL +b = gLL, ρ ≡ gR∗ +a gR +b = gRR, +ζ ≡ gL∗ +a gR +b = gLR, and v ≡ gR∗ +a gL +b = gRL. As a result, we can identify that: +k1 = mbXB +2 , k2 = maXB +1 , k3 = mF(c1 + c2) = mFXB +3 , +¯k1 = mbXf +2 , ¯k2 = mbXf +1 , k3 = −mFXf +3 . +(B1) +21 + +For a gauge boson Bµ, the one-loop form factors relate to the following notations: +y1 = mb +� +2Xf +02 + m2 +F(Xf +2 − Xf +3 ) + m2 +aXf +1 +m2 +B +� +, y2 = ma +� +2Xf +01 + m2 +F(Xf +1 − Xf +3 ) + m2 +bX2 +m2 +B +� +, +y3 = mF +� +−4Xf +0 + m2 +FXf +3 + m2 +aXf +1 + m2 +bXf +2 +m2 +B +� +, y4 = −mambmF(Xf +12 − Xf +3 ) +m2 +B +, +(B2) +and +¯y1 = mb +� +2(Xf +2 − Xf +3 ) + m2 +FXf +02 + m2 +aXf +1 +m2 +B +� +, ¯y2 = ma +� +2(Xf +1 − Xf +3 ) + m2 +FXf +01 + m2 +bXf +2 +m2 +B +� +, +¯y3 = mF +� +4Xf +3 + −m2 +FX0 − m2 +aX1 − m2 +bX2 +m2 +B +� +, ¯y4 = −mambmF +m2 +B +X012. +(B3) +Appendix C: Important steps to derive C(ab)L,R and D(ab)L,R by hand +The notations for calculating the amplitude corresponding to all diagrams of both Higgs +and gauge boson exchanges are shown in Fig. 2. Although all the internal momenta have +(1) +eb +ea +γ +p1 +q +p2 +k1 +k2 +k +(2) +F +h+ +eb +ea +ea +eb +eb +(3) +k +k1 +(4) +ea +ea +eb +k +k2 +FIG. 2: Momneta notations to derive the one-loop contributions +opposite signgs with those denoted following LoopTools, the PV-functions are defined with +the same values. The relations relevant with momenta are: +ki = k − pi, p2 +1 = m2 +a, p2 = q + p1, p2 +2 = m2 +b, q2 = 0, +q.ε∗ = 0, p1.ε∗ = p2.ε∗, +(C1) +Only four diagrams (1), (2), (5), and (6) in Fig. 1 give non zero contributions to C(ab)L,R, +hence we firstly derive C(ab)L,R as the factors of (2p1.ε∗) in the amplitudes arising from +these diagrams. For convenience in detailed calculations, we use simple notations for all +the couplings factors gaL,R +FB +→ gL,R +a +. For integrals containing divergences, we use the regular +dimensional regularization defined by the following replacement: +� +d4k +(2π)4 → +i +16π2 × (2πµ)4−d +iπ2 +� +ddk ≡ +� +Dk. +22 + +The final results now are written in terms of the PV functions. In many intermediate steps, +we use many results for products of gamma matrices in the dimension d [51], namely +γµγµ = d, +γµγνγµ = (2 − d)γν → γµ/pγµ = (2 − d)/p, +γµγνγργµ = 4gνρ + (d − 4)γνγρ → γµ/p1/p2γµ = 4p1.p2 + (d − 4)/p1/p2, +γµγνγργσγµ = −2γσγργν − (d − 4)γνγργσ → γµ/p1/p2/p3γµ = −2/p3/p2/p1 − (d − 4)/p1/p2/p3, . . . +1. +Scalar contributions +We list here 8 formulas of amplitudes corresponding to 8 particular diagrams shown in +Fig. 1. Namely, for three diagrams (1), (3), and (4) we have +iM1 = − eQH +� +Dk × ua[gR +a +∗PL + gL +a +∗PR](mF + /k) +D0D1D2 +[gL +b PL + gR +b PR]ub × (2k1.ε∗), +(C2) +iM3 = −eQe +m2 +a − m2 +b +� +Dk × ua[gR +a +∗PL + gL +a +∗PR](mF + /k) +D0D1 +[gL +b PL + gR +b PR](mb + /p1)/ε∗ub, +(C3) +iM4 = +eQe +m2 +a − m2 +b +� +Dk × ua/ε∗(ma + /p2)[gR +a +∗PL + gL +a +∗PR](mF + /k) +D0D2 +[gL +b PL + gR +b PR]ub, +(C4) +where D0 = k2 − m2 +F and Di = k2 +i − m2 +h. The amplitude for the diagram (2) is: +iM2 = −eQF +� +Dk × ua[gR +a +∗PL + gL +a +∗PR](mF − /k1)/ε∗(mF − /k2) +D0D1D2 +[gL +b PL + gR +b PR]ub, +(C5) +where D0 = k2 − m2 +h and Di = k2 +i − m2 +F. +In the next calculation, we use the following simple notations: +gLL ≡ gL∗ +a gL +b , gRR ≡ gR∗ +a gR +b , gRL ≡ gR∗ +a gL +b , gLR ≡ gL∗ +a gR +b , +A1 = gL∗ +a gR +b PR + gR∗ +a gL +b PL, A2 = gL∗ +a gL +b PL + gR∗ +a gR +b PR, +(C6) +where gL,R +a +≡ gL,R +a,Fh and gL,R +b +≡ gL,R +b,Fh without any confusions with the gauge boson couplings +gL,R +a,FV . It is not hard to write all amplitudes in terms of PV-functions as follows: +M1 =−eQH +16π2 ua +� +−2p1.ε∗ [A1] mFX0 + +� +2Cf +00/ε∗ + +� +Xf +1 /p1 + Xf +2 /p2 +� +(2p1.ε∗) +� +[A2] +� +ub, (C7) +M3 =−eQe +16π2 × +ua[gR +a +∗PL + gL +a +∗PR](mFB(1) +0 +− B(1)f +1 +/p1)[gL +b PL + gR +b PR](mb + /p1)/ε∗ub +(m2 +a − m2 +b) +, +(C8) +M4 = eQe +16π2 × +ua/ε∗(ma + /p2)[gR +a +∗PL + gL +a +∗PR](mFB(2) +0 +− B(2)f +1 +/p2)[gL +b PL + gR +b PR]ub +(m2 +a − m2 +b) +, +(C9) +23 + +and +M2 = − eQF +� +Dk × ua +� +m2 +F/ε∗ + /k1/ε∗/k2 +� +[A2]ub +− eQF(−1)mF +� +Dk × ua +� +2k.ε∗ − /p1/ε∗ − /ε∗/p2 +� +[A1]ub +=−eQF +16π2 ua +�� +m2 +FC0 + (2 − d)C00 +� +/ε∗ + (C11 + C1)/p1/ε∗/p1 + (C22 + C2)/p2/ε∗/p2 ++(X0 + C12)/p1/ε∗/p2 + C12/p2/ε∗/p1 +� +× [A2]ub +− eQFmF +16π2 ua +� +(2p1.ε∗)(C1 + C2) + +� +/p1/ε∗ + /ε∗/p2 +� +C0 +� +[A1]ub. +(C10) +The validation of the WI given in Eq. (12) implies whether f WI +L += 0 is correct with: +f WI +L +≡D(ab)L + maC(ab)L + mbC(ab)R +=gLL +� +�� +Qe +� +m2 +aB(1) +1 +− m2 +bB(2) +1 +�f +m2 +a − m2 +b +− +�1 +2 − 2Ch +00 + m2 +FCh +0 +� +Qf +−Qh +� +m2 +aX1 + m2 +bX2 + 2C00 +�f� ++ gRRmamb +� +� +� +Qe +� +B(1) +1 +− B(2) +1 +�f +m2 +a − m2 +b +− QfXh +012 − QhXf +12 +� +�� . +(C11) +We have used many formulas listed in Eqs. (A2) and (A10) to show that +0 = Xf +12 + Xh +12 + X0 → Xh +012 = −Xf +12, +bf +1 = − +� +m2 +aX1 + m2 +bX2 + 2C00 +�f = 1 +2 − 2Ch +00 + m2 +FCh +0 . +(C12) +Finally, the electric charge conservation QF = Qe + Qh must be satisfied so that Eq. (C11) +resulting in f WI +L += 0. On the other word, the WI is valid for only one-loop Higgs contribu- +tions arising from the set of four diagrams (1)-(4) in Fig. 1. +2. +Vector contributions +To calculate the one-loop contributions from gauge boson exchanges corresponding to +Lagrangian (4), we denote gL,R +a +≡ gL,R +a,FV and gL,R +b +≡ gL,R +b,FV then use the notations given in +Eq. (C6). The amplitudes relevant with gauge boson exchanges are: +iM5 = +� +Dk × uaiγα[gL∗ +a PL + gR∗ +a PR]i(mF + /k) +D0 +iγβ[gL +b PL + gR +b PR]ub +24 + +× −i +D1 +� +gαα′ − kα +1 kα′ +1 +m2 +V +� +[−ieQV Γµα′β′ (−q, k1, −k2) ε∗µ] −i +D2 +� +gββ′ − kβ +2 kβ′ +2 +m2 +V +� +=eQV +� +d4k +(2π)4uaγα[gL∗ +a PL + gR∗ +a PR](mF + /k) +D0D1D2 +γβ[gL +b PL + gR +b PR]ub +× [Γµα′β′ (−q, k1, −k2) ε∗µ] +� +gαα′ − kα +1 kα′ +1 +m2 +V +� � +gββ′ − kβ +2 kβ′ +2 +m2 +V +� +, +(C13) +iM7 = +eQe +m2 +a − m2 +b +� +Dk × +1 +D0D1 +× +� +gαβ − kα +1 kβ +1 +m2 +V +� +× uaγα[gL∗ +a PL + gR∗ +a PR](mF + /k)γβ[gL +b PL + gR +b PR] +� +mb + /p1 +� +/ε∗ub, +(C14) +iM8 = − +eQe +m2 +a − m2 +b +� +Dk × +1 +D0D2 +× +� +gαβ − kα +2 kβ +2 +m2 +V +� +× ua/ε∗ � +ma + /p2 +� +γα[gL∗ +a PL + gR∗ +a PR](mF + /k)γβ[gL +b PL + gR +b PR]ub, +(C15) +where D0 = k2 − m2 +F, Di = k2 +i − m2 +V , and +Γµα′β′ (−q, k1, −k2) = gα′β′ (k1 + k2)µ + gβ′µ (−k2 + q)α′ + gµα′ (−q − k1)β′ . +(C16) +The amplitude for the diagram (6) is: +iM6 =eQF +� +Dk × +1 +D0D1D2 +× +� +gαβ − kαkβ +m2 +V +� +× uaγα[gL∗ +a PL + gR∗ +a PR](mF − /k1)/ε∗(mF − /k2)γβ[gL +b PL + gR +b PR]ub, +(C17) +where D0 = k2 − m2 +V and Di = k2 +i − m2 +F. +Considering diagram (7), we have: +iM7 = +eQe +m2 +a − m2 +b +� +Dk × +ua [γαγβmF [A1] + γα/kγβ [A2]] +� +mb + /p1 +� +/ε∗ub +D0D1 +× +� +gαβ − kα +1 kβ +1 +m2 +V +� += +eQe +m2 +a − m2 +b +� +Dk × +1 +D0D1 +× ua +� +mF +� +d − k2 +1 +m2 +V +� +[A1] + +� +(2 − d)/k − /k1/k/k1 +m2 +V +� +[A2] +� � +mb + /p1 +� +/ε∗ub += +ieQe +16π2(m2 +a − m2 +b)ua +� +mF [A1] +� +(d − 1)B(1) +0 +− A0(m2 +F) +m2 +V +� ++ma +�� +−(2 − d) + m2 +F + m2 +a +m2 +V +� +B(1) +1 ++ A0(m2 +V ) + 2m2 +FB(1) +0 +m2 +V +� +[A2] +� +× +� +mb + /p1 +� +/ε∗ub, +(C18) +25 + +where we have used the following results +/k1/k/k1 = +� +D0 + m2 +F +� /k − 2 +� +D0 + m2 +F +� +/p1 + /p1/k/p1, +� +d4k +(2π)4 × kµ +D1 += A0(m2 +V )p1µ. +Then the one-loop contribution form factors from diagram (7) are: +D(ab)L,7 = +eQe +16π2(m2 +a − m2 +b) +�� +gRLma + gLRmb +� +mF +� +(d − 1)B(1) +0 +− A0(m2 +F) +m2 +V +� ++ ma +� +magLL + mbgRR� +�� +−(2 − d) + m2 +F + m2 +a +m2 +V +� +B(1) +1 ++ A0(m2 +V ) + 2m2 +FB(1) +0 +m2 +V +�� +, +D(ab)R,7 =D(ab)L,7 +� +gL +a ↔ gR +a , gL +b ↔ gR +b +� +. +(C19) +The same calculation for diagram (8) gives the following one-loop contribution form factor: +D(ab)L,8 = − +eQe +16π2(m2 +a − m2 +b) +�� +gRLma + gLRmb +� +mF +� +(d − 1)B(2) +0 +− A0(m2 +F) +m2 +V +� ++ mb +� +magRR + mbgLL� +�� +−(2 − d) + m2 +F + m2 +b +m2 +V +� +B(2) +1 ++ A0(m2 +V ) + 2m2 +FB(2) +0 +m2 +V +�� +, +D(ab)R,8 = D(ab)L,8 +� +gL +a ↔ gR +a , gL +b ↔ gR +b +� +. +(C20) +Using d = 4 − 2ϵ and the divergent parts of PV-functions given in Eq. (A8), we get the +formulas of D(ab)L,78 given in Eq. (22). +Diagram (5) +From equalities q2 = 0, q.ε∗ = 0, and k1 = q + k2, it is easy to prove that +[Γµα′β′ (−q, k1, −k2) ε∗µ] kα +1 kα′ +1 kβ +2 kβ′ +2 += kα +1 kβ +2 {(k1.k2) [(k1 + k2).ε∗] + (k2.ε∗) [k1.(−k2 + q)] + (k1.ε∗) [k2.(−q − k1)]} +∼ (k1.k2) [2k1.ε∗] + (k1.ε∗) +� +q2 − k2 +2 +� ++ (k1.ε∗) +� +q2 − k2 +1 +� += 0. +(C21) +As a result, the amplitude (C13) is written as follows: +iM5 = eQV +� +Dkuaγα [A] γβub +D0D1D2 +[Γµα′β′ (−q, k1, −k2) ε∗µ] +� +gαα′gββ′ − gββ′kα +1 kα′ +1 + gαα′kβ +2 kβ′ +2 +m2 +V +� +, +(C22) +where +A = mF +� +gL∗ +a gR +b PL + gR∗ +a gL +b PR +� ++ [A2] /k. +(C23) +26 + +The first term in the integrand is +(1) =ua +� +4(k1 + k2).ε∗ + (−/k + 2/p2 − /p1)/ε∗ + /ε∗(−/k + 2/p1 − /p2) +� +× mF [A1] ub ++ ua +� +(2 − d)(2k1.ε∗)/k + (−/k + 2/p2 − /p1)/k/ε∗ + /ε∗/k(−/k + 2/p1 − /p2) +� +× [A2] ub +=ua +� +6k.ε∗ − 3(/p1/ε∗ + /ε∗/p2) +� +mF [A1] ub ++ ua +� +(2 − d)(2k1.ε∗)/k + (2/p2 − /p1)/k/ε∗ + /ε∗/k(2/p1 − /p2) − 2k2/ε∗� +[A2] ub. +(C24) +After integrating out, the formulas is +(1) =ua +� +(2p1.ε∗) × (−3mF)X3 − 3mFC0(/p1/ε∗ + /ε∗/p2) +� +[A1] ub ++ ua +� +(2 − d)2εα∗(Cαβ − Cβp1α)γβ + Cα +� +(2/p2 − /p1)γα/ε∗ + /ε∗γα(2/p1 − /p2) +� +−2 +� +B(0) +0 ++ m2 +FC0 +� +/ε∗� +× [A2] ub +=ua(−3mF) × +� +(2p1.ε∗)X3 + C0(/p1/ε∗ + /ε∗/p2) +� +[A1] ub ++ ua +� +/ε∗ � +2(2 − d)C00 − 2(B(0) +0 ++ m2 +FC0) − (3m2 +a + 2m2 +b)C1 − (2m2 +a + 3m2 +b)C2 +� ++/p1/ε∗/p2(−3X3) +� +[A2] ub ++ ua(2p1.ε∗) +� +[−2(C11 + C12) + C2] /p1 + [−2(C12 + C22) + C1] /p2 +� +[A2] ub. +(C25) +The second term in the integrand is +� +− 1 +m2 +V +�−1 +× (2) +=Γµα′β′ (−q, k1, −k2) ε∗µ � +gββ′kα +1 kα′ +1 + gαα′kβ +2 kβ′ +2 +� +× uaγα [A] γβub += ua/k1 [A] +� +(k1.ε∗)/k2 − /ε∗k2 +2 +� +ub + ua +� +(k1.ε∗)/k1 − /ε∗k2 +1 +� +[A] /k2ub += uamF [A1] +� +2(k1.ε∗)/k1/k2 − k2 +2/k1/ε∗ − k2 +1/ε∗/k2 +� +ub ++ ua +� +2(k1.ε∗)/k1/k/k2 − k2 +2/k1/k/ε∗ − k2 +1/ε∗/k/k2 +� +[A2] ub += uamF [A1] +�/k1/ε∗/q/k2 +� +ub + ua +� +2(k1.ε∗)/k1/k/k2 − k2 +2/k1/k/ε∗ − k2 +1/ε∗/k/k2 +� +[A2] ub. +(C26) +The first term in Eq. (C26) gives +/k1/ε∗/q/k2 = +� +/k − /p1 +� +/ε∗/q +� +/k − /p2 +� += /k/ε∗/q/k − /p1/ε∗/q/k − /k/ε∗/q/p2 + /p1/ε∗/q/p2 += Cαβγα/ε∗/qγβ − Cα/p1/ε∗/qγα − Cαγα/ε∗/q/p2 + C0/p1/ε∗/q/p2 ++ +� +C1/p1 + C2/p2 +� +/ε∗/q/p2 + /p1/ε∗/q +� +C1/p1 + C2/p2 +� ++ C0/p1/ε∗/q/p2 +27 + += C00 +� +ε∗.q − (4 − d)/ε∗/q +� ++ +� +C12/p2 + C11/p1 + C1/p1 +� +/ε∗/q/p1 ++ +� +C12/p1 + C22/p2 + C1/p1 + C2/p2 + C2/p1 + C0/p1 +� +/ε∗/q/p2. +(C27) +Because the divergent part C00 = ∆ϵ/4 = 1/(4ϵ), which d = 4 − 2ϵ, hence C00(4 − d) = 1/2. +The result is: +/k1/ε∗/q/k2 = − 1 +2/ε/q + +� +C12 +� +/p1 + /q +� ++ (C11 + C1) /p1 +� +/ε∗/q +� +/p2 − /q +� ++ +� +(C12 + X0) /p1 + (C22 + C2) +� +/p1 + /q +�� +/ε∗/q/p2 += − 1 +2/ε∗/q + X012/p1/ε∗/q/p2, +(C28) +where we have used ε∗.q = q2 = 0 and /q/ε∗/q = 2ε∗.q/q − q2/ε∗ = 0. The final result is +uamF [A1] +�/k1/ε∗/q/k2 +� +ub =uamF +� +/p1/ε∗ +� +−1 +2 + m2 +bX012 +� ++ /ε∗/p2 +� +−1 +2 + m2 +aX012 +� ++(2p1.ε∗) +�1 +2 − X012/p1/p2 +�� +[A1] ub. +(C29) +Consider the last two terms in the last line of the formula (C26) +− k2 +2/k1/k/ε∗ − k2 +1/ε∗/k/k2 += − k2 +2 +� +k2 − /p1/k +� +/ε∗ − /ε∗ � +k2 − /k/p2 +� +k2 +1 += − k2 � +k2 +1 + k2 +2 +� +/ε∗ + +� +D2 + m2 +V +� +/p1/k/ε∗ + +� +D1 + m2 +V +� +/ε∗/k/p2 +→ − /ε∗(D0 + m2 +F)(D1 + D2 + 2m2 +V ) +D0D1D2 ++ /p1/k/ε∗ +D0D1 ++ /ε∗/k/p2 +D0D2 ++ m2 +V +� +/p1/k/ε∗ +D0D1D2 ++ +/ε∗/k/p2 +D0D1D2 +� += − /ε∗ � +2m2 +V (B(0) +0 ++ 1) + 2m2 +V B(0) +0 ++ m2 +F(B(1) +0 ++ B(2) +0 ++ 2m2 +V C0) +� +− /p1 +� +B(1) +1 /p1 + m2 +V +� +C1/p1 + C2/p2 +�� +/ε∗ − /ε∗ � +B(2) +1 /p2 + m2 +V +� +C1/p1 + C2/p2 +�� +/p2 += − /ε∗ � +m2 +V (4B(0) +0 ++ 2 + 2m2 +FC0 + m2 +aC1 + m2 +bC2) + m2 +aB(1) +1 ++ m2 +bB(2) +1 ++ m2 +F(B(1) +0 ++ B(2) +0 ) +� +− m2 +V +� +C2/p1/p2/ε∗ + C1/ε∗/p1/p2 +� += − /ε∗ � +m2 +V (4B(0) +0 ++ 2 + 2m2 +FC0 + m2 +aC1 + m2 +bC2) + m2 +aB(1) +1 ++ m2 +bB(2) +1 ++ m2 +F(B(1) +0 ++ B(2) +0 ) +� ++ m2 +V X3/p1/ε∗/p2 + (2p1.ε∗)(−m2 +V ) +� +C2/p1 + C1/p2 +� +. +(C30) +Lastly, consider the first term in the last line of the formula (C26): +2(k1.ε∗)/k1/k/k2 = (k.ε∗ − 2p1.ε∗) × +� +/k − /p1 +� +/k +� +/k − /p2 +� +28 + += (−2p1.ε∗ + 2k.ε∗) × +� +k2/k − k2/p1 − k2/p2 + /p1/k/p2 +� +→ (−2p1.ε∗ + 2k.ε∗) × +�� +1 +D1D2 ++ +m2 +F +D0D1D2 +� � +/k − /p1 − /p2 +� ++ +/p1/k/p2 +D0D1D2 +� += (−2p1.ε∗) × +�� +−1 +2B(0) +0 +− m2 +FC0 +� +(/p1 + /p2) − m2 +F(C1/p1 + C2/p2) − /p1(C1/p1 + C2/p2)/p2 +� ++ +� +2ε∗ +µ +� +�� +1 +D1D2 ++ +m2 +F +D0D1D2 +� +/kkµ − +� +1 +D1D2 ++ +m2 +F +D0D1D2 +� +(/p1 + /p2)kµ + /p1/k/p2kµ +D0D1D2 +� += (2p1.ε∗) +�� +B(0) +0 +2 ++ m2 +FC0 +� � +/p1 + /p2 +� ++ +� +m2 +FC1 + m2 +bC2 +� +/p1 + +� +m2 +FC2 + m2 +aC1 +� +/p2 +� ++ +� +2ε∗ +µ +� +× +�� +Bµν + m2 +FCµν� +γν − +� +Bµ + m2 +FCµ� � +/p1 + /p2 +� ++ Cµν/p1γν/p2 +� +, +(C31) +where Bµ = Bµ(0, m2 +V , m2 +V ) and Bµν = Bµν(0, m2 +V , m2 +V ). The last line in Eq. (C31) is +expressed in terms of the PV functions as follows +� +2ε∗ +µ +� � +γν +�gµν +2 +� +B(0) +0 ++ 1 +� ++ 1 +6B(0) +0 +(2pµ +1pν +1 + pµ +1pν +2 + pµ +2pν +1 + 2pµ +2pν +2) +� ++ m2 +Fγν [C00gµν + C11pµ +1pν +1 + C12pµ +1pν +2 + C12pµ +2pν +1 + C22pµ +2pν +2] +− +�1 +2B(0) +0 (p1 + p2)µ − m2 +F (C1pµ +1 + C2pµ +2) +� � +/p1 + /p2 +� ++ [C00gµν + C11pµ +1pν +1 + C12pµ +1pν +2 + C12pµ +2pν +1 + C22pµ +2pν +2] /p1γν/p2 +� +=m2 +V /ε∗ � +B(0) +0 ++ 1 +� ++ (p1.ε∗)B(0) +0 +� +/p1 + /p2 +� ++ m2 +F +� +2C00/ε∗ + (2p1.ε∗) +� +C11/p1 + C12/p2 + C12/p1 + C22/p2 +�� +− +� +B(0) +0 +(2p1.ε∗) − m2 +F (2p1.ε∗) (C1 + C2) +� � +/p1 + /p2 +� ++ /p1 +� +2/ε∗C00 + (2p1.ε∗) +� +C11/p1 + C12/p1 + C12/p2 + C22/p2 +�� +/p2. +(C32) +Hence the final result of Eq. (C31) is +2(k1.ε∗)/k1/k/k2 =/ε∗ � +m2 +V (B(0) +0 ++ 1) + 2m2 +FC00 +� ++ /p1/ε∗/p2 (2C00) ++ (2p1.ε∗) +� +m2 +FX01 + m2 +bX2 +� +/p1 + (2p1.ε∗) +� +m2 +FX01 + m2 +aX1 +� +/p2. +(C33) +The sum of three terms given in Eqs. (C25), (C30), and (C33) gives C(ab)L,R corresponding +to the diagrams (5) given in Eqs. (18) and (19). The formulas of D(ab)L,5 and D(ab)R,5 are +given in Eq. (23). +Regarding to the case of photon couplings in Eq. (27), the equality given in Eq. (C21) +is still valid because the new part ∆Γµα′β′ = Γµα′β′ − Γ′ +µα′β′ = δkv (gµα′qβ′ − gβ′µqα′) satisfies +29 + +(gµα′qβ′ − gβ′µqα′) ε∗µkα′ +1 kβ′ +2 = q2(ε∗.k2) − (q.k2)(ε∗.q) = 0. The other relevant part of M5 is: +− γα [A] γβ × ∆Γµα′β′εµ∗ +� +gαα′gββ′ − gββ′kα +1 kα′ +1 + gαα′kβ +2 kβ′ +2 +m2 +V +� += +� +/q/ε∗ − /ε∗/q +� +mFA1 + +� +/q/k/ε∗ − /ε∗/k/q +� +A2 +− +1 +m2 +V +�� +(k1.q)(/k1/ε∗ − /ε∗/k2) + (k1.ε∗)(/q/k2 − /k1/q) +� +mFA1 ++ +� +(k1.q)(−/p1/k/ε∗ + /ε∗/k/p2) + (k1.ε∗)(/p1/k/q − /q/k/p2) +� +A2 +� +. +(C34) +The final result of new contributions to iδM5 is: +iδM5 = +� +d4k +(2π)4 +uaγα [A] γβub +D0D1D2 +× ∆Γµα′β′ +� +gαα′gββ′ − gββ′kα +1 kα′ +1 + gαα′kβ +2 kβ′ +2 +m2 +V +� += − ieQV δv +16π2 +� +ua +�� +4p1.ε∗ − 2/p1/ε∗ − 2/ε∗/p2 +� +C0mFA1 +� +ub ++ua +� +(2p1.ε∗)(/p1 + /p2) − (m2 +a + m2 +b)/ε∗ − 2/p1/ε∗/p2 +� +X3A2ub +− 1 +m2 +V +ua +� +C00 +� +/q/ε∗ − /ε∗/q +� ++ (C11 + C12) +� +−2(p1.q)/ε∗/p1 + 2(p1.ε∗)/q/p1 +� ++(C22 + C12) +� +−2(p2.q)/ε∗/p2 + 2(p2.ε∗)/q/p2 +� ++(p1.q) +� +−X0 +� +−/p1/ε∗ + /ε∗/p2 +� ++ (2p1.ε∗)(C2 − C1) + 2 +� +C1/p1/ε∗ − C2/ε∗/p2 +�� +−(p1.ε∗) +� +2/p1/p2 − m2 +a − m2 +b +� +(2X3 + C0) +� +mFA1ub +− 1 +m2 +V +ua +� +C00 +� +2(m2 +a + m2 +b)/ε∗ + 4/p1/ε∗/p2 − 2(/p1 + /p2) × (2p1.ε∗) +� ++m2 +b − m2 +a +2 +× X1 +� +−m2 +a/ε∗ − /p1/ε∗/p2 + (2p1.ε∗)/p1 +� ++m2 +b − m2 +a +2 +× X2 +� +m2 +b/ε∗ + /p1/ε∗/p2 − (2p1.ε∗)/p2 +�� +A2ub +� +. +(C35) +Ignoring the factor eQV δkv +16π2 , the form factors are: +−δCFV V +(ab)L =gLLma +� +X3 + 4C00 − (m2 +b − m2 +a)X1 +2m2 +V +� ++ gRRmb +� +X3 + 4C00 + (m2 +b − m2 +a)X2 +2m2 +V +� ++ gRLmF +� +2C0 − 8C00 + m2 +a(2X1 + X0) + m2 +b(2X2 + X0) +2m2 +V +� ++ gLRmFmambX012 +m2 +V +, +−δCFV V +(ab)R =δCL,5 +� +gL +a ↔ gR +a , gL +b ↔ gR +b +� +, +−δDFV V +(ab)L =gLL +� +−(m2 +a + m2 +b)X3 − 4(m2 +a + m2 +b)C00 + (m2 +b − m2 +a)(−m2 +aX1 + m2 +bX2) +2m2 +V +� ++ gRRmamb +� +−2X3 − 8C00 + (m2 +b − m2 +a)(−X1 + X2) +2m2 +V +� +30 + ++ gRLmamF +� +−2C0 − −8C00 + (m2 +b − m2 +a)(2X1 + X0) +2m2 +V +� ++ gLRmbmF +� +−2C0 + 8C00 + (m2 +b − m2 +a)(2X2 + X0) +2m2 +V +� +, +δDFV V +(ab)R =δDFV V +(ab)L +� +gL +a ↔ gR +a , gL +b ↔ gR +b +� +. +(C36) +All results given in Eq. (C36) were cross checked using FORM package [48]. All formulas +in Eq. (C36) satisfy automatically the WI, namely δDFV V +(ab)L + maδCFV V +(ab)L + mbδCFV V +(ab)R = 0. +Diagram (6) +After using the property of chiral operators PL,R, the amplitude (C17) is written as +iM6 =eQF +� +d4k +(2π)4 +1 +D0D1D2 +× ua +�� +m2 +Fγα/ε∗γβ + γα/k1/ε∗/k2γβ +� +[A2] +−mF [A1] +� +gL∗ +a gR +b PR + gR∗ +a gL +b PL +� +(γα/k1/ε∗γβ + γα/ε∗/k2γβ) +� +ub. +(C37) +The numerator is divided into the two parts N1 ∼ gαβ and N2 ∼ −kαkβ/m2 +V . After extract- +ing gαβ, the first part is +N1 =ua +�� +(2 − d)m2 +F/ε∗ − 2/k2/ε∗/k1 + (4 − d)/k1/ε∗/k2 +� +[A2] +−mF [A1] [4ε∗.(k1 + k2) − (4 − d) (/k1/ε∗ + /ε∗/k2)]} ub. +(C38) +Ignoring the overall factor eQF/(16π2), the formula in terms of tensor notations is +N1 =ua/ε∗ [A2] ub +� +−2m2 +FC0 + (d − 4)(d − 2)C00 +� ++ (2p1.ε∗) ua [A1] (4mFX0) ub ++ ua +� +(2 − d)Cαβγα/ε∗γβ + 2Cα +� +/p2/ε∗γα + γα/ε∗/p1 +� +− 2C0/p2/ε∗/p1 +� +× [A2] ub. +(C39) +After expanding the tensors in terms of scalar PV-functions, the final result is +N1 =ua/ε∗ [A2] ub +� +−2m2 +FC0 + (d − 2)2C00 + 2m2 +aX01 + m2 +bX02 +� ++ ua/p1/ε∗/p2 [A2] ub × (2X0) ++ (2ε∗.p1) ua(−2) +� +X01/p1 + X02/p2 +� +[A2] ub + (2ε∗.p1) ua {4mFX0 [A1]} ub. +(C40) +Considering the second term proportional to kαkβ, we have +−m2 +V N2 = ua +� +m2 +F /k/ε∗/k + /k/k1/ε∗/k2/k +� +[A2] ub − mFua (/k/k1/ε∗/k + /k/ε∗/k2/k) [A1] ub. +(C41) +The two relations /k/k1 = D1 + m2 +F − m2 +a + /p1/k and /k2/k = D2 + m2 +F − m2 +b + /k/p2 give +N2 ∼ua +� +m2 +F /k/ε∗/k +� +[A2] ub + ua +� +D1 + m2 +F − m2 +1 + /p1/k +� +/ε∗ � +D2 + m2 +F − m2 +2 + /k/p2 +� +[A2] ub +31 + +− mFua +�� +D1 + m2 +F − m2 +a + /p1/k +� +/ε∗/k + /k/ε∗ � +D2 + m2 +F − m2 +b + /k/p2 +�� +[A1] ub +≡ua [(L1 + L2) [A2] − mF [A1] L3] ub, +(C42) +where +L1 =m2 +F +� +/ε∗ � +(2 − d)C00 − m2 +a (C11 + C12) − m2 +b (C12 + C22) +� ++(2p1.ε∗) +� +(C11 + C12) /p1 + (C12 + C22) /p2 +�� +, +L2 = +1 +D0D1D2 +� +D1 + m2 +F − m2 +a + /p1/k +� +/ε∗ � +D2 + m2 +F − m2 +b + /k/p2 +� +=/ε∗ +� +1 +D0 ++ +m2 +F − m2 +b + /k/p2 +D0D2 ++ m2 +F − m2 +a +D0D1 ++ (m2 +F − m2 +a) (m2 +F − m2 +b) +D0D1D2 +� ++ /p1/k/ε∗ +D0D1 ++ /p1/k/ε∗/k/p2 +D0D1D2 ++ /p1/k/ε∗ (m2 +F − m2 +b) +D0D1D2 ++ +(m2 +F − m2 +a) /ε∗/k/p2 +D0D1D2 +=/ε∗ � +A0(m2 +V ) + +� +m2 +F − m2 +b +� +B(2) +0 +− m2 +bB(2) +1 ++ +� +m2 +F − m2 +a +� +B(1) +0 +− m2 +aB(1) +1 ++ +� +m2 +F − m2 +a +� � +m2 +F − m2 +b +� +C0 +� ++ Cαβ(/p1γα/ε∗γβ/p2) + Cα(/p1γα/ε∗) +� +m2 +F − m2 +b +� ++ Cα(/ε∗γα/p2) +� +m2 +F − m2 +a +� +, +L3 = /ε∗/k +D0D2 ++ +/k/ε∗ +D0D1 ++ m2 +F(2k.ε∗) − m2 +a/ε∗/k − m2 +b/k/ε∗ +D0D1D2 ++ /p1/k/ε∗/k + /k/ε∗/k/p2 +D0D1D2 += − B(2) +1 /ε∗/p2 − B(1) +1 /p1/ε∗ − (2p1.ε∗)(C1 + C2)m2 +F +− Cα +� +m2 +a/ε∗γα + m2 +bγα/ε∗� ++ Cαβ +� +/p1γα/ε∗γβ + γα/ε∗γβ/p2 +� +. +It can be proved that: +Cαβ(/p1γα/ε∗γβ/p2) = /p1/ε∗/p2 +� +(2 − d)C00 − m2 +a(C11 + C12) − m2 +b(C22 + C12) +� ++ (2p1.ε∗) +� +m2 +b(C22 + C12)/p1 + m2 +a(C11 + C12)/p2 +� +, +Cα(/p1γα/ε∗) = −m2 +aC1/ε∗ − C2 +� +(2p1.ε∗)/p1 − /p1/ε∗/p2 +� +, +Cα(/ε∗γα/p2) = −m2 +bC2/ε∗ − C1 +� +(2p1.ε∗)/p2 − /p1/ε∗/p2 +� +, +Cα +� +m2 +a/ε∗γα + m2 +bγα/ε∗� +=(2p1.ε∗) +� +−m2 +aC1 − m2 +bC2 +� ++ /p1/ε∗(m2 +a − m2 +b)C1 + /ε∗/p2(m2 +b − m2 +a)C2, +Cαβ +� +/p1γα/ε∗γβ + γα/ε∗γβ/p2 +� +=(2p1.ε∗) +� +m2 +a(C11 + C12) + m2 +b(C22 + C12) + /p1/p2(C11 + 2C12 + C22) +� ++ (/p1/ε∗ + /ε∗/p2) +� +(2 − d)C00 − m2 +a(C11 + C12) − m2 +b(C22 + C12) +� +. +(C43) +32 + +Final results are: +L1 =m2 +F +� +/ε∗ � +(2 − d)C00 − m2 +a (C11 + C12) − m2 +b (C12 + C22) +� ++(2p1.ε∗) +� +(C11 + C12) /p1 + (C12 + C22) /p2 +�� +, +L2 =/ε∗ � +m2 +V (B(0) +0 ++ 1) + m2 +F(B(1) +0 ++ B(2) +0 ) − m2 +a(B(1) +0 ++ B(1) +1 ) − m2 +b(B(2) +0 ++ B(2) +1 ) ++m4 +FC0 − m2 +F +� +(m2 +a + m2 +b)C0 + m2 +aC1 + m2 +bC2 +� ++ m2 +am2 +bX0 +� ++ /p1/ε∗/p2 +� +(2 − d)C00 + m2 +FX3 − m2 +aX1 − m2 +bX2 +� ++ (2p1.ε∗) +�� +m2 +bX2 − m2 +FC2 +� +/p1 + +� +m2 +aX1 − m2 +FC1 +� +/p2 +� +, +L3 =/p1/ε∗ � +−B(1) +1 ++ (2 − d)C00 − m2 +aX1 + m2 +b(X3 − X2) +� ++ /ε∗/p2 +� +−B(2) +1 ++ (2 − d)C00 − m2 +bX2 + m2 +a(X3 − X1) +� ++ (2p1.ε∗) +� +m2 +aX1 + m2 +bX2 − m2 +FX3 + /p1/p2 (X1 + X2 − X3) +� +. +(C44) +The above calculation is enough to derive relevant contributions to CV FF +L,R +given in Eqs. (20) +and (21), and DV FF +L,R +given in (24). +Ward identity for the only gauge boson exchanges +Before coming to discuss the WI, we use the relations given in Eq. (A10) to write all the +one-loop factors (22), (23), and (24) from gauge boson exchanges in the following simple +forms, ignoring the overall factor e/(16π2): +DFV +(ab)L,78 =Qe +� +gRLma + gLRmb +� +(−3mFX0) + QegRRmamb +� +−2Xf +12 + bf +1 − m2 +F(2X0 + Xf +12) +m2 +V +� ++ QegLL +�� +2 + m2 +F + m2 +a + m2 +b +m2 +V +� +bf +1 + 1 + A0(m2 +V ) + m2 +am2 +bXf +12 +m2 +V ++2m2 +F(m2 +aB(1) +0 +− m2 +bB(2) +0 ) +(m2 +a − m2 +b)m2 +V +� +. +(C45) +The WI for the FV V and FV V diagrams are f WI +FV V ≡ DFV V +(ab)L + maCFV V +(ab)L + mbCFV V +(ab)R and +f WI +V FF ≡ DV FF +(ab)L + maCV FF +(ab)L + mbCV FF +(ab)R, respectively. The relations given in Eq. (A10) give: +2Cf +00 + m2 +aXf +1 + m2 +bXf +2 = −bf +1, +Xv +012 = −Xf +12, +m2 +F(Xv +12 − Xv +3) + m2 +aXv +1 + m2 +bXv +2 + bv +1 + 1/2 = m2 +F(Xv +12 − Xv +3) − m2 +FCv +0 +33 + += m2 +F(Xv +012 − 2X0) = −m2 +F(Xf +12 + 2X0), +1/2 + m2 +aXv +1 + m2 +bXv +2 − m2 +FXv +3 = (m2 +F − m2 +V )X0 − m2 +FCv +0 − m2 +FXv +3 = −m2 +V X0. +(C46) +Combining the above formulas and results of Ci,ij functions listed in Ref. [39], the WI of all +diagrams with boson exchanges is derived as follows +f WI +V +=D(ab)L,78 + f WI +FV V + f WI +V FF +∼ (Qe + QV − QF) +× +� +gLL +� +3 − B(0) +0 (m2 +V ) +2 ++ A0(m2 +F) +2m2 +V +− (m2 +a + m2 +F − 2m2 +V )m2 +V + (m2 +a − m2 +F)2 +2(m2 +a − m2 +b)m2 +V +× B(1) +0 ++(m2 +b + m2 +F − 2m2 +V )m2 +V + (m2 +b − m2 +F)2 +2(m2 +a − m2 +b)m2 +V +× B(2) +0 +� ++gRR +� +(m2 +V B(0) +0 +− A0(m2 +F)) × (m2 +F + 2m2 +V ) − (m2 +F − 4m2 +V )m2 +V +2m2 +V +−m2 +b [(m2 +a + m2 +F − 2m2 +V )m2 +V + (m2 +a − m2 +F)2] +2(m2 +a − m2 +b)m2 +V +× B(1) +0 ++m2 +a [(m2 +b + m2 +F − 2m2 +V )m2 +V + (m2 +b − m2 +F)2] +2(m2 +a − m2 +b)m2 +V +× B(2) +0 +� ++ +� +gRLma + gLRmb +� +(3mFX0) +� +. +(C47) +The final result is f WI +V +∼ QF − (Qe + QV ) = 0. In conclusion, the contributions from the +four diagrams with only gauge boson exchanges satisfy the WI when the electric charge +conversation is valid. +Appendix D: Ward Identity for the diagrams of FSV-type in the unitary gauge +This type of diagrams were mentioned firstly in Ref. [34] for the general case of their +contributions to BSM. The γ − S − V vertices come the kinetic terms of the scalars: +LD(S) = (∂µS − iPµS)† (∂µS − iP µS) = +� +gγSV gµνS−QAµV Qν + h.c. +� ++ . . . , +(D1) +where Pµ containing the photon Aµ and Vµ is the covariant part of the covariant derivative +of the Higgs multiplets. The contributions of these two diagrams are: +iM9 =gγSV +� +d4k +(2π)4 × ua[gL∗ +a PR + gR∗ +a PL](mF + /k)γα[gL +b PL + gR +b PR]ub +D0D1D2 +� +ε∗α − (ϵ.k2)kα +2 +m2 +V +� +34 + +=gγSV +� +d4k +(2π)4 × +1 +D0D1D2 +× ua +� +� +�/ε∗ [mF [A2] + /k [A1]] − /k2(k2.ε∗) +m2 +V +mF [A2] − [A1] +� +D0 + m2 +F − /k/p2 +� +(k2.ε∗) +m2 +V +� +� +� ub +=igγSV +16π2 ua +� +/ε∗ � +C0mF [A2] − (C1/p1 + C2/p2) [A1] +� +−mF +m2 +V +� +(γµε∗ν)Cµν − (Cµγµ)(p2.ε∗) + /p2X0(p1.ε∗) +� +[A2] ++ 1 +m2 +V +[A1] +� +C00/ε∗/p2 + (p1.ε∗) +� +m2 +FX0 + X1/p1/p2 + m2 +bX2 +��� +ub, +(D2) +where we have used k2.ε∗/(D1D2) → 0. The formulas of DL,R and CL,R are: +eDFhv +(ab)L,9 × +�gγSV +16π2 +�−1 +=gLLmF +� +C0 − C00 +m2 +V +� +− gRLmaC1 + gLRmb +� +C2 + C00 +m2 +V +� +, +DFhv +(ab)R,9 =DFhv +(ab)L,9 +� +gL +a ↔ gR +a , gL +b ↔ gR +b +� +, +eCFvh +(ab)L,9 × +�gγSV +16π2 +�−1 += − gRLC2 − mF +2m2 +V +� +gLLmaX1 + gRRmbX02 +� ++ +1 +2m2 +V +� +gRL(m2 +FX0 + m2 +bX2) + gLRmambX1 +� +, +CFhv +(ab)R,9 =C(ab)L +� +gL +a ↔ gR +a , gL +b ↔ gR +b +� +, +(D3) +where XFhv +i +≡ Xi(m2 +a, 0, m2 +b; m2 +F, m2 +h, m2 +V ). Similarly, the results for diagram (10) are: +eDFvh +(ab)L,10 × +�gγSV +16π2 +�−1 +=gLLmF +� +C0 − C00 +m2 +V +� +− gLRmbC2 + gRLma +� +C1 + C00 +m2 +V +� +, +DFvh +(ab)R,10 =DFvh +(ab)L,10 +� +gL +a ↔ gR +a , gL +b ↔ gR +b +� +, +eCFvh +(ab)L,10 × +�gγSV +16π2 +�−1 += − gR∗ +a gL +b C1 − mF +2m2 +V +� +gR∗ +a gR +b mbX2 + gL∗ +a gL +b maX01 +� ++ +1 +2m2 +V +� +gR∗ +a gL +b (m2 +FX0 + m2 +aX1) + gL∗ +a gR +b mambX2 +� +, +eCFvh +(ab)R,10 × +�gγSV +16π2 +�−1 +=CFvh +(ab)L +� +gL +a ↔ gR +a , gL +b ↔ gR +b +� +, +(D4) +where XFvh +i +≡ Xi(m2 +a, 0, m2 +b; m2 +F, m2 +V , m2 +h). The above formulas are consistent with calcula- +tion using FORM. The corresponding formulas of WI are +f Fhv +WI +kγSV += gLLmF +� +2(m2 +V C0 − C00) + mambX012 +�fhv − gRRmF +� +m2 +aX1 + m2 +bX02 +�fhv ++ gRL � +−2m2 +V (maC1 + mbC2) + mb +� +m2 +FX0 + m2 +aX1 + m2 +bX2 +��fhv ++ gLR � +2m2 +V (mbC2 − maC1) + 2mbC00 + ma +� +m2 +fX0 + m2 +bX12 +��fhv , +35 + +f Fvh +WI +kγSV += gLLmF +� +2(m2 +V C0 − C00) − mambX012 +� +− gRRmF +� +m2 +aX01 + m2 +bX2 +�fvh ++ gRL � +2m2 +V (maC1 − mbC2) + 2maC00 + mb +� +m2 +fX0 + m2 +aX12 +��fvh ++ gLR � +−2m2 +V (maC1 + mbC2) + ma +� +m2 +fX0 + m2 +aX1 + m2 +bX2 +��fvh , +(D5) +where kγSV = gγSV /(32π2m2 +V ). The WI valid if only f Fhv +WI + f Fvh +WI = 0. We can see crudely +that all C(ab)L,9, C(ab)R,9, C(ab)L,10, and C(ab)R,10 are convergent. In contrast, all D(ab)L,9, +D(ab)R,9, D(ab)L,10, and D(ab)R,10 contain divergent terms. Therefore, the necessary condition +to guarantee the validation of the WI given in Eq. (12) is that all of these divergent terms +must vanish. 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C 1 (1998), 163-175 [arXiv:hep- +ph/9701342 [hep-ph]]. +39 + diff --git a/aNE5T4oBgHgl3EQfDg77/content/tmp_files/load_file.txt b/aNE5T4oBgHgl3EQfDg77/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..96904027703950864cc3a30c316cc63e344e388c --- /dev/null +++ b/aNE5T4oBgHgl3EQfDg77/content/tmp_files/load_file.txt @@ -0,0 +1,1855 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf,len=1854 +page_content='One-loop contributions to decays eb → eaγ and (g − 2)ea anomalies, and Ward identity L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Hue,1, 2, ∗ H.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Vietnam 5Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Can Tho University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 3/2 Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Can Tho 94000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Vietnam Abstract In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' we will present analytic formulas to express one-loop contributions to lepton flavor violating decays eb → eaγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' which are also relevant to the anomalous dipole magnetic moments of charged leptons ea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' These formulas were computed in the unitary gauge, using the well-known Passarino-Veltman notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' We also show that our results are consistent with those calculated previously in the ’t Hooft-Veltman gauge, or in the limit of zero lepton masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' At the one-loop level, we show that the appearance of fermion-scalar-vector type diagrams in the unitary gauge will violate the Ward Identity relating to an external photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' As a result, the validation of the Ward Identity guarantees that the photon always couples with two identical particles in an arbitrary triple coupling vertex containing a photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' PACS numbers: ‡ corresponding author ∗Electronic address: lethohue@vlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='vn †Electronic address: hoangngoclong@vlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='vn §Electronic address: thanhphong@ctu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='vn 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='05407v1 [hep-ph] 13 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' INTRODUCTION The lepton sector is one of the most interesting object for experiments to search for new physics (NP) beyond the prediction of the standard model (SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' For example, the evidence of neutrino oscillation confirms that the SM must be extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Recently, the experimental data of anomalous magnetic moments (AMM) of charged leptons (g −2)ea/2 ≡ aea has been updated, where the deviation between SM prediction and the lasted experiment data for muon is [1] ∆aNP µ ≡ aexp µ − aSM µ = (251 ± 59) × 10−11, (1) corresponding to the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='2σ deviation from standard model (SM) prediction [2] combined from various contributions [3–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' For the electron anomaly, the deviation between SM and experiment is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='6σ discrepancy [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' On the other hand, ∆ae,µ are strongly constrained by the experimental data obtained from searching for the charged lepton flavor violating (cLFV) decays eb → eaγ are [25, 26]: Br(τ → µγ) < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='4 × 10−8, Br(τ → eγ) < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='3 × 10−8, Br(µ → eγ) < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='2 × 10−13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (2) This important property was discussed previously, for example see discussions for a general estimation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [27], and many particular models beyond the standard model (BSM) [28–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' General formulas expressing simultaneously both one-loop contributions to AMM and cLFV amplitudes were introduced in the limits of new heavy scalar and/or gauge boson exchanges m2 B ≫ m2 a with ma being the mass of a charged lepton ea = e, µ, τ [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Other calculations in the unitary gauge were discussed [34, 35] for the one-loop contributions to aea with ma ̸= 0, without the relations with the cLFV amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The analytic one-loop formulas for cLFV amplitudes calculated in the ’t Hooft Feynman (HF) gauge were also shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [36], using the notations of the Passarino-Veltman (PV) functions [37, 38] with ma ̸= mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The approximate formulas with ma = mb = 0 were introduced and consistent with those given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [27], as shown particularly in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [39] for 3-3-1 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The general analytic formulas of these PV functions were introduced for numerical investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' They are consistent with the results generated by LoopTools [40], which can be transformed into other PV notations implemented in the Fortran numerical package Collier [41], used to investigate cLFV decays in a two Higgs doublet model (2HDM) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Many particular expressions to compute the AMM and/or cLFV decay amplitudes predicted by different 2 particular BSM were constructed [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The relations among them can be checked by using suitable transformations, starting from the set of particular PV notations in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' On the other hand, in a discussion on analytic formulas for one-loop contributions to AMM, a class of fermion-scalar-vector (FSV ) diagrams consisting of a photon coupling with two different physical particles, namely one scalar and one gauge boson, were considered even in the unitary gauge [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' It leads us to a question whether the Ward identity (WI) for the external photon is still valid with the presence of this diagram type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' We emphasize that the general results for one-loop contributions to decays eb → eaγ and AMM of leptons introduced in many previous works do not include this FSV diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Moreover, they imply the existence of the triple photon coupling with two distinguishable physical particles that has never been mentioned previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In particular, many works introducing general one-loop contributions for AMM of charged leptons [27, 28, 35], or decays relating with photon such as cLFV decays eb → eaγ [27, 28, 36], loop-induced Higgs decays h → γγ [43, 44], h → Zγ, f ¯fγ [44–47], quark decays q → q′γ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Excluding the FSV vertex type will reduce a huge number of related one- and two-loop diagrams as well as confirm the validation of general one-loop calculation introduced previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In this work, we will show precisely the important steps to derive the one-loop contri- butions to both AMM and cLFV decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The calculation is performed by hand, which is consistent with another cross-checking using FORM package [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The final formulas are expressed exactly in terms of the PV functions defined by LoopTools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The results are then easily to change into all of the other available forms using suitable transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The convention of the PV-functions are very convenient to derive the exact formulas before solv- ing particular pure mathematical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' We also determine contributions arising from a new form of photon coupling with vector bosons such as leptoquarks and confirm the consistence between our results and those introduced in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [44, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Section I explains our aim of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Section II introduces notations and important formulas to establish the relations between AMM and cLFV amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Section III shows discussions to confirm the consistence of our results and previous works, and the validation of the WI for the relevant analyitic formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Section IV summarizes main features of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Finally, we provide many appendices showing precisely many intermediate steps and notations to derive the final results mentioned in this work, including the analytic formulas of the PV functions consistent with LoopTools given 3 in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' GENERAL AMPLITUDES AND NOTATIONS It is well-known that analytic formulas of one-loop contributions to the cLFV amplitudes eb(p2) → ea(p1)γ(q) and AMM of SM charged leptons ea can be presented in the same expressions, see for example Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [27] corresponding to the presence of new heavy particles in BSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Possible one-loop Feynman diagrams contributing to aea and cLFV decay amplitudes eb → eaγ in BSM are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1, where F is a fermion coupling with the SM charged lepton ea = e, µ, τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' and the boson B = h, V is a scalar or gauge boson, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' We note (1) h+ F eb ea γ p1 q p2 (2) F h+ eb ea ea F eb eb (3) (4) ea ea F eb (5) eb ea γ V + F (6) F V + eb ea (7) (8) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1: Feynman diagrams for one-loop contribution to aea and cLFV amplitudes eb → eaγ in the unitary gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' here that Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [34] argues another type of FSV one-loop diagrams giving new contributions to the AMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' They will be discussed in details in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Firstly, we adopt the Lagrangian generating one-loop diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1, namely [27] Lh = F(gL a,FhPL + gR a,FhPR)eah + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=', (3) LV = Fγµ(gL a,FV PL + gR a,FV PR)eaVµ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=', (4) where the fermion F and the boson B = Vµ, h have electric charges QF and QB, and masses mF and mB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' These Lagrangians (3) and (4) are consistent with those in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Moreover, the photon couplings with all physical particles should be mentioned clearly, as given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [36], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=', we will adopt the Feynman rules that the photon alway couples with two identical physical particles, as given in table I, where Γµνλ(p0, p+, p−) = 4 Vertex Coupling Vertex Couplings Vertex Couplings Aµ(p0)V ν(p+)V ∗λ(p−) −ieQV Γµνλ(p0, p+, p−) Aµh(p+)h∗(p−) ieQh(p+ − p−)µ AµFF ieQF γµ TABLE I: Feynman rules for cubic couplings of photon Aµ, where p0,± are incoming momenta into the relevant vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' gµν(p0 − p+)λ + gνλ(p+ − p−)µ + gλµ(p− − p0)ν is the standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The more general form of Γµνλ(p0, p+, p−) introduced in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [44, 49, 50] will be discussed in details later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' All couplings listed in Lagrangians (3), (4), and table I result in the following form factors relevant with one-loop contributions: cab RB = e 16π2gL∗ a,FBgR b,FBmF × fB(xB) + QFgB(xB) m2 B + e 16π2 � mbgL∗ a,FBgL b,FB + magR∗ a,FBgR b,FB � × ˜fB(xB) + QF ˜gB(xB) m2 B , (5) where xB ≡ m2 F/m2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The four scalar functions fB(x), gB(x), ˜fB(x), and ˜gB(x) are listed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A22) of appendix A, as the approximate formulas in the limit ma, mb ≪ mB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (5) does not contain contributions from the FSV diagrams mentioned in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [34], because of the absence of photon coupling AV h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The corresponding formulas of AMM and cLFV decay rates are: aea ≡ −2ma e (caa R + caa∗ R ) = −4ma e Re[caa R ], (6) Br(eb → eaγ) = m3 b 4πΓb ���cab R ��2 + ��cba R ��2� , (7) where ma, mb, and Γb are the masses and total decay width of the leptons ea, eb, and cab R ≡ � B,F cab RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (8) The amplitude for a vertex ¯eaeaAµ in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [51] is consistent with the following form presenting both AMM and cLFV amplitudes [52, 53] iM = −ieua(p1) � γµF1 − σµνqν 2ma � iF2 + γ5F3 �� ub(p2)ε∗ µ, (9) where σµν ≡ i 2 [γµγν − γνγµ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' F1,2,3 are scalar form factors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' ε∗ µ and qν is the polarized vector of the external photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The form factor F2,3 gets contribution only from loop corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' They relate with the well-known experimental quantities called the anomalous magnetic 5 moment aea and electric dipole moment dea for b = a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Specifically we have F1 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' aea = F2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' dea = − e 2ma F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (10) Regarding to the LFV decay eb → eaγ the amplitude can also be written in the same form [36, 54], suggesting that F2 can be calculated based on the one-loop corrections to LFV decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In particular, the second term of the amplitude (9) can be expanded as follows [39] M = (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)ua � C(ab)LPL + C(ab)RPR � ub + ua � D(ab)L/ε∗PL + D(ab)R/ε∗PR � ub, (11) where ma = mb and we can prove that C(ab)LPL + C(ab)RPR = e 2ma(F2 − iγ5F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The WI for the external photon gives D(ab)L = −(mbC(ab)R + maC(ab)L), D(ab)R = −(mbC(ab)L + maC(ab)R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (12) The hermiticity that C(aa)R = C∗ (aa)L [53] gives aea = ma(C(aa)L + C(aa)R) e = 2maRe[C(aa)L,R] e , dea = i(C(aa)R − C(aa)L) = Im[C(aa)L] = −Im[C(aa)R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (13) Hence, the following relations between two different notations must be satisfied: cab R = −1 2C(ab)R and cba R = −1 2C(ab)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (14) From the above discussion, we see that one-loop contributions to the aea and dea can be written in terms of well-known PV functions, see detailed discussions in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [39] or general formula introduced for calculations the LFV decay rates of charged leptons [36], with the identification that σL,R ≡ −C(ab)L,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In the limit of 0 ≃ ma, mb ≪ mB, the numerical values of aea can be evaluated using the numerical packages such as LoopTools [40] or Collier [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Although the exact analytic formulas of one-loop three point functions presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [39] can not be applied to calculate aea, but the limit of mb → ma can be used to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The analytic formulas of aea were introduced completely in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Because of the relations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (12), only C(ab)L,R is needed to determine aea and Br(eb → eaγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Because all two-point diagrams give contributions to just D(ab)L,R, C(ab)L,R are calculated by considering only three-point diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In this work, the analytic formulas of D(ab)L,R will be determined directly from all diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1 to check the validation of the WI in the presence of the FSV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 6 The analytic formulas for one-loop contributions to the cLFV decay amplitudes presented in this work are more general than the results introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [39] for general 3-3-1 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Many important steps in our calculations were shown in appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Using this unitary gauge, the assumption for a particular form of the Goldstone boson couplings given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [36] is unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In contrast, we use the same photon couplings to other physical particles in an arbitrary BSM, as given in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Namely, a tree-level photon coupling always contains two identical physical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' This implies that the contributions from the FSV diagrams are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Using the notations of PV-functions defined in appendix A, the Fhh contributions from diagram (1) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1 are: CFhh (ab)L =−eQh 16π2 � magL∗ a,FhgL b,FhXf 1 + mbgR∗ a,FhgR b,FhXf 2 − mFgR∗ a,FhgL b,FhXf 0 � , CFhh (ab)R =−eQh 16π2 � magR∗ a,FhgbR FhXf 1 + mbgL∗ a,FhgL b,FhXf 2 − mFgL∗ a,FhgR b,FhXf 0 � , (15) where Xf 0 , Xf 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' are linear combinations of the PV-functions C0,00,i,ij defined precisely in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The diagram (2) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1 gives hFF contributions as follows: ChFF (ab)L =−eQF 16π2 � magL∗ a,FhgbL b,FhXh 1 + mbgR∗ a,FhgR b,FhXh 2 + mFgR∗ a,FhgL b,FhXh 3 � , ChFF (ab)R =−eQF 16π2 � magR∗ a,FhgR b,FhXh 1 + mbgL∗ a,FhgL b,FhXh 2 + mFgL∗ a,FhgR b,FhXh 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (16) where Xh 1,2,3 are linear combinations of C0,i,ij(m2 h, m2 F, m2 F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The above result are completely consistent with the results introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [36], except an overall sign and the signs before the PV-functions ¯c1,2, arising from the different definitions of the external momenta pi in the denominators of the one-loop integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' We also give the analytic formulas of DFhh (ab)L,R and DhFF (ab)L,R, used to confirm the WI given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (12) for the only-scalar contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The PV-functions derived from the diagram (2) defined as Xh i are different from Xf i defined for three diagrams (1), (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In contrast, the equal functions are denoted as follows: B(i) 0 ≡ B(i)f 0 = B(i)h 0 , X0 ≡ Xf 0 = Xh 0 , i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The form factors D(ab)L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='R originated from scalar contributions are: DFhh (ab)L =−eQH 16π2 � gL∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FhgL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fh × 2Cf 00 � 7 + −eQe 16π2(m2 a − m2 b) �� mbgL∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FhgR b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fh + magR∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FhgL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fh � mF � B(1) 0 − B(2) 0 � − gL∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FhgL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fh � m2 aB(1)f 1 − m2 bB(2)f 1 � − mambgR∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FhgR b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fh � B(1)f 1 − B(2)f 1 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' DFhh (ab)R =DFHH (ab)L � gL a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fh ↔ gR a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' gL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fh ↔ gR b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fh � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' DhFF (ab)L = − eQF 16π2 � gL∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FhgL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fh � m2 FCh 0 + (2 − d)Ch 00 − m2 aXh 1 − m2 bXh 2 � +gR∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FhgR b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FhmambX0 + � gR∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FhgL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fhma + gL∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FhgR b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fhmb � mFCh 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' DhFF (ab)R =DhFF (ab)L � gL a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fh ↔ gR a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' gL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fh ↔ gR b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fh � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (17) It is noted that the Fhh contributions are the sum of three diagrams (1), (3), and (4), while the hFF contributions are from the only diagram (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' We emphasize that the electric charge conversation QF = Qh + Qe is one of the necessary requirements to guarantee the WI given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (12), see a detailed proof in appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' We can see this crudely from the necessary condition that div[DhFF (ab)L] + div[DFhh (ab)L] ∼ gL∗ a gL b (Qe + Qh − QF) = 0 and div[DhFF (ab)R] + div[DFhh (ab)R] ∼ gR∗ a gR b (Qe + Qh − QF) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' This conclusion supports completely the only case of electric conversation among the remaining ones mentioned in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Regarding Lagrangian (4), which results in four diagrams in the second line of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' diagram (5) gives the following FV V contributions: CFV V (ab)L = − eQV 16π2 � gR∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV mF � 3Xf 3 + 1 2m2 V � − gL∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gR b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV mF × mamb m2 V Xf 012 + gL∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV ma � 2(Xf 1 − Xf 3 ) + m2 FXf 01 + m2 bXf 2 m2 V � +gR∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gR b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV mb � 2(Xf 2 − Xf 3 ) + m2 FXf 02 + m2 aXf 1 m2 V �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (18) where Xf i = Xi(m2 F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' m2 V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' m2 V ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' and CFV V (ab)R = − eQV 16π2 � gL∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gR b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV mF � 3Xf 3 + 1 2m2 V � − gR∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV mF × mamb m2 V Xf 012 + gR∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gR b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV ma � 2(Xf 1 − Xf 3 ) + m2 FXf 01 + m2 bXf 2 m2 V � +gL∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV mb � 2(Xf 2 − Xf 3 ) + m2 FXf 02 + m2 aXf 1 m2 V �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (19) Diagram (6) gives V FF contributions: CV FF (ab)L = − eQF 16π2 � magL∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV � 2Xv 01 + m2 F (Xv 1 − Xv 3) + m2 bXv 2 m2 V � 8 + mbgR∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gR b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV � 2Xv 02 + m2 F (Xv 2 − Xv 3) + m2 aXv 1 m2 V � − gR∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV mF � 4X0 + m2 aXv 1 + m2 bXv 2 − m2 FXv 3 m2 V � −gL∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gR b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV mamb m2 V × mF(Xv 12 − Xv 3) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (20) where all Xv i are expressed in terms of PV functions CV FF 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ij = C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ij(m2 V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' m2 F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' m2 F),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' and CV FF (ab)R = − eQF 16π2 � magR∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gR b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV � 2Xv 01 + m2 F (Xv 1 − Xv 3) + m2 bXv 2 m2 V � + mbgL∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV � 2Xv 02 + m2 F (Xv 2 − Xv 3) + m2 aXv 1 m2 V � − gL∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gR b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV mF � 4Xv 0 + m2 aXv 1 + m2 bXv 2 − m2 FXv 3 m2 V � −gR∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV gL b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV mbma m2 V × mF(Xv 12 − Xv 3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (21) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' using the simple notations gL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='R a ≡ gL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='R a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' the formulas of D(ab)L and D(ab)R are D(78) (ab)L =D(7) (ab)L + D(8) (ab)L = eQe 16π2(m2 a − m2 b) � � gL∗ a gR b mb + gR∗ a gL b ma � 3mF � B(1) 0 − B(2) 0 � − mb � magR∗ a gR b + mbgL∗ a gL b � �� 2 + m2 F + m2 b m2 V � B(2)v 1 + A0(m2 V ) + 2m2 FB(1) 0 m2 V + 1 � + ma � mbgR∗ a gR b + magL∗ a gL b � �� 2 + m2 F + m2 a m2 V � B(1)v 1 + A0(m2 V ) + 2m2 FB(2) 0 m2 V + 1 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (22) D(78) (ab)R =D(78) (ab)L � gL a ↔ gR a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' gL b ↔ gR b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' DFV V (ab)L = − eQV 16π2 � gL∗ a gL b � 2(d − 2)Cf 00 + 2(m2 a + m2 b)Xf 3 − 1 m2 V � m2 F(B(1) 0 + B(2) 0 − 2Cf 00) + A0(m2 V ) + m2 aB(1)f 1 + m2 bB(2)f 1 �� + gR∗ a gR b mamb � 4Xf 3 + 2Cf 00 m2 V � + gR∗ a gL b × mamF � 3Cf 0 − 1 2m2 V + m2 bXf 012 m2 V � +gL∗ a gR b × mbmF � 3Cf 0 − 1 2m2 V + m2 aXf 012 m2 V �� , DFV V (ab)R =CFV V (ab)L � gL a ↔ gR a , gL b ↔ gR a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (23) 9 The remaining formulas of D(ab)L,R from diagram (6) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1 are ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='DV FF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='(ab)L = eQF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='16π2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='gL∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a gL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='− 2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FC0 + (d − 2)2Cv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='00 + 2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='aXv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='01 + 2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='bXv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='02 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='(2 − d)m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FCv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='00 + A0(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ) + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='B(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ B(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='−m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='B(1)v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ B(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='− m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='B(2)v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ B(2)v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='am2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F)C0 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='aXv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='bXv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+gR∗ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='(2 − d)Cv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='00 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FXv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='3 − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='aXv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='bXv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+gR∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a gL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b mamF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='−B(1)v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ (2 − d)C00 − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='aXv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 + m2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='bXv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='2 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a(Xv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='3 − Xv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' DV FF (ab)R =DV FF (ab)L � gL a ↔ gR a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' gL b ↔ gR b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (24) We note that all results presented here are crosschecked by FORM package [48], using intermediate steps given in appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' There is a property that CX (ab)R = CX (ab)L � gL a ↔ gR a , gL b ↔ gR b � for all X = Fhh, hFF, FV V, V FF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The above results of one- loop contribution to C(ab)L,R are totally consistent with those introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [36], after some transformations of notations presented in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In the limit of m2 h, m2 V ≫ m2 a, m2 b, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=', m2 a/m2 B, m2 b/m2 h ≃ 0 with B = h, V , we get consistent results with those given in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [27, 55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' To derive the above results for gauge boson exchanges, we start with many important features different from those mentioned in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [36], namely: i) we do not use the typical form of couplings relating with Goldstone bosons going along with the pres- ence of new gauge bosons, ii) we have to use the massless property of the on-shell photon q2 = 0, iii) to confirm the WI for all diagrams given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1, we need the charge conver- sation law corresponding to the Lagrangian (1): QF = QV + Qe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Therefore, our calculation is another independent approach to confirm the result given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The details of the calculation to confirm the WI for all one-loop contributions are given in appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' We remind that our results are derived from the photon couplings listed in the table I, and do not contain the contributions from the FSV diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In the following, we pay attention to the possibility of adding the FSV diagrams or the new forms of the photon couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 10 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' DISCUSSION ON WI AND PREVIOUS RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' WI to constrain the form of photon couplings Now we focus on the feature that the WI of the on-shell photon will constrain strongly the forms of the cubic photon couplings with two physical particles in a renormalized Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Now we consider the existence of the photon couplings type at tree level: LγXX =eQFAµ � F1γµF2 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' � + eQhAµ [(h∗ 1∂µh2 − h2∂µh∗ 1) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='] − � eQV AµV ν 1 V λ∗ 2 Γµνλ(p0, p+p−) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' � + � gγhV gµνh−QAµV Qν + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' � , (25) where all couplings are more general than those well-known as the standard forms given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The last term corresponds to the photon couplings with a scalar h and a gauge boson V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The above Lagrangian results in the following decays from the heavy particle to lighter one: i) F2 → F1γ, ii) h2 → h1γ, iii) V2 → V1γ, and iv) V → hγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The WI for these decay amplitudes at tree level is Mµ(X1 → X2γ)p0µ = 0 with p0µ being the external photon momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' It can be derived that: Using the same convention of external momenta given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1, we have Mµ(F2 → F1γ)qµ ∼ (mF2 − mF1)uF2(p2)uF1(p1) = 0, where p0 ≡ −q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Therefore, mF2 = mF1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' This case is automatically satisfied for the tree level AMM amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Mµ(h2 → h1γ)p0µ ∼ (p2−p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (p2+p1) = (m2 h2 −m2 h1) = 0, where all on-shell momenta are incoming the vertex Aµh∗ 1h2, implying that p0 = −(p1 + p2) and p2 1,2 = m2 h1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The consequence is mh1 = mh2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Mµ(V → hγ)p0µ ∼ εv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='p0 = 0, where εv and p0 are the polarization of gauge boson V and the external momentum of the photon Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Hence the presence of a AhV vertex does not automatically satisfy the WI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' One-loop contributions for all diagrams arising from this vertex must be checked for the validation of WI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Mµ(V1 → V2γ)pµ 0 ∼ εν 1ελ∗ 2 pµ 0Γµνλ(p0, p1, p2) = 0, where ε1,2, and p1,2,0 are the polariza- tion of the gauge boson V1,2 and the external momentum of the gauge bosons V1,2 and photon Aµ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' We will use the following properties of the external gauge bosons Vi(i = 1, 2) and photon: εi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='pi = 0, p2 0 = 0, p2 i = m2 Vi, and the momentum 11 conversation p0 + p1 + p2 = 0 following notations in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' After some intermediate steps of calculation, we have: Mµ(V1 → V2γ)pµ 0 ∼(p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε1) [(p0 − p1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ 2] + (ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ 2) [(p1 − p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='p0] + (p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ 2) [(p2 − p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε1] =(ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ 2) � m2 V2 − m2 V1 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (26) Hence, mV1 = mV2 is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' From this, we consider the more general photon coupling with a gauge boson [49] describing the couplings of a leptoquark field [50] Γ′ µνλ(p0, p1, p2) =gµν(kvp0 − p1)λ + gνλ(p1 − p2)µ + gλµ(p2 − kvp0)ν =Γµνλ(p0, p1, p2) + δkv (gµνp0λ − gλµp0ν) , (27) with δkv = kv − 1 showing the deviation from the standard vertex listed in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' This may change the one-loop contributions of the diagram (5) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1, hence change the formulas of CFV V (ab)L,R given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (18) and (19), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' One can prove immediately that the vertex deviation δΓµνλ(p0, p1, p2) ≡ Γ′ µνλ(p0, p1, p2) − Γµνλ(p0, p1, p2) = δkv (gµνp0λ − gλµp0ν) (28) guarantees the WI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The new one-loop contributions arising from δΓ are also satisfied the WI, see analytic formulas given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Now we start from the point that all results of one loop contributions given from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (15) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (24) based on the standard forms of photon couplings given in table I, where a photon always couples with two identical physical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' On the other hand, a recent work [34] assumed the existence of a new photon coupling kind ASV , which may appear in some BSM, in which the photon couples with one gauge boson V and one scalar S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The appearance of a boson V or S will generate by itself the one-loop contributions that always guarantee the WI by the respective set of four diagrams given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Hence, the two FSV diagrams must give contributions satisfying the WI themselves, namely DFSV (ab)L + maCFSV (ab)L + mbCFSV (ab)R = DFSV (ab)R + maCFSV (ab)R + mbCFSV (ab)L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (29) As a result, the divergent parts for both L and R parts give: 0 =gγSV � 2gL∗ ahgL bV mF − gL∗ ahgR bV mb − gR∗ ah gL bV ma � 12 =gγSV � 2gR∗ ah gR bV mF − gR∗ ah gL bV mb − gL∗ ahgR bV ma � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (30) Considering the case of gγSV ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Then, all quantities gL ah, gR bh, gL bV , and gR bV are zero if at least one of them is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' More strictly, we require that the two Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (29) must be hold for both divergent and finite parts arising from D(ab)L,R and C(ab)L,R given in appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Consequently, gγSV = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=', the FSV diagram type does not satisfy the WI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Regarding to the vertex deviation of the AV V couplings defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (28), the new one-loop contributions relating to CFV V (ab)L,R and DFV V (ab)L,R are shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C36) of appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Our results are consistent with previous works [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Although they satisfy the WI, they contain divergent parts, for example div � −δCFV V L � = δkveQV 32π2m2 V � gL∗ a gL b ma + gR∗ a gR b mb − 2gR∗ a gL b mF � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (31) Hence, δkv = 0 is equivalent to the renormalizable condition of the theory, see a more detailed explanation in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' This confirms that the AV V coupling listed in table I is still valid for a general UV-complete model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Consequently, δCFV V L = 0, implying that the results of CFV V (ab)L,R given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (18) and (19) are unchanged for many renormalizable theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Discussions on previous results It is easy to derive that C(ab)L,R = σL,R corresponding to the notations given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [36], see a detailed explanation in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' This confirms a perfect consistence of the two results obtaining from different original assumptions that we have indicted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In addition, these results are also consistent with those given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [27] in the limit of heavy boson masses in the loops, which are very useful for studying the correlations of AMM and cLFV decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In some BSM, SM light quark may play role of the light fermions u, d ≡ F in the Yukawa couplings [29], hence the condition m2 F ≫ m2 a, m2 b is not held.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' But numerical illustrations [39] to investigate cLFV decays eb → eaγ with very light neutrinos show that the case of m2 F ≪ m2 a are also valid for approximation formulas with m2 a = m2 b = 0, provided m2 a, m2 b ≪ m2 h, m2 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' An analytic approximation to explain this result was given in, for example Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 13 For analytic formulas of cLFV and aea introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [28], They can be changed into the form of PV-functions consistent with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' An exceptional case mentioned there is the couplings of a double charged boson with two identical leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' For example, the Lagrangian containing couplings of a doubly charged Higgs boson is [28]: Lint = gij s3φ++ℓC i ℓj + gij p3φ++γ5ℓC i ℓj + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=', (32) where we can identify that gR a,Fh = gij s3 +gij p3 and gR a,Fh = gij s3 −gij p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' But the Feynman rules for the a vertex ℓC i ℓjφ++ containing two identical leptons gives an extra factor 2, implying that C(ab)L,R given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (15) and (16) must be added a factor 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Instead of many particular formulas to calculate one-loop contributions relating to different charged particles, the one- loop results for (g − 2)ea and eb → eaγ decays can be generalized for aea with an arbitrary electric charge QF of a new fermion and the boson with QB = QF − Qe with B = h, V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' the aea formulas are aea(h) =Qhma 16π2 � 1 0 dx × x(x − 1) � 2Re[gRL]mF + (gLL + gRR)max � (1 − x)m2 F + x [m2 h + m2 a(x − 1)] + QFma 16π2 � 1 0 dx × x2 � −2gRL[gRL]mF + (gLL + gRR)ma(x − 1) � (1 − x)m2 h + x [m2 F + m2 a(x − 1)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='(33) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='aea(V ) = − QV ma ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='16π2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='dx × ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='�Re[gRL]mF [m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F(x − 1) + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V x(6x − 1) + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ax(3 − 5x + 2x2)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='(1 − x)m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F + x [m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a(x − 1)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='−ma(gLL + gRR) [m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F(2 − 3x + x2) + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V 2x(x + 1) + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ax(x − 1)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='(1 − x)m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F + x [m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a(x − 1)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ QFma ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='16π2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='�2gRL[gRL]mFx [m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fx − 4m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V (1 − x) + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ax(2x − 1)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='(1 − x)m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V + x [m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a(x − 1)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+(gLL + gRR)max [m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Fx(1 + x) + 2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V (2 − 3x + x2) + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ax(x − 1)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='(1 − x)m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V + x [m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a(x − 1)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (34) where gRL = gR∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FBgL a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' gLL = gL∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FBgL a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' and gRR = gR∗ a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FBgR a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FB with B = h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The coupling identifications are gR a,Fh = gaa sk + gaa pk and gR a,Fh = gaa sk − gaa pk for k = 1, 2, 3 relating to neutral, singly, and doubly charged Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Similarly for the gauge bosons, gR a,FV = gaa vk + gaa ak and gR a,FV = gaa vk − gaa ak for QV = 1, 0, −1, 2 corresponding k = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The two formulas (33) and (34) are derived by inserting the PV functions given in appendix A in the limit p2 1 = p2 2 = m2 a into C(ab)L,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' We have checked that our results are consistent with all HFF, FHH, and V FF contributions relating to the diagrams (1), (2), and (6) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' For the one-loop FV V contributions arising from the diagram (5), there is a 14 difference between our result and that in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [28], namely δ(aea)(FV V ) = QV mamF 16π2m2 V (|gaa vk|2 − |gaa ak|2) � 1 0 dx(2x + 1) = QV mamF 8π2m2 V (|gaa vk|2 − |gaa ak|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' It shows that the two results are consistent if gaa vk = ±gaa ak,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=', gL a,FBgR a,FB = 0, which appears in many BSM such as the SM, 3-3-1 models,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' We also see that the FV V contribution to aea of the doubly gauge boson given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [28] has an opposite sign with our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' We note that our results are also valid as the exact solutions for studying the AMM and eb → eaγ decay in BSM consisting of very light bosons mB ≪ m2 a, m2 b such as an axion-like particle (ALP) [58, 59], or a new scalar singlet [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' CONCLUSION Using the unitary gauge, we confirm the exact results of analytic formulas in terms of PV functions for one-loop contributions to the cLFV decay rates eb → eaγ given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [36], which are also applicable to compute the AMM of charged leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' These results are con- sistent with those given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [27] in the limit of heavy bosons mB ≫ ma, mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The general expressions in terms of PV-functions are very convenient to change into available forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Our calculations here are in many new feature as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Our calculation is independent with the Goldstone boson couplings of the new gauge bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The Ward Identity of the external photon constrains allows only the couplings of photon with two identical physical particles, as given in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' At tree-level, the ASV couplings do not satisfy the WI if εv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='p0 ̸= 0, where εv and p0 are the polarization of gauge boson V and the external momentum of the photon, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The one-loop FSV contributions arising from this vertex type to cLFV amplitudes and AMM do violate the WI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Therefore, the results given in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [27, 36] are valid in all renormalizable BSM respecting the WI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' They are still applied for other similar decays of quarks q → q′γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The photon-scalar-vector ASV vertex does not appear in BSM satisfying the WI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Our conclusion is very useful for constructing loop calculations relating to photon couplings, where only the vertex types listed in Table I are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Acknowledgments This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the grant number 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='01-2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Hue is thankful 15 to Van Lang University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Appendix A: PV functions for one loop contributions defined by LoopTools 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' General notations The PV-functions used here were listed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [39], namely A0(m2) = (2πµ)4−d iπ2 � ddk k2 − m2 + iδ, B{0,µ,µν}(p2 i , M 2 1, M 2 2) = (2πµ)4−d iπ2 � ddk × {1, kµ, kµkν} D0Di , i = 1, 2, C0,µ,µν = (2πµ)4−d iπ2 � ddk{1, kµ, kµkν} D0D1D2 , Cµ = (−p1µ) C1 + (−p2µ) C2, Cµν = gµνC00 + p1µp1νC11 + p2µp2νC22 + (p1µp2ν + p2µp1ν)C12, (A1) where D0 ≡ k2 − M 2 1 + iδ, Di ≡ (k − pi)2 − M 2 2 + iδ, C0,µ,µν = C0,µ,µν(p2 1, 0, p2 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' M 2 1, M 2 2, M 2 2), µ is an arbitrary mass parameter introduced via dimensional regularization [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In this work, we discuss only the case of external photon q2 = (p2 − p1)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The scalar functions A0, B0, C0, C00, Ci, Cij (i, j = 1, 2) are well-known PV functions, which are consistent with those defined by LoopTools [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The well-known relations are: B(i) 0 ≡ B(i) 0 (p2 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' M 2 1, M 2 2) = B(i) 0 (p2 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' M 2 2, M 2 1), B(i) 1 ≡ B(i) 1 (p2 i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' M 2 1, M 2 2) = − 1 2p2 i � A0(M 2 2) − A0(M 2 1) + fiB(i) 0 � , (A2) where fi = p2 i + M 2 2 − M 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The scalar functions A0, B0, C0 can be calculated using the techniques of [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Other PV functions needed in this work are B0,µ,µν(M2) = (2πµ)4−d iπ2 � ddk {1, kµ, kµkν} D1D2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A3) For simplicity, we define the following notations appearing in many important formulas: X0 ≡ C0 + C1 + C2, X1 ≡ C11 + C12 + C1, X2 ≡ C12 + C22 + C2, 16 X3 ≡ C1 + C2 = X0 − C0, X012 ≡ X0 + X1 + X2, Xij = Xi + Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A4) Depending on the form of the PV-functions, we have Xf i = Xi(m2 F, m2 B, m2 B), Xh i ∼ Xi(m2 B, m2 F, m2 F), and Xv i ∼ Xi(m2 V , m2 F, m2 F), corresponds to the diagrams FBB, HFF, and V FF with B = h, V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' p2 1 ̸= p2 2 ̸= 0 and p2 1, p2 2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' From the definitions of PV-functions given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A1), it can be proved that: B(0) 0 ≡ B(0) 0 (M2) ≡ B0(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' M2) = CUV − ln(M 2 2) + O(ϵ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' A0(M) = M 2 � B(0) 0 (M) + 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A5) Bµ(M2) = 1 2B(0) 0 (pµ 1 + pµ 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A6) Bµν(M2) = gµν 2 M 2 2 � B(0) 0 + 1 � + 1 6B(0) 0 (2pµ 1pν 1 + pµ 1pν 2 + pµ 2pν 1 + 2pµ 2pν 2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' C00 = 1 4 � 2M 2 2C0 + (M 2 2 − M 2 1 + m2 a) B(1) 0 − (M 2 2 − M 2 1 + m2 b) B(2) 0 m2 a − m2 b + 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A7) where CUV is defined as the divergent part of the PV functions when D → 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' CUV = 1/ϵ − γE + log(4πµ2) with γE being Euler’s constant and D = 4 − 2ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' It is well-known that the PV-functions having non-zero divergent parts are: div � B(0) 0 � = div � B(1) 0 � = div � B(2) 0 � = −2div � B(1) 1 � = −2div � B(2) 1 � = 4div [C00] = CUV , div [A0(M)] = M 2CUV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A8) As mentioned in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [39], we can derive all formulas of Ci, and Cij as functions of A0, B(i) 0 , and C0 consistent with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [39],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' using the following relations: 2m2 aC1 + (m2 a + m2 b)C2 = −faC0 − B(0) 0 + B(2) 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (m2 a + m2 b)C1 + 2m2 bC2 = −fbC0 − B(0) 0 + B(1) 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 2C00 + 2m2 aC11 + (m2 a + m2 b)C12 = 1 2B(0) 0 − faC1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 2m2 aC12 + (m2 a + m2 b)C22 = 1 2B(0) 0 + B(2) 1 − faC2 2C00 + (m2 a + m2 b)C12 + 2m2 bC22 = 1 2B(0) 0 − fbC2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 17 (m2 a + m2 b)C11 + 2m2 bC12 = 1 2B(0) 0 + B(1) 1 − fbC1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 4C00 − 1 2 + m2 aC11 + (m2 a + m2 b)C12 + m2 bC22 = B(0) 0 + M 2 1C0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A9) where fa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b = M 2 2 −M 2 1 +m2 a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' and C12 = C21 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In this work, we need just combinations of these PV-functions for our immediate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' we can prove that: X0 = −B(1) 0 − B(2) 0 m2 a − m2 b ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' X12 = −B(1) 1 − B(2) 1 m2 a − m2 b = A0(M 2 1) − A0(M 2 2) 2m2 am2 b + (M 2 1 − M 2 2) 2(m2 a − m2 b) � B(1) 0 m2 a − B(2) 0 m2 b � − 1 2X0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' m2 aB(1) 1 − m2 bB(2) 1 = −1 2 �� m2 a + M 2 2 − M 2 1 � B(1) 0 − � m2 b + M 2 2 − M 2 1 � B(2) 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' b1 ≡ m2 aB(1) 1 − m2 bB(2) 1 (m2 a − m2 b) = −(2C00 + m2 aX1 + m2 bX2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (2 − d)C00 + M 2 2C0 = −2C00 + 1 2 + M 2 2C0 = −(m2 a + M 2 1 − M 2 2) B(1) 0 − (m2 b + M 2 1 − M 2 2) B(2) 0 2(m2 a − m2 b) = b1 + (M 2 2 − M 2 1)X0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A10) where A0(M 2 2) = M 2 2(B(0) 0 + 1) and A0(M 2 1) = M 2 1(B(0) 0 + 1 + ln(M 2 2/M 2 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' It was proved previously, for example [39], that B0(p2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' M 2 1, M 2 2) = B0(p2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' M 2 2, M 2 1) = CUV − ln(M 2 2) + 2 − � σ=± (1 − 1 xσ ) ln (1 − xσ) , C0(m2 a, 0, m2 b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' M 2 1, M 2 2, M 2 2) = − 1 m2 a − m2 b � σ=± [Li2(yaσ) − Li2(ybσ)] , (A11) where p = pa, pb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' and x± = 1 2M 2 2 � (M 2 2 − M 2 1 + p2) ± � (M 2 2 − M 2 1 + p2)2 − 4M 2 2p2 � , ya± = 1 2M 2 2 � (M 2 2 − M 2 1 + m2 a) ± Λ � , yb± = xa± [b → a] (A12) with Λ = (M 4 1 + M 4 2 + m4 a − 2M 2 1M 2 2 − 2M 2 1m2 a − 2M 2 2m2 a)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The above formula of C0 is also consistent with that introduced in loop-induced decay amplitude of h → Zγ [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' m2 a = p2 a = p2 b ̸= 0 Formulas for AMM in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [34] require that analytic formulas of PV functions with mb = ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' It seems that the results of PV-functions listed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [39] are not valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' But the limit mb = ma can be derived mathematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' For example, the result of C0 given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A11) leads to a consequence that C0(m2 a, 0, m2 a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' M 2 1, M 2 2, M 2 2) = lim mb→ma C0(m2 a, 0, m2 b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' M 2 1, M 2 2, M 2 2) = − ∂ ∂(m2 a) � σ=± Li2(yaσ) = � σ=± y′ aσ ln(1 − yaσ) yaσ = � σ=± ln(1 − yaσ) 2M 2 2yaσ × � 1 − σ × M 2 1 + M 2 2 − m2 a Λ � , (A13) where f ′ ≡ ∂f/(∂m2 a) denotes a well-known derivative notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In addition, B(1) 0 = B(2) 0 and B(1) 1 = B(2) 1 is automatically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Many formulas containing (m2 a − m2 b) in the denominators corresponding a derivative in the limit ma → mb: X0 = −B(1)′ 0 = � σ=± y′ aσ [yaσ + ln(1 − yaσ)] y2 aσ , X12 = −B(1)′ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' , (A14) In this way, we can confirm all results introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' There is another way to calculate form factors, using the Feynman trick: 1 D0D1D2 = Γ(3) � 1 0 dx dy dz δ(1 − x − y − z) D3 , (A15) where D = [k − (yp1 + zp2)]2 − M 2 + iδ, M 2 = y(y + z − 1)p2 1 + z(y + z − 1)p2 2 + xM 2 1 + (1 − x)M 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A16) With M 2 0 = (p2 2 − p2 1)xy − x(1 − x)p2 2 + xM 2 1 + (1 − x)M 2 2, the PV functions are: C{0,1,2,11,22,12} = − � 1 0 dx � 1−x 0 dy {1, −y, −(1 − x − y), y2, (1 − x − y)y, (1 − x − y)2} M 2 0 , X0,1,2,3 = − � 1 0 dx � 1−x 0 dy × {x, −xy, −x(1 − x − y), −(1 − x)} M 2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A17) 19 The expressions of Xi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A17) are very convenient for the case of (g − 2) anomaly, where p2 1 = p2 2 = m2 a results in M 2 0 = −x(1−x)m2 a +xM 2 1 +(1−x)M 2 2, which is independent with y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Consequently, the X0,1,2,3 = − � 1 0 dx{x(1 − x), −x(1 − x)2/2, −x(1 − x)2/2, −(1 − x)2} M 2 0 = − � 1 0 dx{x(1 − x), −(1 − x)x2/2, −(1−)x2/2, −x2} M 2 0 , (A18) Formulas of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A18) are enough to check the consistence between our results with those of (g − 2) anomalies and cLFV amplitudes mentioned in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Using the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A18), we can write the general formulas of aµ as shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (33) and (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In deed, all integrals in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (33) and (34) can be solved analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Starting from the general formulas of M 2 0 as a functions of x: M 2 0(x) = m2 a(x − x+)(x − x−) corresponding to the two solutions x±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' All numerators in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (33) and (34) are always written in the following forms: ax2 + bx2 + c = a1M 2 0 + b1 dM 2 0 dx + c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A19) The consequence is � 1 0 dx × ax2 + bx2 + c M 2 0 = a1 + b1 ln M 2 1 M 2 2 + c1 √ Λ ln �(1 − x−)x+ (1 − x+)x− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A20) The result in this way must be consistent with those discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [34], hence we do not show precisely here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' p2 a = p2 b = 0 Results for the case of p2 a = p2 b = 0 were provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [36], namely C0 = a = M 2 1 − M 2 2 + M 2 1 ln � M2 2 M2 1 � (M 2 1 − M 2 2)2 , C1 = C2 = c = − 3M 4 1 − 4M 2 1M 2 2 + M 4 2 + 2M 4 1 ln � M2 2 M2 1 � 4(M 2 1 − M 2 2)3 , C11 = C22 = 2C12 = d = 11M 6 1 − 18M 4 1M 2 2 + 9M 2 1M 4 2 − 2M 6 2 + 6M 6 1 ln � M2 2 M2 1 � 18(M 2 1 − M 2 2)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A21) 20 This approximate formulas of PV functions give results consistent with those given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [27], namely fh(x) = 2˜gh(x) = x2 − 1 − 2x log x 4(x − 1)3 , gh(x) = x − 1 − log x 2(x − 1)2 , ˜fh(x) = 2x3 + 3x2 − 6x + 1 − 6x2 log x 24(x − 1)4 , (A22) fV (x) = x3 − 12x2 + 15x − 4 + 6x2 log x 4(x − 1)3 , gV (x) = x2 − 5x + 4 + 3x log x 2(x − 1)2 , ˜fV (x) = −4x4 + 49x3 − 78x2 + 43x − 10 − 18x3 log x 24(x − 1)4 , ˜gV (x) = −3(x3 − 6x2 + 7x − 2 + 2x2 log x) 8(x − 1)3 , where x ≡ m2 F/m2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The diagrams FBB and BFF corresponds to different identifications that {M1, M2} = {mF, mB} or and {M1, M2} = {mB, mF}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Appendix B: Notations in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [36] Corresponding to the two one-loop diagram classes FV V and V FF, we have the following equivalence between two classes of notations {a, c1, c2, d1, d2, f, g} ≡ {C0, C2, C1, C22, C11, C12, C00}B , � ¯a, −¯c1, −¯c2, ¯d1, ¯d2, ¯f, ¯g � ≡ {C0, C2, C1, C11, C22, C12, C00}f , where B = h, V are gauge bosons in the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In addtion, the diferent notations in the definitions of the one-loop integrals given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A1), we have {m1, m2} ≡ {mb, ma} while {p1, p2} ≡ {−p2, −p1} and {p1, p2} ≡ {p2, p1} for the diagrams V FF and FV V respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The couplings in the Yukawa Lagrangians of physical bosons are L1 ≡ gL b , R1 ≡ gR b , L2 ≡ gL a , and R2 ≡ ga R, which result in the following equvalences: λ ≡ gL∗ a gL b = gLL, ρ ≡ gR∗ a gR b = gRR, ζ ≡ gL∗ a gR b = gLR, and v ≡ gR∗ a gL b = gRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' As a result, we can identify that: k1 = mbXB 2 , k2 = maXB 1 , k3 = mF(c1 + c2) = mFXB 3 , ¯k1 = mbXf 2 , ¯k2 = mbXf 1 , k3 = −mFXf 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (B1) 21 For a gauge boson Bµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' the one-loop form factors relate to the following notations: y1 = mb � 2Xf 02 + m2 F(Xf 2 − Xf 3 ) + m2 aXf 1 m2 B � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' y2 = ma � 2Xf 01 + m2 F(Xf 1 − Xf 3 ) + m2 bX2 m2 B � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' y3 = mF � −4Xf 0 + m2 FXf 3 + m2 aXf 1 + m2 bXf 2 m2 B � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' y4 = −mambmF(Xf 12 − Xf 3 ) m2 B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (B2) and ¯y1 = mb � 2(Xf 2 − Xf 3 ) + m2 FXf 02 + m2 aXf 1 m2 B � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' ¯y2 = ma � 2(Xf 1 − Xf 3 ) + m2 FXf 01 + m2 bXf 2 m2 B � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' ¯y3 = mF � 4Xf 3 + −m2 FX0 − m2 aX1 − m2 bX2 m2 B � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' ¯y4 = −mambmF m2 B X012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (B3) Appendix C: Important steps to derive C(ab)L,R and D(ab)L,R by hand The notations for calculating the amplitude corresponding to all diagrams of both Higgs and gauge boson exchanges are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Although all the internal momenta have (1) eb ea γ p1 q p2 k1 k2 k (2) F h+ eb ea ea eb eb (3) k k1 (4) ea ea eb k k2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 2: Momneta notations to derive the one-loop contributions opposite signgs with those denoted following LoopTools, the PV-functions are defined with the same values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The relations relevant with momenta are: ki = k − pi, p2 1 = m2 a, p2 = q + p1, p2 2 = m2 b, q2 = 0, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ = 0, p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ = p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗, (C1) Only four diagrams (1), (2), (5), and (6) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1 give non zero contributions to C(ab)L,R, hence we firstly derive C(ab)L,R as the factors of (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) in the amplitudes arising from these diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' For convenience in detailed calculations, we use simple notations for all the couplings factors gaL,R FB → gL,R a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' For integrals containing divergences, we use the regular dimensional regularization defined by the following replacement: � d4k (2π)4 → i 16π2 × (2πµ)4−d iπ2 � ddk ≡ � Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 22 The final results now are written in terms of the PV functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In many intermediate steps, we use many results for products of gamma matrices in the dimension d [51], namely γµγµ = d, γµγνγµ = (2 − d)γν → γµ/pγµ = (2 − d)/p, γµγνγργµ = 4gνρ + (d − 4)γνγρ → γµ/p1/p2γµ = 4p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='p2 + (d − 4)/p1/p2, γµγνγργσγµ = −2γσγργν − (d − 4)γνγργσ → γµ/p1/p2/p3γµ = −2/p3/p2/p1 − (d − 4)/p1/p2/p3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Scalar contributions We list here 8 formulas of amplitudes corresponding to 8 particular diagrams shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Namely, for three diagrams (1), (3), and (4) we have iM1 = − eQH � Dk × ua[gR a ∗PL + gL a ∗PR](mF + /k) D0D1D2 [gL b PL + gR b PR]ub × (2k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗), (C2) iM3 = −eQe m2 a − m2 b � Dk × ua[gR a ∗PL + gL a ∗PR](mF + /k) D0D1 [gL b PL + gR b PR](mb + /p1)/ε∗ub, (C3) iM4 = eQe m2 a − m2 b � Dk × ua/ε∗(ma + /p2)[gR a ∗PL + gL a ∗PR](mF + /k) D0D2 [gL b PL + gR b PR]ub, (C4) where D0 = k2 − m2 F and Di = k2 i − m2 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The amplitude for the diagram (2) is: iM2 = −eQF � Dk × ua[gR a ∗PL + gL a ∗PR](mF − /k1)/ε∗(mF − /k2) D0D1D2 [gL b PL + gR b PR]ub, (C5) where D0 = k2 − m2 h and Di = k2 i − m2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In the next calculation, we use the following simple notations: gLL ≡ gL∗ a gL b , gRR ≡ gR∗ a gR b , gRL ≡ gR∗ a gL b , gLR ≡ gL∗ a gR b , A1 = gL∗ a gR b PR + gR∗ a gL b PL, A2 = gL∗ a gL b PL + gR∗ a gR b PR, (C6) where gL,R a ≡ gL,R a,Fh and gL,R b ≡ gL,R b,Fh without any confusions with the gauge boson couplings gL,R a,FV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' It is not hard to write all amplitudes in terms of PV-functions as follows: M1 =−eQH 16π2 ua � −2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ [A1] mFX0 + � 2Cf 00/ε∗ + � Xf 1 /p1 + Xf 2 /p2 � (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � [A2] � ub, (C7) M3 =−eQe 16π2 × ua[gR a ∗PL + gL a ∗PR](mFB(1) 0 − B(1)f 1 /p1)[gL b PL + gR b PR](mb + /p1)/ε∗ub (m2 a − m2 b) , (C8) M4 = eQe 16π2 × ua/ε∗(ma + /p2)[gR a ∗PL + gL a ∗PR](mFB(2) 0 − B(2)f 1 /p2)[gL b PL + gR b PR]ub (m2 a − m2 b) , (C9) 23 and M2 = − eQF � Dk × ua � m2 F/ε∗ + /k1/ε∗/k2 � [A2]ub − eQF(−1)mF � Dk × ua � 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ − /p1/ε∗ − /ε∗/p2 � [A1]ub =−eQF 16π2 ua �� m2 FC0 + (2 − d)C00 � /ε∗ + (C11 + C1)/p1/ε∗/p1 + (C22 + C2)/p2/ε∗/p2 +(X0 + C12)/p1/ε∗/p2 + C12/p2/ε∗/p1 � × [A2]ub − eQFmF 16π2 ua � (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)(C1 + C2) + � /p1/ε∗ + /ε∗/p2 � C0 � [A1]ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C10) The validation of the WI given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (12) implies whether f WI L = 0 is correct with: f WI L ≡D(ab)L + maC(ab)L + mbC(ab)R =gLL � �� Qe � m2 aB(1) 1 − m2 bB(2) 1 �f m2 a − m2 b − �1 2 − 2Ch 00 + m2 FCh 0 � Qf −Qh � m2 aX1 + m2 bX2 + 2C00 �f� + gRRmamb � � � Qe � B(1) 1 − B(2) 1 �f m2 a − m2 b − QfXh 012 − QhXf 12 � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C11) We have used many formulas listed in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A2) and (A10) to show that 0 = Xf 12 + Xh 12 + X0 → Xh 012 = −Xf 12, bf 1 = − � m2 aX1 + m2 bX2 + 2C00 �f = 1 2 − 2Ch 00 + m2 FCh 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C12) Finally, the electric charge conservation QF = Qe + Qh must be satisfied so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C11) resulting in f WI L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' On the other word, the WI is valid for only one-loop Higgs contribu- tions arising from the set of four diagrams (1)-(4) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Vector contributions To calculate the one-loop contributions from gauge boson exchanges corresponding to Lagrangian (4), we denote gL,R a ≡ gL,R a,FV and gL,R b ≡ gL,R b,FV then use the notations given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The amplitudes relevant with gauge boson exchanges are: iM5 = � Dk × uaiγα[gL∗ a PL + gR∗ a PR]i(mF + /k) D0 iγβ[gL b PL + gR b PR]ub 24 × −i D1 � gαα′ − kα 1 kα′ 1 m2 V � [−ieQV Γµα′β′ (−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' −k2) ε∗µ] −i D2 � gββ′ − kβ 2 kβ′ 2 m2 V � =eQV � d4k (2π)4uaγα[gL∗ a PL + gR∗ a PR](mF + /k) D0D1D2 γβ[gL b PL + gR b PR]ub × [Γµα′β′ (−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' −k2) ε∗µ] � gαα′ − kα 1 kα′ 1 m2 V � � gββ′ − kβ 2 kβ′ 2 m2 V � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C13) iM7 = eQe m2 a − m2 b � Dk × 1 D0D1 × � gαβ − kα 1 kβ 1 m2 V � × uaγα[gL∗ a PL + gR∗ a PR](mF + /k)γβ[gL b PL + gR b PR] � mb + /p1 � /ε∗ub,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C14) iM8 = − eQe m2 a − m2 b � Dk × 1 D0D2 × � gαβ − kα 2 kβ 2 m2 V � × ua/ε∗ � ma + /p2 � γα[gL∗ a PL + gR∗ a PR](mF + /k)γβ[gL b PL + gR b PR]ub,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C15) where D0 = k2 − m2 F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Di = k2 i − m2 V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' and Γµα′β′ (−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' −k2) = gα′β′ (k1 + k2)µ + gβ′µ (−k2 + q)α′ + gµα′ (−q − k1)β′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C16) The amplitude for the diagram (6) is: iM6 =eQF � Dk × 1 D0D1D2 × � gαβ − kαkβ m2 V � × uaγα[gL∗ a PL + gR∗ a PR](mF − /k1)/ε∗(mF − /k2)γβ[gL b PL + gR b PR]ub, (C17) where D0 = k2 − m2 V and Di = k2 i − m2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Considering diagram (7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' we have: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='iM7 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='eQe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Dk × ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ua [γαγβmF [A1] + γα/kγβ [A2]] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='mb + /p1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='/ε∗ub ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D0D1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='gαβ − kα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 kβ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='eQe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Dk × ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D0D1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='× ua ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='mF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='d − k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='[A1] + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='(2 − d)/k − /k1/k/k1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='[A2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='mb + /p1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='/ε∗ub ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ieQe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='16π2(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b)ua ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='mF [A1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='(d − 1)B(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='− A0(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ma ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='−(2 − d) + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='B(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ A0(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ) + 2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FB(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='[A2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='mb + /p1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='/ε∗ub,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C18) 25 where we have used the following results /k1/k/k1 = � D0 + m2 F � /k − 2 � D0 + m2 F � /p1 + /p1/k/p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' � d4k (2π)4 × kµ D1 = A0(m2 V )p1µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Then the one-loop contribution form factors from diagram (7) are: D(ab)L,7 = eQe 16π2(m2 a − m2 b) �� gRLma + gLRmb � mF � (d − 1)B(1) 0 − A0(m2 F) m2 V � + ma � magLL + mbgRR� �� −(2 − d) + m2 F + m2 a m2 V � B(1) 1 + A0(m2 V ) + 2m2 FB(1) 0 m2 V �� , D(ab)R,7 =D(ab)L,7 � gL a ↔ gR a , gL b ↔ gR b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C19) The same calculation for diagram (8) gives the following one-loop contribution form factor: D(ab)L,8 = − eQe 16π2(m2 a − m2 b) �� gRLma + gLRmb � mF � (d − 1)B(2) 0 − A0(m2 F) m2 V � + mb � magRR + mbgLL� �� −(2 − d) + m2 F + m2 b m2 V � B(2) 1 + A0(m2 V ) + 2m2 FB(2) 0 m2 V �� , D(ab)R,8 = D(ab)L,8 � gL a ↔ gR a , gL b ↔ gR b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C20) Using d = 4 − 2ϵ and the divergent parts of PV-functions given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A8), we get the formulas of D(ab)L,78 given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Diagram (5) From equalities q2 = 0, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ = 0, and k1 = q + k2, it is easy to prove that [Γµα′β′ (−q, k1, −k2) ε∗µ] kα 1 kα′ 1 kβ 2 kβ′ 2 = kα 1 kβ 2 {(k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='k2) [(k1 + k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗] + (k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) [k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (−k2 + q)] + (k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) [k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (−q − k1)]} ∼ (k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='k2) [2k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗] + (k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � q2 − k2 2 � + (k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � q2 − k2 1 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C21) As a result, the amplitude (C13) is written as follows: iM5 = eQV � Dkuaγα [A] γβub D0D1D2 [Γµα′β′ (−q, k1, −k2) ε∗µ] � gαα′gββ′ − gββ′kα 1 kα′ 1 + gαα′kβ 2 kβ′ 2 m2 V � , (C22) where A = mF � gL∗ a gR b PL + gR∗ a gL b PR � + [A2] /k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C23) 26 The first term in the integrand is (1) =ua � 4(k1 + k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ + (−/k + 2/p2 − /p1)/ε∗ + /ε∗(−/k + 2/p1 − /p2) � × mF [A1] ub + ua � (2 − d)(2k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/k + (−/k + 2/p2 − /p1)/k/ε∗ + /ε∗/k(−/k + 2/p1 − /p2) � × [A2] ub =ua � 6k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ − 3(/p1/ε∗ + /ε∗/p2) � mF [A1] ub + ua � (2 − d)(2k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/k + (2/p2 − /p1)/k/ε∗ + /ε∗/k(2/p1 − /p2) − 2k2/ε∗� [A2] ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C24) After integrating out, the formulas is (1) =ua � (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) × (−3mF)X3 − 3mFC0(/p1/ε∗ + /ε∗/p2) � [A1] ub + ua � (2 − d)2εα∗(Cαβ − Cβp1α)γβ + Cα � (2/p2 − /p1)γα/ε∗ + /ε∗γα(2/p1 − /p2) � −2 � B(0) 0 + m2 FC0 � /ε∗� × [A2] ub =ua(−3mF) × � (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)X3 + C0(/p1/ε∗ + /ε∗/p2) � [A1] ub + ua � /ε∗ � 2(2 − d)C00 − 2(B(0) 0 + m2 FC0) − (3m2 a + 2m2 b)C1 − (2m2 a + 3m2 b)C2 � +/p1/ε∗/p2(−3X3) � [A2] ub + ua(2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � [−2(C11 + C12) + C2] /p1 + [−2(C12 + C22) + C1] /p2 � [A2] ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C25) The second term in the integrand is � − 1 m2 V �−1 × (2) =Γµα′β′ (−q, k1, −k2) ε∗µ � gββ′kα 1 kα′ 1 + gαα′kβ 2 kβ′ 2 � × uaγα [A] γβub = ua/k1 [A] � (k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/k2 − /ε∗k2 2 � ub + ua � (k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/k1 − /ε∗k2 1 � [A] /k2ub = uamF [A1] � 2(k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/k1/k2 − k2 2/k1/ε∗ − k2 1/ε∗/k2 � ub + ua � 2(k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/k1/k/k2 − k2 2/k1/k/ε∗ − k2 1/ε∗/k/k2 � [A2] ub = uamF [A1] �/k1/ε∗/q/k2 � ub + ua � 2(k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/k1/k/k2 − k2 2/k1/k/ε∗ − k2 1/ε∗/k/k2 � [A2] ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C26) The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C26) gives /k1/ε∗/q/k2 = � /k − /p1 � /ε∗/q � /k − /p2 � = /k/ε∗/q/k − /p1/ε∗/q/k − /k/ε∗/q/p2 + /p1/ε∗/q/p2 = Cαβγα/ε∗/qγβ − Cα/p1/ε∗/qγα − Cαγα/ε∗/q/p2 + C0/p1/ε∗/q/p2 + � C1/p1 + C2/p2 � /ε∗/q/p2 + /p1/ε∗/q � C1/p1 + C2/p2 � + C0/p1/ε∗/q/p2 27 = C00 � ε∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='q − (4 − d)/ε∗/q � + � C12/p2 + C11/p1 + C1/p1 � /ε∗/q/p1 + � C12/p1 + C22/p2 + C1/p1 + C2/p2 + C2/p1 + C0/p1 � /ε∗/q/p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C27) Because the divergent part C00 = ∆ϵ/4 = 1/(4ϵ), which d = 4 − 2ϵ, hence C00(4 − d) = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The result is: /k1/ε∗/q/k2 = − 1 2/ε/q + � C12 � /p1 + /q � + (C11 + C1) /p1 � /ε∗/q � /p2 − /q � + � (C12 + X0) /p1 + (C22 + C2) � /p1 + /q �� /ε∗/q/p2 = − 1 2/ε∗/q + X012/p1/ε∗/q/p2, (C28) where we have used ε∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='q = q2 = 0 and /q/ε∗/q = 2ε∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='q/q − q2/ε∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The final result is uamF [A1] �/k1/ε∗/q/k2 � ub =uamF � /p1/ε∗ � −1 2 + m2 bX012 � + /ε∗/p2 � −1 2 + m2 aX012 � +(2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) �1 2 − X012/p1/p2 �� [A1] ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='(C29) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='Consider the last two terms in the last line of the formula (C26) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='− k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='2/k1/k/ε∗ − k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1/ε∗/k/k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='= − k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='k2 − /p1/k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='/ε∗ − /ε∗ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='k2 − /k/p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='= − k2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 + k2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='/ε∗ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D2 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='/p1/k/ε∗ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D1 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='/ε∗/k/p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='→ − /ε∗(D0 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F)(D1 + D2 + 2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D0D1D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ /p1/k/ε∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D0D1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ /ε∗/k/p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D0D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='/p1/k/ε∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D0D1D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='/ε∗/k/p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D0D1D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='= − /ε∗ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V (B(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ 1) + 2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V B(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F(B(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ B(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ 2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V C0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='− /p1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='B(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 /p1 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='C1/p1 + C2/p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='/ε∗ − /ε∗ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='B(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 /p2 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='C1/p1 + C2/p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='/p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='= − /ε∗ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V (4B(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ 2 + 2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FC0 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='aC1 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='bC2) + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='aB(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='bB(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F(B(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ B(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='− m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='C2/p1/p2/ε∗ + C1/ε∗/p1/p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='= − /ε∗ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V (4B(0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ 2 + 2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FC0 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='aC1 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='bC2) + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='aB(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='bB(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F(B(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ B(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V X3/p1/ε∗/p2 + (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)(−m2 V ) � C2/p1 + C1/p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C30) Lastly, consider the first term in the last line of the formula (C26): 2(k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/k1/k/k2 = (k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ − 2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) × � /k − /p1 � /k � /k − /p2 � 28 = (−2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ + 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) × � k2/k − k2/p1 − k2/p2 + /p1/k/p2 � → (−2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ + 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) × �� 1 D1D2 + m2 F D0D1D2 � � /k − /p1 − /p2 � + /p1/k/p2 D0D1D2 � = (−2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) × �� −1 2B(0) 0 − m2 FC0 � (/p1 + /p2) − m2 F(C1/p1 + C2/p2) − /p1(C1/p1 + C2/p2)/p2 � + � 2ε∗ µ � �� 1 D1D2 + m2 F D0D1D2 � /kkµ − � 1 D1D2 + m2 F D0D1D2 � (/p1 + /p2)kµ + /p1/k/p2kµ D0D1D2 � = (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) �� B(0) 0 2 + m2 FC0 � � /p1 + /p2 � + � m2 FC1 + m2 bC2 � /p1 + � m2 FC2 + m2 aC1 � /p2 � + � 2ε∗ µ � × �� Bµν + m2 FCµν� γν − � Bµ + m2 FCµ� � /p1 + /p2 � + Cµν/p1γν/p2 � , (C31) where Bµ = Bµ(0, m2 V , m2 V ) and Bµν = Bµν(0, m2 V , m2 V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The last line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C31) is expressed in terms of the PV functions as follows � 2ε∗ µ � � γν �gµν 2 � B(0) 0 + 1 � + 1 6B(0) 0 (2pµ 1pν 1 + pµ 1pν 2 + pµ 2pν 1 + 2pµ 2pν 2) � + m2 Fγν [C00gµν + C11pµ 1pν 1 + C12pµ 1pν 2 + C12pµ 2pν 1 + C22pµ 2pν 2] − �1 2B(0) 0 (p1 + p2)µ − m2 F (C1pµ 1 + C2pµ 2) � � /p1 + /p2 � + [C00gµν + C11pµ 1pν 1 + C12pµ 1pν 2 + C12pµ 2pν 1 + C22pµ 2pν 2] /p1γν/p2 � =m2 V /ε∗ � B(0) 0 + 1 � + (p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)B(0) 0 � /p1 + /p2 � + m2 F � 2C00/ε∗ + (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � C11/p1 + C12/p2 + C12/p1 + C22/p2 �� − � B(0) 0 (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) − m2 F (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) (C1 + C2) � � /p1 + /p2 � + /p1 � 2/ε∗C00 + (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � C11/p1 + C12/p1 + C12/p2 + C22/p2 �� /p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C32) Hence the final result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C31) is 2(k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/k1/k/k2 =/ε∗ � m2 V (B(0) 0 + 1) + 2m2 FC00 � + /p1/ε∗/p2 (2C00) + (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � m2 FX01 + m2 bX2 � /p1 + (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � m2 FX01 + m2 aX1 � /p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C33) The sum of three terms given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C25), (C30), and (C33) gives C(ab)L,R corresponding to the diagrams (5) given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (18) and (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The formulas of D(ab)L,5 and D(ab)R,5 are given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Regarding to the case of photon couplings in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (27), the equality given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C21) is still valid because the new part ∆Γµα′β′ = Γµα′β′ − Γ′ µα′β′ = δkv (gµα′qβ′ − gβ′µqα′) satisfies 29 (gµα′qβ′ − gβ′µqα′) ε∗µkα′ 1 kβ′ 2 = q2(ε∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='k2) − (q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='k2)(ε∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The other relevant part of M5 is: − γα [A] γβ × ∆Γµα′β′εµ∗ � gαα′gββ′ − gββ′kα 1 kα′ 1 + gαα′kβ 2 kβ′ 2 m2 V � = � /q/ε∗ − /ε∗/q � mFA1 + � /q/k/ε∗ − /ε∗/k/q � A2 − 1 m2 V �� (k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='q)(/k1/ε∗ − /ε∗/k2) + (k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)(/q/k2 − /k1/q) � mFA1 + � (k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='q)(−/p1/k/ε∗ + /ε∗/k/p2) + (k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)(/p1/k/q − /q/k/p2) � A2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C34) The final result of new contributions to iδM5 is: iδM5 = � d4k (2π)4 uaγα [A] γβub D0D1D2 × ∆Γµα′β′ � gαα′gββ′ − gββ′kα 1 kα′ 1 + gαα′kβ 2 kβ′ 2 m2 V � = − ieQV δv 16π2 � ua �� 4p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗ − 2/p1/ε∗ − 2/ε∗/p2 � C0mFA1 � ub +ua � (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)(/p1 + /p2) − (m2 a + m2 b)/ε∗ − 2/p1/ε∗/p2 � X3A2ub − 1 m2 V ua � C00 � /q/ε∗ − /ε∗/q � + (C11 + C12) � −2(p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='q)/ε∗/p1 + 2(p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/q/p1 � +(C22 + C12) � −2(p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='q)/ε∗/p2 + 2(p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/q/p2 � +(p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='q) � −X0 � −/p1/ε∗ + /ε∗/p2 � + (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)(C2 − C1) + 2 � C1/p1/ε∗ − C2/ε∗/p2 �� −(p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � 2/p1/p2 − m2 a − m2 b � (2X3 + C0) � mFA1ub − 1 m2 V ua � C00 � 2(m2 a + m2 b)/ε∗ + 4/p1/ε∗/p2 − 2(/p1 + /p2) × (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � +m2 b − m2 a 2 × X1 � −m2 a/ε∗ − /p1/ε∗/p2 + (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/p1 � +m2 b − m2 a 2 × X2 � m2 b/ε∗ + /p1/ε∗/p2 − (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/p2 �� A2ub � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C35) Ignoring the factor eQV δkv 16π2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' the form factors are: −δCFV V (ab)L =gLLma � X3 + 4C00 − (m2 b − m2 a)X1 2m2 V � + gRRmb � X3 + 4C00 + (m2 b − m2 a)X2 2m2 V � + gRLmF � 2C0 − 8C00 + m2 a(2X1 + X0) + m2 b(2X2 + X0) 2m2 V � + gLRmFmambX012 m2 V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' −δCFV V (ab)R =δCL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='5 � gL a ↔ gR a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' gL b ↔ gR b � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' −δDFV V (ab)L =gLL � −(m2 a + m2 b)X3 − 4(m2 a + m2 b)C00 + (m2 b − m2 a)(−m2 aX1 + m2 bX2) 2m2 V � + gRRmamb � −2X3 − 8C00 + (m2 b − m2 a)(−X1 + X2) 2m2 V � 30 + gRLmamF � −2C0 − −8C00 + (m2 b − m2 a)(2X1 + X0) 2m2 V � + gLRmbmF � −2C0 + 8C00 + (m2 b − m2 a)(2X2 + X0) 2m2 V � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' δDFV V (ab)R =δDFV V (ab)L � gL a ↔ gR a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' gL b ↔ gR b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C36) All results given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C36) were cross checked using FORM package [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' All formulas in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C36) satisfy automatically the WI, namely δDFV V (ab)L + maδCFV V (ab)L + mbδCFV V (ab)R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Diagram (6) After using the property of chiral operators PL,R, the amplitude (C17) is written as iM6 =eQF � d4k (2π)4 1 D0D1D2 × ua �� m2 Fγα/ε∗γβ + γα/k1/ε∗/k2γβ � [A2] −mF [A1] � gL∗ a gR b PR + gR∗ a gL b PL � (γα/k1/ε∗γβ + γα/ε∗/k2γβ) � ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C37) The numerator is divided into the two parts N1 ∼ gαβ and N2 ∼ −kαkβ/m2 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' After extract- ing gαβ, the first part is N1 =ua �� (2 − d)m2 F/ε∗ − 2/k2/ε∗/k1 + (4 − d)/k1/ε∗/k2 � [A2] −mF [A1] [4ε∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (k1 + k2) − (4 − d) (/k1/ε∗ + /ε∗/k2)]} ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C38) Ignoring the overall factor eQF/(16π2), the formula in terms of tensor notations is N1 =ua/ε∗ [A2] ub � −2m2 FC0 + (d − 4)(d − 2)C00 � + (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) ua [A1] (4mFX0) ub + ua � (2 − d)Cαβγα/ε∗γβ + 2Cα � /p2/ε∗γα + γα/ε∗/p1 � − 2C0/p2/ε∗/p1 � × [A2] ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C39) After expanding the tensors in terms of scalar PV-functions, the final result is N1 =ua/ε∗ [A2] ub � −2m2 FC0 + (d − 2)2C00 + 2m2 aX01 + m2 bX02 � + ua/p1/ε∗/p2 [A2] ub × (2X0) + (2ε∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='p1) ua(−2) � X01/p1 + X02/p2 � [A2] ub + (2ε∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='p1) ua {4mFX0 [A1]} ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C40) Considering the second term proportional to kαkβ, we have −m2 V N2 = ua � m2 F /k/ε∗/k + /k/k1/ε∗/k2/k � [A2] ub − mFua (/k/k1/ε∗/k + /k/ε∗/k2/k) [A1] ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C41) The two relations /k/k1 = D1 + m2 F − m2 a + /p1/k and /k2/k = D2 + m2 F − m2 b + /k/p2 give N2 ∼ua � m2 F /k/ε∗/k � [A2] ub + ua � D1 + m2 F − m2 1 + /p1/k � /ε∗ � D2 + m2 F − m2 2 + /k/p2 � [A2] ub 31 − mFua �� D1 + m2 F − m2 a + /p1/k � /ε∗/k + /k/ε∗ � D2 + m2 F − m2 b + /k/p2 �� [A1] ub ≡ua [(L1 + L2) [A2] − mF [A1] L3] ub, (C42) where L1 =m2 F � /ε∗ � (2 − d)C00 − m2 a (C11 + C12) − m2 b (C12 + C22) � +(2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � (C11 + C12) /p1 + (C12 + C22) /p2 �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='L2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D0D1D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D1 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a + /p1/k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='/ε∗ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D2 + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b + /k/p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='=/ε∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b + /k/p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D0D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F − m2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D0D1D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ /p1/k/ε∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D0D1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ /p1/k/ε∗/k/p2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D0D1D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ /p1/k/ε∗ (m2 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='D0D1D2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='=/ε∗ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='A0(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='B(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='− m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='bB(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='C0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ Cαβ(/p1γα/ε∗γβ/p2) + Cα(/p1γα/ε∗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ Cα(/ε∗γα/p2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' L3 = /ε∗/k D0D2 + /k/ε∗ D0D1 + m2 F(2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) − m2 a/ε∗/k − m2 b/k/ε∗ D0D1D2 + /p1/k/ε∗/k + /k/ε∗/k/p2 D0D1D2 = − B(2) 1 /ε∗/p2 − B(1) 1 /p1/ε∗ − (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)(C1 + C2)m2 F − Cα � m2 a/ε∗γα + m2 bγα/ε∗� + Cαβ � /p1γα/ε∗γβ + γα/ε∗γβ/p2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' It can be proved that: Cαβ(/p1γα/ε∗γβ/p2) = /p1/ε∗/p2 � (2 − d)C00 − m2 a(C11 + C12) − m2 b(C22 + C12) � + (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � m2 b(C22 + C12)/p1 + m2 a(C11 + C12)/p2 � , Cα(/p1γα/ε∗) = −m2 aC1/ε∗ − C2 � (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/p1 − /p1/ε∗/p2 � , Cα(/ε∗γα/p2) = −m2 bC2/ε∗ − C1 � (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗)/p2 − /p1/ε∗/p2 � , Cα � m2 a/ε∗γα + m2 bγα/ε∗� =(2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � −m2 aC1 − m2 bC2 � + /p1/ε∗(m2 a − m2 b)C1 + /ε∗/p2(m2 b − m2 a)C2, Cαβ � /p1γα/ε∗γβ + γα/ε∗γβ/p2 � =(2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � m2 a(C11 + C12) + m2 b(C22 + C12) + /p1/p2(C11 + 2C12 + C22) � + (/p1/ε∗ + /ε∗/p2) � (2 − d)C00 − m2 a(C11 + C12) − m2 b(C22 + C12) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C43) 32 Final results are: L1 =m2 F � /ε∗ � (2 − d)C00 − m2 a (C11 + C12) − m2 b (C12 + C22) � +(2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � (C11 + C12) /p1 + (C12 + C22) /p2 �� , L2 =/ε∗ � m2 V (B(0) 0 + 1) + m2 F(B(1) 0 + B(2) 0 ) − m2 a(B(1) 0 + B(1) 1 ) − m2 b(B(2) 0 + B(2) 1 ) +m4 FC0 − m2 F � (m2 a + m2 b)C0 + m2 aC1 + m2 bC2 � + m2 am2 bX0 � + /p1/ε∗/p2 � (2 − d)C00 + m2 FX3 − m2 aX1 − m2 bX2 � + (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) �� m2 bX2 − m2 FC2 � /p1 + � m2 aX1 − m2 FC1 � /p2 � , L3 =/p1/ε∗ � −B(1) 1 + (2 − d)C00 − m2 aX1 + m2 b(X3 − X2) � + /ε∗/p2 � −B(2) 1 + (2 − d)C00 − m2 bX2 + m2 a(X3 − X1) � + (2p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � m2 aX1 + m2 bX2 − m2 FX3 + /p1/p2 (X1 + X2 − X3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C44) The above calculation is enough to derive relevant contributions to CV FF L,R given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (20) and (21), and DV FF L,R given in (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Ward identity for the only gauge boson exchanges Before coming to discuss the WI, we use the relations given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A10) to write all the one-loop factors (22), (23), and (24) from gauge boson exchanges in the following simple forms, ignoring the overall factor e/(16π2): DFV (ab)L,78 =Qe � gRLma + gLRmb � (−3mFX0) + QegRRmamb � −2Xf 12 + bf 1 − m2 F(2X0 + Xf 12) m2 V � + QegLL �� 2 + m2 F + m2 a + m2 b m2 V � bf 1 + 1 + A0(m2 V ) + m2 am2 bXf 12 m2 V +2m2 F(m2 aB(1) 0 − m2 bB(2) 0 ) (m2 a − m2 b)m2 V � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C45) The WI for the FV V and FV V diagrams are f WI FV V ≡ DFV V (ab)L + maCFV V (ab)L + mbCFV V (ab)R and f WI V FF ≡ DV FF (ab)L + maCV FF (ab)L + mbCV FF (ab)R, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The relations given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (A10) give: 2Cf 00 + m2 aXf 1 + m2 bXf 2 = −bf 1, Xv 012 = −Xf 12, m2 F(Xv 12 − Xv 3) + m2 aXv 1 + m2 bXv 2 + bv 1 + 1/2 = m2 F(Xv 12 − Xv 3) − m2 FCv 0 33 = m2 F(Xv 012 − 2X0) = −m2 F(Xf 12 + 2X0), 1/2 + m2 aXv 1 + m2 bXv 2 − m2 FXv 3 = (m2 F − m2 V )X0 − m2 FCv 0 − m2 FXv 3 = −m2 V X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C46) Combining the above formulas and results of Ci,ij functions listed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [39],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' the WI of all diagrams with boson exchanges is derived as follows f WI V =D(ab)L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='78 + f WI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='FV V + f WI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V FF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='∼ (Qe + QV − QF) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='gLL ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='2(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b)m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='× B(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='−m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b [(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F − 2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V )m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V + (m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F)2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='2(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b)m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='× B(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a [(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b + m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F − 2m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V )m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V + (m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='F)2] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='2(m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='a − m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='b)m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='× B(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='gRLma + gLRmb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='(3mFX0) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (C47) The final result is f WI V ∼ QF − (Qe + QV ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In conclusion, the contributions from the four diagrams with only gauge boson exchanges satisfy the WI when the electric charge conversation is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Appendix D: Ward Identity for the diagrams of FSV-type in the unitary gauge This type of diagrams were mentioned firstly in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [34] for the general case of their contributions to BSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The γ − S − V vertices come the kinetic terms of the scalars: LD(S) = (∂µS − iPµS)† (∂µS − iP µS) = � gγSV gµνS−QAµV Qν + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' , (D1) where Pµ containing the photon Aµ and Vµ is the covariant part of the covariant derivative of the Higgs multiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The contributions of these two diagrams are: iM9 =gγSV � d4k (2π)4 × ua[gL∗ a PR + gR∗ a PL](mF + /k)γα[gL b PL + gR b PR]ub D0D1D2 � ε∗α − (ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='k2)kα 2 m2 V � 34 =gγSV � d4k (2π)4 × 1 D0D1D2 × ua � � �/ε∗ [mF [A2] + /k [A1]] − /k2(k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) m2 V mF [A2] − [A1] � D0 + m2 F − /k/p2 � (k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) m2 V � � � ub =igγSV 16π2 ua � /ε∗ � C0mF [A2] − (C1/p1 + C2/p2) [A1] � −mF m2 V � (γµε∗ν)Cµν − (Cµγµ)(p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) + /p2X0(p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � [A2] + 1 m2 V [A1] � C00/ε∗/p2 + (p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗) � m2 FX0 + X1/p1/p2 + m2 bX2 ��� ub, (D2) where we have used k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='ε∗/(D1D2) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The formulas of DL,R and CL,R are: eDFhv (ab)L,9 × �gγSV 16π2 �−1 =gLLmF � C0 − C00 m2 V � − gRLmaC1 + gLRmb � C2 + C00 m2 V � , DFhv (ab)R,9 =DFhv (ab)L,9 � gL a ↔ gR a , gL b ↔ gR b � , eCFvh (ab)L,9 × �gγSV 16π2 �−1 = − gRLC2 − mF 2m2 V � gLLmaX1 + gRRmbX02 � + 1 2m2 V � gRL(m2 FX0 + m2 bX2) + gLRmambX1 � , CFhv (ab)R,9 =C(ab)L � gL a ↔ gR a , gL b ↔ gR b � , (D3) where XFhv i ≡ Xi(m2 a, 0, m2 b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' m2 F, m2 h, m2 V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' the results for diagram (10) are: eDFvh (ab)L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='10 × �gγSV 16π2 �−1 =gLLmF � C0 − C00 m2 V � − gLRmbC2 + gRLma � C1 + C00 m2 V � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' DFvh (ab)R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='10 =DFvh (ab)L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='10 � gL a ↔ gR a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' gL b ↔ gR b � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' eCFvh (ab)L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='10 × �gγSV 16π2 �−1 = − gR∗ a gL b C1 − mF 2m2 V � gR∗ a gR b mbX2 + gL∗ a gL b maX01 � + 1 2m2 V � gR∗ a gL b (m2 FX0 + m2 aX1) + gL∗ a gR b mambX2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' eCFvh (ab)R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='10 × �gγSV 16π2 �−1 =CFvh (ab)L � gL a ↔ gR a ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' gL b ↔ gR b � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (D4) where XFvh i ≡ Xi(m2 a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' m2 b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' m2 F, m2 V , m2 h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The above formulas are consistent with calcula- tion using FORM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The corresponding formulas of WI are f Fhv WI kγSV = gLLmF � 2(m2 V C0 − C00) + mambX012 �fhv − gRRmF � m2 aX1 + m2 bX02 �fhv + gRL � −2m2 V (maC1 + mbC2) + mb � m2 FX0 + m2 aX1 + m2 bX2 ��fhv + gLR � 2m2 V (mbC2 − maC1) + 2mbC00 + ma � m2 fX0 + m2 bX12 ��fhv ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 35 f Fvh WI kγSV = gLLmF � 2(m2 V C0 − C00) − mambX012 � − gRRmF � m2 aX01 + m2 bX2 �fvh + gRL � 2m2 V (maC1 − mbC2) + 2maC00 + mb � m2 fX0 + m2 aX12 ��fvh + gLR � −2m2 V (maC1 + mbC2) + ma � m2 fX0 + m2 aX1 + m2 bX2 ��fvh ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (D5) where kγSV = gγSV /(32π2m2 V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' The WI valid if only f Fhv WI + f Fvh WI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' We can see crudely that all C(ab)L,9, C(ab)R,9, C(ab)L,10, and C(ab)R,10 are convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' In contrast, all D(ab)L,9, D(ab)R,9, D(ab)L,10, and D(ab)R,10 contain divergent terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Therefore, the necessary condition to guarantee the validation of the WI given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' (12) is that all of these divergent terms must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Strictly, the WI is valid if only gγSV =0 or gL a = gR a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Because at least one of gL a or gR a must be non-zero, the condition gγSV =0 is the only valid choice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=', the vertex- type γ-S-V does not appear in the all BSM guaranteeing the WI for the external photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' This conclusions is also true for the case a = b, corresponding to the one-loop contribution to the AMM of the leptons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' Abi et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} +page_content=' 39' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE5T4oBgHgl3EQfDg77/content/2301.05407v1.pdf'} diff --git a/b9E2T4oBgHgl3EQfwQjN/content/tmp_files/2301.04100v1.pdf.txt b/b9E2T4oBgHgl3EQfwQjN/content/tmp_files/2301.04100v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ab522ced462345337433bba42542e314e4e5d021 --- /dev/null +++ b/b9E2T4oBgHgl3EQfwQjN/content/tmp_files/2301.04100v1.pdf.txt @@ -0,0 +1,1253 @@ +Triggered Superradiance and Spin Inversion Storage in a Hybrid Quantum System +Wenzel Kersten,1, ∗ Nikolaus de Zordo,1 Oliver Diekmann,2 Tobias Reiter,2 Matthias +Zens,2 Andrew N. Kanagin,1 Stefan Rotter,2 J¨org Schmiedmayer,1 and Andreas Angerer1 +1Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, A-1020 Vienna, Austria +2Institute for Theoretical Physics, TU Wien, Wiedner Hauptstraße 8-10/136, A-1040 Vienna, Austria +(Dated: January 11, 2023) +We study the superradiant emission of an inverted spin ensemble strongly coupled to a supercon- +ducting cavity. After fast inversion, we detune the spins from the cavity and store the inversion for +tens of milliseconds, during which the remaining transverse spin components disappear. Switching +back on resonance enables to study the onset of superradiance. A weak trigger pulse of a few +hundred photons shifts the superradiant burst to earlier times and imprints its phase onto the +emitted radiation. For long hold times, the inversion decreases below the threshold for spontaneous +superradiance. There the energy stored in the ensemble can be used to amplify microwave pulses +passing through the cavity. +Superradiance is the process by which an ensemble +of excited two-level systems synchronizes to produce a +short, highly coherent burst of light [1]. The build-up +of correlations during the collective decay, mediated by +an enhanced coupling to a common mode, gives rise to +non-linear scaling of the decay rate with the number +of emitters [2]. Superradiant (SR) emission is not only +fundamental to many fields of physics, but also attracts +increasing interest for applications in metrology [3], laser +physics [4–6] and quantum technology in general [7–12]. +SR phenomena are at the heart of the transition from a +genuine quantum regime, where individual fluctuations +of the vacuum field will jump-start the collective decay +of inverted emitters, to the classical regime, where the +emission is akin to that of a macroscopic radiating dipole. +While experiments on superradiance have recently been +successfully transferred from atomic ensembles to solid- +state spin systems [13, 14], the possibilities this opens up +for controlling and exploiting superradiance for applica- +tions have been very little explored so far. Progress in +this direction has primarily been hindered by the fact that +systems giving rise to superradiance are fundamentally +unstable, reacting to the slightest disturbance. Although +this extreme sensitivity even to weak signals poses a great +challenge for experimental implementation, it also offers +very promising possibilities for applications in sensor and +detector technology [3, 15]. +In this work we report on a novel experimental platform +that allows us not only to invert a large ensemble of +nitrogen-vacancy (NV) spins, but also to hold and stabilize +the stored inversion for up to 20 ms – four orders of +magnitude longer than the timescale of the SR decay +process. This stabilization lets us study and control the +emission of a SR burst that releases the energy stored in +the ensemble. To trigger the SR emission, we use weak +microwave (MW) pulses. Additionally, we investigate a +regime with reduced inversion, where the spins act as a +gain medium for the amplification of a series of short MW +pulses. +In order to coherently manipulate spins with high fi- +delity, we use a newly developed MW resonator [see +Fig. 1(a)]. Our setup is based on two opposing super- +conducting chips that exhibit a small mode volume with +homogeneous coupling strength, while retaining a high +quality factor of Q = 3000 (unloaded Q = 30000). This +design allows us to reach the regime of strong collective +spin-cavity coupling already with a number of NVs that +is reduced by three orders of magnitude compared to +earlier works [16, 17]. Moreover, owing to the resonator’s +compact design, we are able to add a small loop of super- +conducting wire in order to magnetically tune the spins +in and out of the cavity resonance faster than the SR +timescale. +The resonator is composed of two sapphire chips with +a 200 nm thin layer of 16 × 16 mm2 superconducting Nio- +bium mounted in a copper housing. The identical pat- +terns on both chips feature a hole in the center from +which a 4 µm slit reaches outwards, resembling a split +ring resonator [18]. +The chips are stacked, with the +roughly cube shaped diamond sample placed between +the center holes. The hole radii, the distance between +the chips and the sample size are all of similar dimen- +sion d ∼ 200 µm. This configuration results in a reso- +nance frequency of ωc/2π = 3.1 GHz and linewidth of +κ/2π = 0.5 MHz (HWHM). The resonator’s oscillating +magnetic field couples homogeneously to all spins with a +collective coupling strength of gcoll/2π = 5.2 MHz. The +NV sample has a number of 3.6 × 1013 spins with an +effective linewidth for the inhomogeneously broadened +spin ensemble of Γ⊥/2π = 4.3 MHz [19], a value that +takes the spin frequency distribution, modeled by a q- +Gaussian function [20], into account (see Supplemental +Material). The resulting cooperativity parameter of our +coupled system is C = g2 +coll/(κΓ⊥) ≈ 12.2. +We begin our explorations with all NV spins as effective +two-level systems thermally relaxed to the ground state. +The spins are then inverted using a 400 ns modified chirp +pulse with a Gaussian envelope. Details of the initializa- +tion procedure are given in the Supplemental Material. +Following the inversion pulse, we rapidly switch off the +arXiv:2301.04100v1 [quant-ph] 10 Jan 2023 + +2 +- - - ++++ +- - - ++++ +Bext +Bloop +Bosc +1K +Pump +Probe +Out +-70dB +κ2 +κ1 +25mK +400 ns +4 μs +t = 0 +|a| +I/Q +δ +2 ms +hold time +2 +3 +4 +5 +6 +t (µs) +0 +1 +tD + I +Q +|a| +sim +0.15 +0.2 +inversion p +1 +2 +3 +4 +t (µs) +65 µs +150 µs +400 µs +850 µs +2.2 ms +5.7 ms +13 ms +hold time +0 +2 +4 +|a| (arb. u.) +0 +10 +20 10-1 +100 +� = 7.6 ms +hold time (ms) +max(|a|) +(a) +(b) +(d) +(c) +FIG. 1. (a) Schematic of the MW cavity located in a dilution refrigerator operating at 25 mK and connected to a homodyne MW +setup. Two sapphire chips with opposing split ring structures and the diamond sample are stacked inside a copper box. Between +the center holes the oscillating magnetic field homogeneously penetrates the sample. The chips are surrounded by a loop made +of 5 windings of superconducting wire, enabling a rapid detuning of the spins. The pin coupler of port 1 is connected to the +pump-line, which can be decoupled from the room temperature thermal bath at the 1 K stage using a solenoid switch. Port 2 is +connected to the out-line for acquiring data, and the attenuated probe-line for injecting weak trigger pulses. (b) Sequence of +the experiment. We use a modified chirp pulse (red/blue) to invert the spin ensemble and subsequently detune the spins by +switching off the current (green) in the loop, thereby storing the inversion. After a certain hold time we bring the spins back +into resonance and see their SR decay (black), optionally triggered by a short probe pulse (orange). (c) SR decays for varying +hold times, triggered by room temperature noise. The inset shows the SR decay maxima together with an exponential fit in a +semi-log plot. (d) Example data and simulation of a SR decay and its associated quadrature values I/Q, together with the +simulated inversion p. The vertical line indicates tD, the time of the maximum cavity amplitude. +loop current in about 200 ns using a semiconductor switch, +causing a detuning of the spins by δ/2π = 26 MHz. This +detuning by more than the ensemble linewidth inhibits +the SR interaction of the spin ensemble with the cavity +mode, thereby storing the inversion [19]. +During the inversion hold time, dephasing processes +eliminate the spins’ collective dipole moment determined +by the transversal component of the collective spin vector, +S− = Sx − iSy. When tuning the ensemble back into +resonance, we thus create a metastable inverted state +whose tipping angle θ = arctan(|S−|/Sz) with respect +to the z-axis in the Bloch sphere is exponentially de- +creased for longer hold times. Under the condition that +the product of the stored ensemble inversion −1 ≤ p ≤ +1 +and cooperativity is above the threshold pC > 1, this +metastable state will decay by emitting a SR photon burst +[19]. Here, the inversion parameter p is implicitly defined +by Sz = 1 +2⟨� +j σj +z⟩ = pN/2. In this state, the presence +of even a single photon in the cavity will stimulate the +collective emission of radiation, starting a self-accelerating +photonic avalanche. During this process, the energy re- +leased in the form of cavity photons gradually builds up, +reaches a maximum and then oscillates back and forth +between the two subsystems, before the process stops due +to the dephasing of the spins and their decoherence. The +full experimental sequence is summarized in Fig. 1(b). +Our first notable result is presented in Fig. 1(c), where +we plot the SR decay pulses for varying inversion hold +times. Here, the SR decay is triggered by noise from the +high power amplifier of the pump-line. The measured SR +dynamics are captured in a semi-classical description using +the Maxwell-Bloch equations [21]. We model the time +evolution starting from an inverted state with a slight +tipping angle accounting for fluctuations that initiate +the SR decay (see Supplemental Material). To simulate +the measured signals of |a| we only adjust the ensemble +inversion p and a time offset, resulting in curves as shown +in Fig. 1(d). The role of fluctuations at the start of the +SR decay process is studied in more detail below. We +find the decay maximum max(|a|), an indirect measure +of the energy stored by the spins, to decrease roughly +exponentially with increasing hold times, exhibiting a +characteristic timescale of τ = 7.6 ms [see inset Fig. 1(c)]. +For hold times longer than 20 ms, the inversion has already +decreased below the threshold pC = 1 for spontanoeus +superradiance. We hypothesize two distinct timescales +for the relaxation of the inverted state; a fast one that +rapidly thermalizes the ensemble, driven by spin-spin +interactions involving NVs with very short lifetimes (so +called fluctuators [22]), and a slow one which brings them +to the ground state over extended times (T1 = 134 s, see +Supplemental Material [23]). + +3 +We now focus on the onset of the SR decay process and +the possibility to trigger it prior to its self-decay. Using a +2 ms hold time, we give the cavity mode enough time to +reach thermal equilibrium after the inversion pulse and +subsequent decoupling from the high power amplifier noise +by the solenoid switch, with an estimated number of n ≈ 2 +thermal photons remaining. The completely dephased +inverted state, that is brought back into resonance, has +zero tipping angle apart from unavoidable quantum and +thermal fluctuations. Another 150 ns after switching back +the detuning current (defined as t = 0), we send a 100 ns +trigger pulse through the highly attenuated MW probe- +line. The pulse is resonant with the cavity and contains a +calibrated number of photons (see Supplemental Material +[24]). The experiment is repeated many times for varying +numbers of trigger photons, and without trigger pulse. +For every run, we extract the delay time tD and the +ID/QD quadrature values of the SR decay maximum +[cf. Fig. 1(d)]. We post-select the runs for max(|a|) = +� +I2 +D + Q2 +D to fall into a narrow window, as there is some +variation in the initial spin inversion between the runs. +The SR decay phases ϕ = arctan(QD/ID) are corrected +for a linear phase drift with tD, caused by a minor constant +detuning of the spins (see Supplemental Material). The +resulting sets of delay times and phases is summarized +in Figs. 2(a) and 2(b). Clearly, stronger trigger pulses +with higher numbers of photons ntrig lead to earlier tD +values and narrower distributions for tD and ϕ. While our +simulation allows to describe the decay process starting +from a slightly tipped initial collective spin vector, it is +the randomness in the initial conditions that leads to +the observed variance in time and phase. These thermal +and quantum fluctuations are not included in our semi- +classical model. To understand these phenomena and +model the distributions of times and phases of the decay +pulses, we split the analysis of the SR decay into two +stages [2, 15, 25]. +The decay process starts with a linear stage, in which +the (optional) trigger pulse leads to a coherent rota- +tion of the collective spin vector about an axis defined +by the phase of the pulse, which is kept identical for +all runs. Prior to this rotation, the initial state is lo- +cated very close to the +z-axis but with a small tip- +ping angle θ = arctan(|S−|/Sz) and random polar an- +gle φ = arg(S−). As cos θ ≃ 1 throughout the linear +phase, we can treat the spin vector to be confined to a +plane with a z-offset corresponding to the initial inver- +sion. The geometric construction of this plane is illus- +trated in Fig. 2(c), mathematical formulas of the distri- +bution functions are given in the Supplemental Material +citerice1945mathematical,angularmarginalization. The +initial state of the spin vector follows a two dimensional +Gaussian distribution of width θ centered at θ = 0. The +influence of the trigger pulse then causes a displacement in +the plane, which we choose to be in the direction of φ = 0. +The parameter η expresses the displacement in units of +�� +� +� +� +��� �� +�� +�� +��� +1 +1.5 +2 +2.5 +3 +tD (µs) +no pulse +180 +570 +1800 +5700 +18k +57k +180k +570k +1.8M +5.7M +number of trigger photons +-� +0 +� +phase � +180 +570 +1800 +5700 +18k +57k +180k +570k +1.8M +5.7M +ntrig +100 +102 +�� +(c) +(a) +(b) +(d) +FIG. 2. Swarm plots of the measured delay times tD (a) and +phases ϕ (b) of the SR decay maxima in over 1200 runs with +varying powers (photon numbers) of the 100 ns trigger pulse, +or no pulse at all. The distributions plotted using solid lines +in (a) correspond to the maximum likelihood estimates for the +displacement parameter η to the recorded tD data, where other +parameters are kept fixed. The dashed lines in (b) are the +expected phase distributions for those estimates of η. (c) Initial +state of the collective spin vector with coordinates (θ, φ) close +to the north pole of the Bloch sphere. In-plane distribution +before (blue) and after (red) the coherent displacement η in +units of the width θ via the trigger pulse. (d) Log-log plot +of the squared displacement η2 over the estimated number of +trigger photons in the cavity. +the width parameter θ. For growing η, i.e., higher trig- +ger pulse powers, the initially randomly distributed polar +angles become increasingly well defined and approach a +narrow distribution around φ = 0 [see Fig. 2(b)]. +After this linear stage, where the collective spin vector +is coherently displaced from its random in-plane starting +position, we enter a nonlinear regime. Now the SR dy- +namics dominate and via a collective process of stimulated +emission the spin vector accelerates its rotation towards +the equator while emitting a considerable burst of MW +radiation. +The phase ϕ of the emitted decay pulse is directly de- +termined by the value of φ at the start of the nonlinear +stage. Less directly, we can infer the initial tipping angles +θ from the delay times tD, which result via the relation + +-4 +20 +22 +24 +26 +28 +30 +32 +34 +36 +t (µs) +30 ms +35 ms +40 ms +50 ms +hold time +0 +0.1 +0.2 + |a| (arb. u.) +|a| +|a| empty +0 +10 +20 +30 +40 +50 +60 +70 +hold time (ms) +0 +0.1 +0.2 +0.3 +inversion p +0 +1 +2 +3 +max(|a|) (arb. u.) +pC = 1 +p(t) +a +T +b += a·exp(-(t/T)b) += 0.36 += 11.8 ms += 0.451 +max(|a|) +self-decay +pulse trains +stretched exp. +(a) +(b) +FIG. 3. (a) Cavity amplitude |a| for a series of 100 ns pulses, +each injecting ntrig ≈ 4.1 × 109 photons, amplified by the +partially inverted spin ensemble in the reduced effective co- +operativity regime pC < 1 for different hold times (red). In +comparison, we plot the signal obtained with an empty cavity +where spins are far detuned (blue). For choosing the parame- +ters in our semi-classical model (black), we ignore noise below +a certain threshold (green line at the top). (b) Ensemble inver- +sion as a function of hold time, extracted by simulations in the +two regimes above and below pC = 1. Above this threshold, +the pulse maxima (right y-axis) follow the values of p from +simulations of the self-decays shown in Fig. 1(c). A stretched +exponential is fitted to the inversion as a guide to the eye. +tD = −2TR log (θ/2) [15]. Here, the parameter TR repre- +sents the timescale for the SR emission process (see Sup- +plemental Material). With this relation and the assumed +distribution for the initial tipping angles θ depending on +η [see Fig. 2(c)], we can reproduce the distributions of the +measured tD data using maximum likelihood estimation +to fit η. For that, we fix the values TR = 212 ns and +θ = 6 × 10−3 using the measurement runs without probe +pulses (η = 0). In Fig. 2(a) we show the resulting tD dis- +tributions from these estimates of η. The corresponding +phase distributions in Fig. 2(b) are not fitted, but result +directly from those η estimates. +As the displacement η is caused by the MW magnetic +field of the trigger pulse, its square is a measure of the +energy imparted onto the spin system during the linear +stage of the SR process. Therefore, the number of photons +that trigger the SR decay is proportional to η2 ∝ ntrig +[see Fig. 2(d)]. Remarkably, a weak MW pulse on the +order of 10−11 photons per spin is sufficient to have an +observable effect on the SR decay. This observation opens +the perspective to operate our system as a detection device +that is not only sensitive to the amplitude, but also the +phase of weak MW signals. +We now investigate a regime of reduced effective co- +operativity pC < 1, where SR emission does not occur +spontaneously [19]. To that end, we employ hold times +longer than 20 ms, thus reducing the polarization below +the threshold for the SR decay. We probe the system by +injecting, at 5 µs intervals, a sequence of resonant MW +pulses of 100 ns duration via the pump-line. Interestingly, +in Fig. 3(a), we find that this results in an amplification +of the pulses as compared to the empty cavity response +(with far detuned spins). Although no spontaneous SR +decay occurs on its own, it is still possible to repeatedly +extract energy from the stored inversion. The incident +pulses hereby effectively supply the necessary coherence +that is otherwise constituent to the SR emission, but +hindered from building up when the stored inversion is +insufficient. Notably, tens of injected MW pulses can +be amplified in succession. We are able to replicate the +measured dynamics using our numerical model with only +the amplitude of the incident pulses (kept fixed for all fits) +and the ensemble inversion p as free parameters. These +results are combined in Fig. 3(b) with the p values at- +tained by simulating the SR self-decays [cf. Fig. 1(d)]. +The semi-classical model seamlessly captures the behavior +of our system in both regimes of high and low effective +cooperativity. +In summary, we present and successfully operate an +experimental platform to store the energy of an inverted +spin ensemble for tens of milliseconds and to release +it via a strong SR burst. By initializing the system to +a fully upright inverted state, we demonstrate a high +sensitivity to weak MW pulses that strongly influence +the subsequent SR dynamics, allowing us to infer both +amplitude and phase of the trigger pulse. The decrease of +inversion over time allows us to study a regime of reduced +cooperativity without spontaneous SR emission, where +the inverted spins effectively act as a gain medium for a +series of short MW pulses. Our setup lays the ground +work for the design of highly sensitive MW detectors on +a chip. +We thank Johannes Majer for discussions and techni- +cal support in the initial phases of the project. We ac- +knowledge support by the Austrian Science Fund (FWF) +projects I3765 (MICROSENS), P34314 (Spins in Quan- +tum Solids) and P32300, and by the European Union’s +Horizon 2020 research and innovation programme (FET- +OPEN project FATMOLS, Grant No. 862893) as well as +by the Studienstiftung des Deutschen Volkes. +∗ wenzel.kersten@tuwien.ac.at +[1] R. H. Dicke, Physical review 93, 99 (1954). +[2] M. Gross and S. Haroche, Physics reports 93, 301 (1982). +[3] M. Koppenh¨ofer, P. Groszkowski, H.-K. Lau, and A. A. +Clerk, PRX Quantum 3, 030330 (2022). +[4] J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. +Holland, and J. K. Thompson, Nature 484, 78 (2012). +[5] Y. Zhang, C. Shan, +and K. Mølmer, Physical Review +Letters 126, 123602 (2021). + +5 +[6] Q. Wu, Y. Zhang, X. Yang, S.-L. Su, C. Shan, +and +K. Mølmer, Science China Physics, Mechanics & Astron- +omy 65, 1 (2022). +[7] A. Kuzmich, W. Bowen, A. Boozer, A. Boca, C. Chou, +L.-M. Duan, and H. 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Farid, “A toolbox for the radial and +angular marginalization of bivariate normal distributions, +arxiv:2005.09696,” (2020). + +S1 +Supplemental Material: +Triggered Superradiance and Spin Inversion Storage in a Hybrid Quantum System +SYSTEM HAMILTONIAN, EQUATIONS OF +MOTION AND NUMERICAL MODELLING +Our system is described by the driven Tavis-Cummings +Hamiltonian in the rotating frame [20], +H =ℏ ∆ca†a + ℏ +2 +� +j +∆j +sσj +z ++ ℏ +� +j +g0 +� +a†σj +− + σj ++a +� ++ iℏ η +� +a† − a +� +, +(S1) +with a† (a) being the creation (annihilation) operator +of the cavity mode and σj +z, σj +± being the Pauli-z and +raising/lowering operators for the jth spin, respectively. +The spins are coupled to the cavity with constant cou- +pling g0 = gcoll/ +√ +N, where gcoll is the collective coupling +strength and N is the number of spins. The spin detun- +ings ∆j +s = [ωj +s + δ(t)] − ωp account for the inhomogeneous +broadening of the spin ensemble and the additional shift +δ(t) caused by the detuning loop, while ∆c = ωc − ωp is +the detuning of the cavity mode. Both detunings are cal- +culated with respect to the driving frequency ωp. Further, +the amplitude of the driving field is determined by η. +Using a Lindblad master equation, we take into account +the loss rate κ for the cavity mode and the spin deco- +herence rate γ⊥, thus yielding a set of coupled equations +describing the dynamics of the operators, +˙a = −(i∆c + κ)a − i +� +j +g0σj +− + η , +(S2) +˙σj +− = −(i∆j +s + γ⊥)σj +− + ig0aσj +z , +(S3) +˙σj +z = 2ig0 +� +a†σj +− − aσj ++ +� +. +(S4) +Identifying these operators with their expectation val- +ues, effectively neglecting correlations between individual +spins and the cavity by separating higher order moments +into products of their first order counterparts, yields the +well known Maxwell-Bloch equations. These equations +of motion represent a semi-classical description of our +system’s dynamics, which can be solved numerically, e.g., +to model the response to external stimuli. In our analysis, +this description only fails to capture the stochastic nature +of the inverted state’s initial conditions. +To perform the numerical simulations, we approxi- +mate the spin frequency distribution ρ(ω), which is quasi- +continuous due to the large number of spins, by sampling +it at 1500 frequencies ωj with equidistant spacing ∆ω. +The resulting weights ρj = ρ(ωj)∆ω are then used to cal- +culate the number of spins, nj = ρjN, for each frequency +bin ωj. Assuming identical initial conditions for all spins +of one bin, also their dynamics according to Eqs. (S3) +and (S4) are identical, thus tremendously reducing the +number of relevant equations. +We determine the parameters describing our system, +namely ωc, κ, gcoll, and Γ⊥ [combining γ⊥ and the spin +frequency distribution in one parameter, see Eq. (S5)] +by fitting the steady state solution of the Maxwell-Bloch +equations to the transmission signals [c.f. Fig. S4(a)], +obtained with a vector network analyzer (VNA). +For +modelling +SR +emission +dynamics +on +short +timescales, we can safely neglect T1 processes, i.e., we do +not include these in the master equation. +SPIN FREQUENCY DISTRIBUTION +We model the inhomogeneously broadened spin distri- +bution ρ(ω) with a q-Gaussian function [20] with shape +parameter q = 1.39 and FWHM of γq/2π = 11 MHz. The +effective linewidth, present in the definition of the cooper- +ativity C = g2 +coll/κΓ⊥ in the main text, can be calculated +using [19] +Γ⊥ = +�� +∞ +−∞ +ρ(ω)dω +γ⊥ + i(ω − ω0) +�−1 +, +(S5) +with ω0 being the spin center frequency, together with +the value for γ⊥/2π = 208 kHz. +INITIALIZATION PROCEDURE AND +INVERSION PULSE +All four NV sub-ensembles are tuned into resonance +with the cavity using a vector magnet together with the +detuning loop. As the third level of the NV ground state +manifold is far away from resonance, we can treat the NVs +as effective two-level systems. For all experiments, the +spins are initialized close to their ground state by waiting +3 min after repeatedly sweeping a MW tone across the +cavity resonance for 30 s, decreasing the cooperativity by +roughly 25% but reducing wait times between different +runs. +Starting from this state, the spins are inverted using an +in-phase and quadrature modulated MW pulse. Similar +to adiabatic fast passage methods for spins in free space, +the starting point for the design of our inversion pulse is +a chirped pulse of length 400 ns with a Gaussian envelope, +that covers a frequency interval of about -8 to +8 widths +γq. As our spins are not in free space but strongly coupled +to a cavity, we cannot use the chirped pulse directly but +need to adapt it for this circumstance. Comparing the + +S2 +Maxwell-Bloch equations above with the optical Bloch +equations, describing a two-level-system in free space +driven by a classical coherent light field with driving +amplitude Ω(t), +˙σ− = −γ⊥σ− + iΩ(t)σz , +(S6) +˙σz = 2i (Ω(t)σ− − Ω∗(t)σ+) , +(S7) +we can see that the role of the driving amplitude in the +case of the coupled cavity-spin system is taken by the +term g0a†. Disregarding g0 as a proportionality constant +that has to be determined experimentally, we can assume +a desired photonic amplitude a(t) = aR + iaI given by +the aforementioned chirped pulse and numerically solve +for MW drive η(t) = I − iQ necessary to create it in +the cavity (see Fig. S1). Strictly speaking, this approach +produces the correct effective inversion drive only for the +spins with the center frequency, as only these are fully +resonant. For other frequencies, the cavity acts as a filter +and reduces the amplitude of the drive. Nevertheless, +as the chirped pulse comes with a certain robustness +to amplitude deviations, we still get a useful inversion +efficiency for the whole spin ensemble. In the experiment, +we scan the inversion pulse power to select the value for +optimum inversion, which is in turn visible as the highest +SR decay maximum for a given hold time. +0 +100 +200 +300 +400 +Time (ns) +Re(a) +Im(a) +I +Q +FIG. S1. Chirped pulse for initial spin inversion, desired ⟨a(t)⟩ +and corresponding I/Q channels for optimal inversion. +DELAY TIME OF THE SUPERRADIANT +EMISSION +To derive Eq. (3) in the main text, we neglect the +inhomogeneous broadening ∆j +s ≈ 0 and describe the +spin ensemble as a giant dipole using S± = � +j σj +± and +Sz = 1 +2 +� +j σj +z. When inserting Eq. (S2) into Eq. (S4) on +resonance ∆c = 0, we get +˙Sz = −2g2 +0 +κ S+S− − ig0 +κ (S+ ˙a − ˙a†S−) , +(S8) +where we now can neglect the second term as it is of lower +order in the number of spins N ∝ S−. Effectively, the +cavity acts similar to a vacuum environment for the SR +burst, although enhancing the coupling of individual spins +to the electromagnetic field. Its ability to store photons +becomes of importance only at later times, when the cavity +amplitude shows revivals, with excitations oscillating back +and forth between cavity and spins. Then, we evaluate +[13] +⟨ ˙Sz⟩ = −2g2 +0 +κ ⟨S+S−⟩ += −2g2 +0 +κ (S + ⟨Sz⟩) (S − ⟨Sz⟩ + 1) . +(S9) +By parametrizing ⟨Sz⟩ = cos(θ)N/2 with a tipping angle +θ and using S = N/2, we can now solve for the delay +time tD where ⟨Sz⟩ = 0, as the SR emission reaches its +maximum when the giant dipole points to the equator. +The resulting expression [2, 15] +tD = t0 − +κ +2g2 +0N log +� +tan2 +�θ +2 +�� +(S10) +already resembles the one given in the main text. Now +we linearize tan θ ≈ θ, neglect the constant offset t0 and +summarize the prefactor as TR = +κ +2g2 +0N , representing the +timescale of the SR emission process. We find good quali- +tative agreement of Eq. (S10) with our results. Quantita- +tively, when using the explicit values for κ and g2 +0N = g2 +coll, +the timescale of the SR emission is underestimated due +to the approximations involved (in particular, neglecting +the inhomogeneous broadening). +POST-SELECTION AND PHASE CORRECTION +OF THE DECAY PULSES +The data collected in all experimental runs has some +variance in the SR decay amplitudes. We assume this +variance comes mostly from the solenoid switch located +at the 1 K stage which is used to disconnect the the pump +line from port 1 of the cavity after the inversion pulse. +The switch opens and closes a mechanical connection by +a latching mechanism, thus leading to slightly different +initial ensemble inversions between the runs. The initial +inversion Sz is what determines the length of the S− +component during the SR decay process, which in turn +directly determines max(|a|) [see Eq. (S2)]. To account +for these inversion imperfections, we post-select the data +such that the maximum decay amplitudes fall in a narrow +interval, as shown in Fig. S2(a). +The N appearing in Eq. (S10) can be taken to +parametrize the initial inversion, and therefore also +max(|a|). +Solving Eq. (S10) for N, we therefore can +deduce the functional relation between the maximum cav- +ity amplitude and the delay time, max(|a|) = C/(t0 −tD), +for a given tipping angle. We confirm this dependency +by fitting the data of the highest power probe pulses in +Fig. S2(a). The dependency of the delay time on the +inversion clearly necessitates a post-selection, as not to + +S3 +conflate this effect with the influence of the trigger pulse +power in shifting the delay time tD. +The linear shift of the SR decay phase with tD [see +Fig. S2(b)] comes from a slight detuning between cavity +and spins and is corrected for in the data presented in +the main text for clarity. +1 +2 +3 +tD (µs) +0.6 +0.8 +1 +1.2 +max(|a|) +(a) +1 +2 +3 +tD (µs) +-: +-:/2 +0 +:/2 +: +phase +(b) +FIG. S2. +(a) All recorded data points of the SR decay +amplitude maxima max(|a|) = +� +I2 +D + Q2 +D and phases ϕ = +arctan(QD/ID) over the delay time tD, with the same color +scheme as in the main text. The dashed line is a fit of the +expected functional dependency to the highest power probe +pulse values. The post-selected data points presented in the +main text lie between the two horizontal lines. (b) Phase drift +over delay time with a linear fit to the six highest probe pulse +powers. This drift is corrected by a linear shift that aligns the +dashed line with the horizontal axis. +DISTRIBUTION FUNCTIONS FOR DELAY +TIME tD AND PHASE φ OF THE SR DECAY +As described in the main text, the initial state of the +inverted collective spin vector is located near the north +pole of the Bloch sphere, close to the z-axis. Now, we +approximate the surface near the north pole as a plane. +Prior to the trigger pulse acting on the spin vector, the +tipping angle θ = arctan(|S−|/Sz) is centered around +θ = 0 but with a finite width of θ. After the trigger +pulse displaces the spin state, the tipping angle follows +the Rician distribution [26] +fΘ(θ, η, θ) = θ +θ +2 exp +� +−1 +2 +�θ2 +θ +2 + η2 +�� +I0 +�θη +θ +� +, (S11) +with the modified Bessel function of the first kind I0. +The parameter η expresses the displacement of the initial +spin vector away from the origin in units of the width +parameter θ, which we assume to be in the direction +φ = 0. This displacement is a result of the spin rotation +caused by the trigger pulse. For η ≫ 1 the distribution +fΘ becomes a Gaussian with mean value ⟨θ⟩ = ηθ and +variance Var(θ) = θ +2. The angular distribution for φ = +arg(S−) of the resulting in-plane vector is given by [27] +fΦ(φ, η) = +η +√ +2π ˜ϕ(η) +� +1 + η cos(φ) +˜Φ(η cos(φ)) +˜ϕ(η cos(φ)) +� +, +(S12) +with the standard normal distribution ˜ϕ and its cumula- +tive distribution function ˜Φ. As η increases, the initially +randomly distributed angle φ becomes more and more +well defined and approaches φ = 0. +We can infer the initial tipping angles θ from the delay +times tD using a simplified expression for the delay time +derived above +tD = −2TR log +�θ +2 +� +, +(S13) +depending only on θ and TR. By applying a change of +variables we arrive at the distribution for the delay times +tD +ftD(tD, η, θ) = fΘ +� +θ(tD, TR), η, θ +� ���� +d θ(tD, TR) +d tD +���� . (S14) +MICROWAVE SETUP +For the generation of our MW inversion pulses, we use +an arbitrary waveform generator to modulate the I/Q- +quadratures onto a carrier wave created by a power source +generator (PSG) at the cavity frequency of 3.1 GHz. The +pulses are gated using a fast MW switch, pass through a +chain of digital attenuators and are amplified using a high +power amplifier (+40 dB), before they enter the pump +MW line, leading into the cryostat. At the 1 K stage +inside the cryostat there is a relay switch, which can be +used to completely decouple the pump line from the lower +stages, blocking the room temperature thermal photons +and the amplifier noise, which takes about 1 ms. +The probe pulses are created with another PSG and gated +using a fast MW switch. Subsequently they pass through +a second chain of variable digital attenuators, after which +they are sent through the probe-line. In the experiment +the probe-line has a fixed attenuation of −72.5 dB, of +which −20 dB are located right outside the cryostat, the +rest distributed among the stages. The probe-line is con- +nected to cavity port 2 using a splitter, together with the +out-line. +Following the out-line upwards, we have two MW isola- +tors with a combined isolation of −20 dB and a −10 dB +attenuator, for reducing thermal noise photons from the +higher stages, before the signal is amplified with a low +noise cryogenic amplifier. The signal is then demodulated +using a homodyne detection setup, with the demodulation +frequency supplied by the probe PSG. The two quadra- +ture channels are finally measured with a high-speed +data-acquisition system. +ESTIMATING THE NUMBER OF PHOTONS +To estimate the number of photons contained in a probe +pulse we do a calibration measurement of the attenuation + +S4 +3.09 +3.095 +3.1 +3.105 +Frequency (GHz) +-60 +-40 +-20 +Transmission (dB) +S11 +S31 +S32 +FIG. S3. S-parameter traces (solid lines) of the cavity with +far detuned spins where the numbers 1, 2 and 3 correspond +to the pump, probe, and out-line and their respective fits +(dashed lines). For these measurements an additional −20 dB +attenuator at the probe-port entry was removed. This does +not change the values for κ1,2 obtained by fitting the dips, as +they appear only relative to the base level. The parameter +κtot manifests itself in the HWHM of the Lorentzian peak of +S31. +A2 = −54.5 dB+2 dB at room-temperature for the probe- +line inside the fridge up to port 2 of the cavity, where +a value of +2 dB is added to account for the decreased +resistance of the lines when cold. Then we determine the +MW power for the strongest probe pulses of the signal +that enters the probe-line outside the fridge using a power +spectrum analyzer, Pmax = −58 µW. The other probe +pulses used in the experiment have variable attenuation +decrements of −5 dB each, so the photon numbers change +accordingly down to −45 dB relative to the highest power +value. +Next, we determine the ratio R2 of MW power reflected +at cavity port 2 versus the incident power. For that, we +measure the S-parameters of our system on resonance +at 25 mK with the spins far detuned using the VNA as +summarized in Fig. S3. By fitting the measured traces +with the expected results from cavity input-output theory +(reproduced for ∆c = 0, i.e., on resonance condition) [24] +we obtain R2. +|S11|2 = A2 +1 (2κ1/κtot − 1)2 +|S31|2 = A1A3 (2√κ1κ2/κtot)2 +|S32|2 = A2A3 (2κ2/κtot − 1)2 +� +�� +� += R2 +(S15) +Here, the subscripts in A1,2,3 refer to the fixed MW line +attenuations inside the cryostat for pump, probe, and +out-line respectively. +TABLE I. Summary of the parameters used to estimate the +number of photons entering the cavity via the 100 ns trigger +pulses through the probe-line. +κ2/2π +κtot/2π +R2 +A2 +P −45 dB +max +59 kHz +586 kHz +0.64 +−52.5 dB +1.83 nW +In a pulse with duration ∆t = 100 ns, as used in the +experiment, the number of photons that enter through +TABLE II. Parameters used to estimate the number of photons +per pulse in the pulse sequences injected via the pump-line. +As this experiment was done in another cool-down of our +cryogenic system, the Q-factor of the resonator, therefore the +κtot value, exhibits some deviations from the ones above. +κ1/2π +κtot/2π +R1 +A1 +PMW +182 kHz +516 kHz +0.086 +−13.6 dB +2.1 µW +TABLE III. Temperatures of the various stages inside the +dilution refrigerator and corresponding attenuations between +the respective stages to estimate the number of thermal cavity +photons, when the solenoid switch at the nominal 1 K stage is +disconnected. +stage i +1 +2 +3 +4 +5 +6 +Ti (K) +296 +42 +4 +0.9 +0.12 +0.025 +Ai,i+1 (dB) +pump +– +– +– +-2.5 +-2 +probe +-2 +-22 +-2 +-12 +-14.5 +out +-2 +-2 +-2 +-2 +-30 +port 2 into the cavity using the lowest probe powers with +−45 dB attenuation is then calculated as (c.f. Table I) +nmin +trig = P −45 dB +max +∆t +ℏωc +A2(1 − R2) ≈ 180. +For the pulse train measurements in the reduced coop- +erativity regime we use the same procedure to calculate +the number of photons per 100 ns pulse entering through +the pump-line (c.f. Table II), +npC<1 +trig += PMW∆t +ℏωc +A1(1 − R1) ≈ 4.1 × 109. +Lastly, we calculate an estimate for the number of +thermal photons in the cavity, when the solenoid switch at +the 1 K stage is open to decouple the higher temperature +stages. +We use the values of Table III and evaluate +according to +ni = n(Ti) + Ai -1,i ni -1 , +n(T) = +1 +exp(ℏωc/kBT) − 1 , +(S16) +going down the stages for all MW lines. The dominant +contribution are thermal photons from the 1 K stage of +the pump-line, which result in a value of n ≈ 2.3 photons. +T1 MEASUREMENTS USING THE VECTOR +NETWORK ANALYZER +In the main text we discuss a fast relaxation of the +stored spin inversion with a characteristic timescale of +7.6 ms. This observed fast decay is contrasted by a slow +relaxation from an approximately thermally mixed state, + +S5 +3.095 +3.1 +3.105 +3.11 +3.115 +Frequency (GHz) +-50 +-40 +-30 +-20 +Transmission (dB) +@ +(a) +scrambled +detuned +on res. +0 +100 +200 +300 +400 +Time (sec) +-0.6 +-0.4 +-0.2 +0 +@ (MHz) +T1 = 134.3 sec +(b) +FIG. S4. (a) VNA transmission measurement of the hybrid +system in its ground state on resonance (blue), with the detun- +ing loop off (spins detuned, yellow) and measured using high +input power to scramble the spins (red). (b) Dispersive shift χ +over time, extracted from Lorentzian fits to the transmission +data. The black fit line corresponds to a simple exponential +decay law. +as shown in Fig. S4. The initial state for measuring this +slow relaxation is created by repeatedly sweeping across +the resonance with the VNA using a high input power for +30 s. This way, we scramble the spins, creating a state +that resembles a thermal ensemble. For a large ensemble +detuning δ ≫ gcoll the dispersive shift χ, represented +graphically in Fig. S4(a), allows a direct way to determine +the long T1 time of the spins with the result +χ(t) = g2 +coll +δ ⟨Sz(t)⟩ , +(S17) +as similarly employed in [23]. +NITROGEN VACANCY CENTER SPINS AND +DIAMOND SAMPLE +The spin ensemble used in this work consists of +negatively charged nitrogen vacancy centers in diamond +(NV), which are made up of a substitutional nitrogen +atom with an adjacent lattice vacancy. This paramagnetic +impurity has an electron spin S = 1 and can be described +by the Hamiltonian H = ℏDS2 +z + µBS, with the zero +field splitting D = 2.88 GHz and µ = 28 MHz/mT. +The diamond symmetry results in four possible orien- +tations of the NV centers. In the present experiment +the external magnetic field orientation was chosen to +tune all four sub-ensembles into resonance with the cavity. +The roughly cube shaped diamond samples with side +length d ∼ 200 µm were cut from a larger sample by +Delaware Diamond Knives. The larger sample was cre- +ated similarly to the one characterized in detail in [23], +referred to as “N1” therein. It was made by irradiating +a commercially available high-pressure high-temperature +diamond with an initial nitrogen concentration of 200 +ppm and naturally abundant 13C isotopes with our in- +house neutron source (TRIGA Mark II reactor) for lattice +vacancy creation. +It was irradiated with a fluence of +5 × 1017cm−2 for 50 h and annealed at 900 °C for 3 h, +resulting in a NV density of 40 ppm. + diff --git a/b9E2T4oBgHgl3EQfwQjN/content/tmp_files/load_file.txt b/b9E2T4oBgHgl3EQfwQjN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7c6b5ba4a0c9ad284bf1a051e4334f089929760b --- /dev/null +++ b/b9E2T4oBgHgl3EQfwQjN/content/tmp_files/load_file.txt @@ -0,0 +1,595 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf,len=594 +page_content='Triggered Superradiance and Spin Inversion Storage in a Hybrid Quantum System Wenzel Kersten,1, ∗ Nikolaus de Zordo,1 Oliver Diekmann,2 Tobias Reiter,2 Matthias Zens,2 Andrew N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Kanagin,1 Stefan Rotter,2 J¨org Schmiedmayer,1 and Andreas Angerer1 1Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, A-1020 Vienna, Austria 2Institute for Theoretical Physics, TU Wien, Wiedner Hauptstraße 8-10/136, A-1040 Vienna, Austria (Dated: January 11, 2023) We study the superradiant emission of an inverted spin ensemble strongly coupled to a supercon- ducting cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' After fast inversion, we detune the spins from the cavity and store the inversion for tens of milliseconds, during which the remaining transverse spin components disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Switching back on resonance enables to study the onset of superradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' A weak trigger pulse of a few hundred photons shifts the superradiant burst to earlier times and imprints its phase onto the emitted radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' For long hold times, the inversion decreases below the threshold for spontaneous superradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' There the energy stored in the ensemble can be used to amplify microwave pulses passing through the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Superradiance is the process by which an ensemble of excited two-level systems synchronizes to produce a short, highly coherent burst of light [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The build-up of correlations during the collective decay, mediated by an enhanced coupling to a common mode, gives rise to non-linear scaling of the decay rate with the number of emitters [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Superradiant (SR) emission is not only fundamental to many fields of physics, but also attracts increasing interest for applications in metrology [3], laser physics [4–6] and quantum technology in general [7–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' SR phenomena are at the heart of the transition from a genuine quantum regime, where individual fluctuations of the vacuum field will jump-start the collective decay of inverted emitters, to the classical regime, where the emission is akin to that of a macroscopic radiating dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' While experiments on superradiance have recently been successfully transferred from atomic ensembles to solid- state spin systems [13, 14], the possibilities this opens up for controlling and exploiting superradiance for applica- tions have been very little explored so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Progress in this direction has primarily been hindered by the fact that systems giving rise to superradiance are fundamentally unstable, reacting to the slightest disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Although this extreme sensitivity even to weak signals poses a great challenge for experimental implementation, it also offers very promising possibilities for applications in sensor and detector technology [3, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' In this work we report on a novel experimental platform that allows us not only to invert a large ensemble of nitrogen-vacancy (NV) spins, but also to hold and stabilize the stored inversion for up to 20 ms – four orders of magnitude longer than the timescale of the SR decay process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' This stabilization lets us study and control the emission of a SR burst that releases the energy stored in the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' To trigger the SR emission, we use weak microwave (MW) pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Additionally, we investigate a regime with reduced inversion, where the spins act as a gain medium for the amplification of a series of short MW pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' In order to coherently manipulate spins with high fi- delity, we use a newly developed MW resonator [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Our setup is based on two opposing super- conducting chips that exhibit a small mode volume with homogeneous coupling strength, while retaining a high quality factor of Q = 3000 (unloaded Q = 30000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' This design allows us to reach the regime of strong collective spin-cavity coupling already with a number of NVs that is reduced by three orders of magnitude compared to earlier works [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Moreover, owing to the resonator’s compact design, we are able to add a small loop of super- conducting wire in order to magnetically tune the spins in and out of the cavity resonance faster than the SR timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The resonator is composed of two sapphire chips with a 200 nm thin layer of 16 × 16 mm2 superconducting Nio- bium mounted in a copper housing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The identical pat- terns on both chips feature a hole in the center from which a 4 µm slit reaches outwards, resembling a split ring resonator [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The chips are stacked, with the roughly cube shaped diamond sample placed between the center holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The hole radii, the distance between the chips and the sample size are all of similar dimen- sion d ∼ 200 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' This configuration results in a reso- nance frequency of ωc/2π = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='1 GHz and linewidth of κ/2π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='5 MHz (HWHM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The resonator’s oscillating magnetic field couples homogeneously to all spins with a collective coupling strength of gcoll/2π = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='2 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The NV sample has a number of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='6 × 1013 spins with an effective linewidth for the inhomogeneously broadened spin ensemble of Γ⊥/2π = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='3 MHz [19], a value that takes the spin frequency distribution, modeled by a q- Gaussian function [20], into account (see Supplemental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The resulting cooperativity parameter of our coupled system is C = g2 coll/(κΓ⊥) ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We begin our explorations with all NV spins as effective two-level systems thermally relaxed to the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The spins are then inverted using a 400 ns modified chirp pulse with a Gaussian envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Details of the initializa- tion procedure are given in the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Following the inversion pulse, we rapidly switch off the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='04100v1 [quant-ph] 10 Jan 2023 2 - - +++ - - +++ Bext Bloop Bosc 1K Pump Probe Out 70dB κ2 κ1 25mK 400 ns 4 μs t = 0 |a| I/Q δ 2 ms hold time 2 3 4 5 6 t (µs) 0 1 tD I Q |a| sim 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='2 inversion p 1 2 3 4 t (µs) 65 µs 150 µs 400 µs 850 µs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='2 ms 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='7 ms 13 ms hold time 0 2 4 |a| (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=') 0 10 20 10-1 100 � = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='6 ms hold time (ms) max(|a|) (a) (b) (d) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (a) Schematic of the MW cavity located in a dilution refrigerator operating at 25 mK and connected to a homodyne MW setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Two sapphire chips with opposing split ring structures and the diamond sample are stacked inside a copper box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Between the center holes the oscillating magnetic field homogeneously penetrates the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The chips are surrounded by a loop made of 5 windings of superconducting wire, enabling a rapid detuning of the spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The pin coupler of port 1 is connected to the pump-line, which can be decoupled from the room temperature thermal bath at the 1 K stage using a solenoid switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Port 2 is connected to the out-line for acquiring data, and the attenuated probe-line for injecting weak trigger pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (b) Sequence of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We use a modified chirp pulse (red/blue) to invert the spin ensemble and subsequently detune the spins by switching off the current (green) in the loop, thereby storing the inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' After a certain hold time we bring the spins back into resonance and see their SR decay (black), optionally triggered by a short probe pulse (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (c) SR decays for varying hold times, triggered by room temperature noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The inset shows the SR decay maxima together with an exponential fit in a semi-log plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (d) Example data and simulation of a SR decay and its associated quadrature values I/Q, together with the simulated inversion p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The vertical line indicates tD, the time of the maximum cavity amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' loop current in about 200 ns using a semiconductor switch, causing a detuning of the spins by δ/2π = 26 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' This detuning by more than the ensemble linewidth inhibits the SR interaction of the spin ensemble with the cavity mode, thereby storing the inversion [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' During the inversion hold time, dephasing processes eliminate the spins’ collective dipole moment determined by the transversal component of the collective spin vector, S− = Sx − iSy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' When tuning the ensemble back into resonance, we thus create a metastable inverted state whose tipping angle θ = arctan(|S−|/Sz) with respect to the z-axis in the Bloch sphere is exponentially de- creased for longer hold times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Under the condition that the product of the stored ensemble inversion −1 ≤ p ≤ +1 and cooperativity is above the threshold pC > 1, this metastable state will decay by emitting a SR photon burst [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Here, the inversion parameter p is implicitly defined by Sz = 1 2⟨� j σj z⟩ = pN/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' In this state, the presence of even a single photon in the cavity will stimulate the collective emission of radiation, starting a self-accelerating photonic avalanche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' During this process, the energy re- leased in the form of cavity photons gradually builds up, reaches a maximum and then oscillates back and forth between the two subsystems, before the process stops due to the dephasing of the spins and their decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The full experimental sequence is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Our first notable result is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 1(c), where we plot the SR decay pulses for varying inversion hold times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Here, the SR decay is triggered by noise from the high power amplifier of the pump-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The measured SR dynamics are captured in a semi-classical description using the Maxwell-Bloch equations [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We model the time evolution starting from an inverted state with a slight tipping angle accounting for fluctuations that initiate the SR decay (see Supplemental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' To simulate the measured signals of |a| we only adjust the ensemble inversion p and a time offset, resulting in curves as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The role of fluctuations at the start of the SR decay process is studied in more detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We find the decay maximum max(|a|), an indirect measure of the energy stored by the spins, to decrease roughly exponentially with increasing hold times, exhibiting a characteristic timescale of τ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='6 ms [see inset Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 1(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' For hold times longer than 20 ms, the inversion has already decreased below the threshold pC = 1 for spontanoeus superradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We hypothesize two distinct timescales for the relaxation of the inverted state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' a fast one that rapidly thermalizes the ensemble, driven by spin-spin interactions involving NVs with very short lifetimes (so called fluctuators [22]), and a slow one which brings them to the ground state over extended times (T1 = 134 s, see Supplemental Material [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 3 We now focus on the onset of the SR decay process and the possibility to trigger it prior to its self-decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Using a 2 ms hold time, we give the cavity mode enough time to reach thermal equilibrium after the inversion pulse and subsequent decoupling from the high power amplifier noise by the solenoid switch, with an estimated number of n ≈ 2 thermal photons remaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The completely dephased inverted state, that is brought back into resonance, has zero tipping angle apart from unavoidable quantum and thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Another 150 ns after switching back the detuning current (defined as t = 0), we send a 100 ns trigger pulse through the highly attenuated MW probe- line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The pulse is resonant with the cavity and contains a calibrated number of photons (see Supplemental Material [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The experiment is repeated many times for varying numbers of trigger photons, and without trigger pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' For every run, we extract the delay time tD and the ID/QD quadrature values of the SR decay maximum [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 1(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We post-select the runs for max(|a|) = � I2 D + Q2 D to fall into a narrow window, as there is some variation in the initial spin inversion between the runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The SR decay phases ϕ = arctan(QD/ID) are corrected for a linear phase drift with tD, caused by a minor constant detuning of the spins (see Supplemental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The resulting sets of delay times and phases is summarized in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 2(a) and 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Clearly, stronger trigger pulses with higher numbers of photons ntrig lead to earlier tD values and narrower distributions for tD and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' While our simulation allows to describe the decay process starting from a slightly tipped initial collective spin vector, it is the randomness in the initial conditions that leads to the observed variance in time and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' These thermal and quantum fluctuations are not included in our semi- classical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' To understand these phenomena and model the distributions of times and phases of the decay pulses, we split the analysis of the SR decay into two stages [2, 15, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The decay process starts with a linear stage, in which the (optional) trigger pulse leads to a coherent rota- tion of the collective spin vector about an axis defined by the phase of the pulse, which is kept identical for all runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Prior to this rotation, the initial state is lo- cated very close to the +z-axis but with a small tip- ping angle θ = arctan(|S−|/Sz) and random polar an- gle φ = arg(S−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' As cos θ ≃ 1 throughout the linear phase, we can treat the spin vector to be confined to a plane with a z-offset corresponding to the initial inver- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The geometric construction of this plane is illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 2(c), mathematical formulas of the distri- bution functions are given in the Supplemental Material citerice1945mathematical,angularmarginalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The initial state of the spin vector follows a two dimensional Gaussian distribution of width θ centered at θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The influence of the trigger pulse then causes a displacement in the plane, which we choose to be in the direction of φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The parameter η expresses the displacement in units of �� � � � ��� �� �� �� ��� 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='5 3 tD (µs) no pulse 180 570 1800 5700 18k 57k 180k 570k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='8M 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='7M number of trigger photons � 0 � phase � 180 570 1800 5700 18k 57k 180k 570k 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='8M 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='7M ntrig 100 102 �� (c) (a) (b) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Swarm plots of the measured delay times tD (a) and phases ϕ (b) of the SR decay maxima in over 1200 runs with varying powers (photon numbers) of the 100 ns trigger pulse, or no pulse at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The distributions plotted using solid lines in (a) correspond to the maximum likelihood estimates for the displacement parameter η to the recorded tD data, where other parameters are kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The dashed lines in (b) are the expected phase distributions for those estimates of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (c) Initial state of the collective spin vector with coordinates (θ, φ) close to the north pole of the Bloch sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' In-plane distribution before (blue) and after (red) the coherent displacement η in units of the width θ via the trigger pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (d) Log-log plot of the squared displacement η2 over the estimated number of trigger photons in the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' the width parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' For growing η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=', higher trig- ger pulse powers, the initially randomly distributed polar angles become increasingly well defined and approach a narrow distribution around φ = 0 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 2(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' After this linear stage, where the collective spin vector is coherently displaced from its random in-plane starting position, we enter a nonlinear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Now the SR dy- namics dominate and via a collective process of stimulated emission the spin vector accelerates its rotation towards the equator while emitting a considerable burst of MW radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The phase ϕ of the emitted decay pulse is directly de- termined by the value of φ at the start of the nonlinear stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Less directly, we can infer the initial tipping angles θ from the delay times tD, which result via the relation 4 20 22 24 26 28 30 32 34 36 t (µs) 30 ms 35 ms 40 ms 50 ms hold time 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='2 |a| (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=') |a| |a| empty 0 10 20 30 40 50 60 70 hold time (ms) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='3 inversion p 0 1 2 3 max(|a|) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=') pC = 1 p(t) a T b = a·exp(-(t/T)b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='36 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='8 ms = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='451 max(|a|) self-decay pulse trains stretched exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (a) Cavity amplitude |a| for a series of 100 ns pulses, each injecting ntrig ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='1 × 109 photons, amplified by the partially inverted spin ensemble in the reduced effective co- operativity regime pC < 1 for different hold times (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' In comparison, we plot the signal obtained with an empty cavity where spins are far detuned (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' For choosing the parame- ters in our semi-classical model (black), we ignore noise below a certain threshold (green line at the top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (b) Ensemble inver- sion as a function of hold time, extracted by simulations in the two regimes above and below pC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Above this threshold, the pulse maxima (right y-axis) follow the values of p from simulations of the self-decays shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' A stretched exponential is fitted to the inversion as a guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' tD = −2TR log (θ/2) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Here, the parameter TR repre- sents the timescale for the SR emission process (see Sup- plemental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' With this relation and the assumed distribution for the initial tipping angles θ depending on η [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 2(c)], we can reproduce the distributions of the measured tD data using maximum likelihood estimation to fit η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' For that, we fix the values TR = 212 ns and θ = 6 × 10−3 using the measurement runs without probe pulses (η = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 2(a) we show the resulting tD dis- tributions from these estimates of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The corresponding phase distributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 2(b) are not fitted, but result directly from those η estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' As the displacement η is caused by the MW magnetic field of the trigger pulse, its square is a measure of the energy imparted onto the spin system during the linear stage of the SR process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Therefore, the number of photons that trigger the SR decay is proportional to η2 ∝ ntrig [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 2(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Remarkably, a weak MW pulse on the order of 10−11 photons per spin is sufficient to have an observable effect on the SR decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' This observation opens the perspective to operate our system as a detection device that is not only sensitive to the amplitude, but also the phase of weak MW signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We now investigate a regime of reduced effective co- operativity pC < 1, where SR emission does not occur spontaneously [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' To that end, we employ hold times longer than 20 ms, thus reducing the polarization below the threshold for the SR decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We probe the system by injecting, at 5 µs intervals, a sequence of resonant MW pulses of 100 ns duration via the pump-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Interestingly, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 3(a), we find that this results in an amplification of the pulses as compared to the empty cavity response (with far detuned spins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Although no spontaneous SR decay occurs on its own, it is still possible to repeatedly extract energy from the stored inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The incident pulses hereby effectively supply the necessary coherence that is otherwise constituent to the SR emission, but hindered from building up when the stored inversion is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Notably, tens of injected MW pulses can be amplified in succession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We are able to replicate the measured dynamics using our numerical model with only the amplitude of the incident pulses (kept fixed for all fits) and the ensemble inversion p as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' These results are combined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 3(b) with the p values at- tained by simulating the SR self-decays [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 1(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The semi-classical model seamlessly captures the behavior of our system in both regimes of high and low effective cooperativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' In summary, we present and successfully operate an experimental platform to store the energy of an inverted spin ensemble for tens of milliseconds and to release it via a strong SR burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' By initializing the system to a fully upright inverted state, we demonstrate a high sensitivity to weak MW pulses that strongly influence the subsequent SR dynamics, allowing us to infer both amplitude and phase of the trigger pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The decrease of inversion over time allows us to study a regime of reduced cooperativity without spontaneous SR emission, where the inverted spins effectively act as a gain medium for a series of short MW pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Our setup lays the ground work for the design of highly sensitive MW detectors on a chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We thank Johannes Majer for discussions and techni- cal support in the initial phases of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We ac- knowledge support by the Austrian Science Fund (FWF) projects I3765 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Gross, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Raimond, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Fabre, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Haroche, Physical Review A 27, 2043 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' [26] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Rice, The Bell System Technical Journal 24, 46 (1945).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' [27] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Cooper and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Farid, “A toolbox for the radial and angular marginalization of bivariate normal distributions, arxiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='09696,” (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' S1 Supplemental Material: Triggered Superradiance and Spin Inversion Storage in a Hybrid Quantum System SYSTEM HAMILTONIAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' EQUATIONS OF MOTION AND NUMERICAL MODELLING Our system is described by the driven Tavis-Cummings Hamiltonian in the rotating frame [20],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' H =ℏ ∆ca†a + ℏ 2 � j ∆j sσj z + ℏ � j g0 � a†σj − + σj +a � + iℏ η � a† − a � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (S1) with a† (a) being the creation (annihilation) operator of the cavity mode and σj z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' σj ± being the Pauli-z and raising/lowering operators for the jth spin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The spins are coupled to the cavity with constant cou- pling g0 = gcoll/ √ N, where gcoll is the collective coupling strength and N is the number of spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The spin detun- ings ∆j s = [ωj s + δ(t)] − ωp account for the inhomogeneous broadening of the spin ensemble and the additional shift δ(t) caused by the detuning loop, while ∆c = ωc − ωp is the detuning of the cavity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Both detunings are cal- culated with respect to the driving frequency ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Further, the amplitude of the driving field is determined by η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Using a Lindblad master equation, we take into account the loss rate κ for the cavity mode and the spin deco- herence rate γ⊥, thus yielding a set of coupled equations describing the dynamics of the operators, ˙a = −(i∆c + κ)a − i � j g0σj − + η , (S2) ˙σj − = −(i∆j s + γ⊥)σj − + ig0aσj z , (S3) ˙σj z = 2ig0 � a†σj − − aσj + � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (S4) Identifying these operators with their expectation val- ues, effectively neglecting correlations between individual spins and the cavity by separating higher order moments into products of their first order counterparts, yields the well known Maxwell-Bloch equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' These equations of motion represent a semi-classical description of our system’s dynamics, which can be solved numerically, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=', to model the response to external stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' In our analysis, this description only fails to capture the stochastic nature of the inverted state’s initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' To perform the numerical simulations, we approxi- mate the spin frequency distribution ρ(ω), which is quasi- continuous due to the large number of spins, by sampling it at 1500 frequencies ωj with equidistant spacing ∆ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The resulting weights ρj = ρ(ωj)∆ω are then used to cal- culate the number of spins, nj = ρjN, for each frequency bin ωj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Assuming identical initial conditions for all spins of one bin, also their dynamics according to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (S3) and (S4) are identical, thus tremendously reducing the number of relevant equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We determine the parameters describing our system, namely ωc, κ, gcoll, and Γ⊥ [combining γ⊥ and the spin frequency distribution in one parameter, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (S5)] by fitting the steady state solution of the Maxwell-Bloch equations to the transmission signals [c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' S4(a)], obtained with a vector network analyzer (VNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' For modelling SR emission dynamics on short timescales, we can safely neglect T1 processes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=', we do not include these in the master equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' SPIN FREQUENCY DISTRIBUTION We model the inhomogeneously broadened spin distri- bution ρ(ω) with a q-Gaussian function [20] with shape parameter q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='39 and FWHM of γq/2π = 11 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The effective linewidth, present in the definition of the cooper- ativity C = g2 coll/κΓ⊥ in the main text, can be calculated using [19] Γ⊥ = �� +∞ −∞ ρ(ω)dω γ⊥ + i(ω − ω0) �−1 , (S5) with ω0 being the spin center frequency, together with the value for γ⊥/2π = 208 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' INITIALIZATION PROCEDURE AND INVERSION PULSE All four NV sub-ensembles are tuned into resonance with the cavity using a vector magnet together with the detuning loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' As the third level of the NV ground state manifold is far away from resonance, we can treat the NVs as effective two-level systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' For all experiments, the spins are initialized close to their ground state by waiting 3 min after repeatedly sweeping a MW tone across the cavity resonance for 30 s, decreasing the cooperativity by roughly 25% but reducing wait times between different runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Starting from this state, the spins are inverted using an in-phase and quadrature modulated MW pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Similar to adiabatic fast passage methods for spins in free space, the starting point for the design of our inversion pulse is a chirped pulse of length 400 ns with a Gaussian envelope, that covers a frequency interval of about -8 to +8 widths γq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' As our spins are not in free space but strongly coupled to a cavity, we cannot use the chirped pulse directly but need to adapt it for this circumstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Comparing the S2 Maxwell-Bloch equations above with the optical Bloch equations, describing a two-level-system in free space driven by a classical coherent light field with driving amplitude Ω(t), ˙σ− = −γ⊥σ− + iΩ(t)σz , (S6) ˙σz = 2i (Ω(t)σ− − Ω∗(t)σ+) , (S7) we can see that the role of the driving amplitude in the case of the coupled cavity-spin system is taken by the term g0a†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Disregarding g0 as a proportionality constant that has to be determined experimentally, we can assume a desired photonic amplitude a(t) = aR + iaI given by the aforementioned chirped pulse and numerically solve for MW drive η(t) = I − iQ necessary to create it in the cavity (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Strictly speaking, this approach produces the correct effective inversion drive only for the spins with the center frequency, as only these are fully resonant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' For other frequencies, the cavity acts as a filter and reduces the amplitude of the drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Nevertheless, as the chirped pulse comes with a certain robustness to amplitude deviations, we still get a useful inversion efficiency for the whole spin ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' In the experiment, we scan the inversion pulse power to select the value for optimum inversion, which is in turn visible as the highest SR decay maximum for a given hold time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 0 100 200 300 400 Time (ns) Re(a) Im(a) I Q FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Chirped pulse for initial spin inversion, desired ⟨a(t)⟩ and corresponding I/Q channels for optimal inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' DELAY TIME OF THE SUPERRADIANT EMISSION To derive Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (3) in the main text, we neglect the inhomogeneous broadening ∆j s ≈ 0 and describe the spin ensemble as a giant dipole using S± = � j σj ± and Sz = 1 2 � j σj z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' When inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (S2) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (S4) on resonance ∆c = 0, we get ˙Sz = −2g2 0 κ S+S− − ig0 κ (S+ ˙a − ˙a†S−) , (S8) where we now can neglect the second term as it is of lower order in the number of spins N ∝ S−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Effectively, the cavity acts similar to a vacuum environment for the SR burst, although enhancing the coupling of individual spins to the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Its ability to store photons becomes of importance only at later times, when the cavity amplitude shows revivals, with excitations oscillating back and forth between cavity and spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Then, we evaluate [13] ⟨ ˙Sz⟩ = −2g2 0 κ ⟨S+S−⟩ = −2g2 0 κ (S + ⟨Sz⟩) (S − ⟨Sz⟩ + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (S9) By parametrizing ⟨Sz⟩ = cos(θ)N/2 with a tipping angle θ and using S = N/2, we can now solve for the delay time tD where ⟨Sz⟩ = 0, as the SR emission reaches its maximum when the giant dipole points to the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The resulting expression [2, 15] tD = t0 − κ 2g2 0N log � tan2 �θ 2 �� (S10) already resembles the one given in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Now we linearize tan θ ≈ θ, neglect the constant offset t0 and summarize the prefactor as TR = κ 2g2 0N , representing the timescale of the SR emission process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We find good quali- tative agreement of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (S10) with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Quantita- tively, when using the explicit values for κ and g2 0N = g2 coll, the timescale of the SR emission is underestimated due to the approximations involved (in particular, neglecting the inhomogeneous broadening).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' POST-SELECTION AND PHASE CORRECTION OF THE DECAY PULSES The data collected in all experimental runs has some variance in the SR decay amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We assume this variance comes mostly from the solenoid switch located at the 1 K stage which is used to disconnect the the pump line from port 1 of the cavity after the inversion pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The switch opens and closes a mechanical connection by a latching mechanism, thus leading to slightly different initial ensemble inversions between the runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The initial inversion Sz is what determines the length of the S− component during the SR decay process, which in turn directly determines max(|a|) [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (S2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' To account for these inversion imperfections, we post-select the data such that the maximum decay amplitudes fall in a narrow interval, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' S2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The N appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (S10) can be taken to parametrize the initial inversion, and therefore also max(|a|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (S10) for N, we therefore can deduce the functional relation between the maximum cav- ity amplitude and the delay time, max(|a|) = C/(t0 −tD), for a given tipping angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We confirm this dependency by fitting the data of the highest power probe pulses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' S2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The dependency of the delay time on the inversion clearly necessitates a post-selection, as not to S3 conflate this effect with the influence of the trigger pulse power in shifting the delay time tD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The linear shift of the SR decay phase with tD [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' S2(b)] comes from a slight detuning between cavity and spins and is corrected for in the data presented in the main text for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 1 2 3 tD (µs) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='2 max(|a|) (a) 1 2 3 tD (µs) : :/2 0 :/2 : phase (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (a) All recorded data points of the SR decay amplitude maxima max(|a|) = � I2 D + Q2 D and phases ϕ = arctan(QD/ID) over the delay time tD, with the same color scheme as in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The dashed line is a fit of the expected functional dependency to the highest power probe pulse values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The post-selected data points presented in the main text lie between the two horizontal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (b) Phase drift over delay time with a linear fit to the six highest probe pulse powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' This drift is corrected by a linear shift that aligns the dashed line with the horizontal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' DISTRIBUTION FUNCTIONS FOR DELAY TIME tD AND PHASE φ OF THE SR DECAY As described in the main text, the initial state of the inverted collective spin vector is located near the north pole of the Bloch sphere, close to the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Now, we approximate the surface near the north pole as a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Prior to the trigger pulse acting on the spin vector, the tipping angle θ = arctan(|S−|/Sz) is centered around θ = 0 but with a finite width of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' After the trigger pulse displaces the spin state, the tipping angle follows the Rician distribution [26] fΘ(θ, η, θ) = θ θ 2 exp � −1 2 �θ2 θ 2 + η2 �� I0 �θη θ � , (S11) with the modified Bessel function of the first kind I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The parameter η expresses the displacement of the initial spin vector away from the origin in units of the width parameter θ, which we assume to be in the direction φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' This displacement is a result of the spin rotation caused by the trigger pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' For η ≫ 1 the distribution fΘ becomes a Gaussian with mean value ⟨θ⟩ = ηθ and variance Var(θ) = θ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The angular distribution for φ = arg(S−) of the resulting in-plane vector is given by [27] fΦ(φ, η) = η √ 2π ˜ϕ(η) � 1 + η cos(φ) ˜Φ(η cos(φ)) ˜ϕ(η cos(φ)) � , (S12) with the standard normal distribution ˜ϕ and its cumula- tive distribution function ˜Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' As η increases, the initially randomly distributed angle φ becomes more and more well defined and approaches φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We can infer the initial tipping angles θ from the delay times tD using a simplified expression for the delay time derived above tD = −2TR log �θ 2 � , (S13) depending only on θ and TR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' By applying a change of variables we arrive at the distribution for the delay times tD ftD(tD, η, θ) = fΘ � θ(tD, TR), η, θ � ���� d θ(tD, TR) d tD ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (S14) MICROWAVE SETUP For the generation of our MW inversion pulses, we use an arbitrary waveform generator to modulate the I/Q- quadratures onto a carrier wave created by a power source generator (PSG) at the cavity frequency of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The pulses are gated using a fast MW switch, pass through a chain of digital attenuators and are amplified using a high power amplifier (+40 dB), before they enter the pump MW line, leading into the cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' At the 1 K stage inside the cryostat there is a relay switch, which can be used to completely decouple the pump line from the lower stages, blocking the room temperature thermal photons and the amplifier noise, which takes about 1 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The probe pulses are created with another PSG and gated using a fast MW switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Subsequently they pass through a second chain of variable digital attenuators, after which they are sent through the probe-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' In the experiment the probe-line has a fixed attenuation of −72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='5 dB, of which −20 dB are located right outside the cryostat, the rest distributed among the stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The probe-line is con- nected to cavity port 2 using a splitter, together with the out-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Following the out-line upwards, we have two MW isola- tors with a combined isolation of −20 dB and a −10 dB attenuator, for reducing thermal noise photons from the higher stages, before the signal is amplified with a low noise cryogenic amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The signal is then demodulated using a homodyne detection setup, with the demodulation frequency supplied by the probe PSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The two quadra- ture channels are finally measured with a high-speed data-acquisition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' ESTIMATING THE NUMBER OF PHOTONS To estimate the number of photons contained in a probe pulse we do a calibration measurement of the attenuation S4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='095 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='105 Frequency (GHz) 60 40 20 Transmission (dB) S11 S31 S32 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' S-parameter traces (solid lines) of the cavity with far detuned spins where the numbers 1, 2 and 3 correspond to the pump, probe, and out-line and their respective fits (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' For these measurements an additional −20 dB attenuator at the probe-port entry was removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' This does not change the values for κ1,2 obtained by fitting the dips, as they appear only relative to the base level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The parameter κtot manifests itself in the HWHM of the Lorentzian peak of S31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' A2 = −54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='5 dB+2 dB at room-temperature for the probe- line inside the fridge up to port 2 of the cavity, where a value of +2 dB is added to account for the decreased resistance of the lines when cold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Then we determine the MW power for the strongest probe pulses of the signal that enters the probe-line outside the fridge using a power spectrum analyzer, Pmax = −58 µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The other probe pulses used in the experiment have variable attenuation decrements of −5 dB each, so the photon numbers change accordingly down to −45 dB relative to the highest power value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Next, we determine the ratio R2 of MW power reflected at cavity port 2 versus the incident power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' For that, we measure the S-parameters of our system on resonance at 25 mK with the spins far detuned using the VNA as summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' By fitting the measured traces with the expected results from cavity input-output theory (reproduced for ∆c = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=', on resonance condition) [24] we obtain R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' |S11|2 = A2 1 (2κ1/κtot − 1)2 |S31|2 = A1A3 (2√κ1κ2/κtot)2 |S32|2 = A2A3 (2κ2/κtot − 1)2 � �� � = R2 (S15) Here, the subscripts in A1,2,3 refer to the fixed MW line attenuations inside the cryostat for pump, probe, and out-line respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Summary of the parameters used to estimate the number of photons entering the cavity via the 100 ns trigger pulses through the probe-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' κ2/2π κtot/2π R2 A2 P −45 dB max 59 kHz 586 kHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='64 −52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='5 dB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='83 nW In a pulse with duration ∆t = 100 ns, as used in the experiment, the number of photons that enter through TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Parameters used to estimate the number of photons per pulse in the pulse sequences injected via the pump-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' As this experiment was done in another cool-down of our cryogenic system, the Q-factor of the resonator, therefore the κtot value, exhibits some deviations from the ones above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' κ1/2π κtot/2π R1 A1 PMW 182 kHz 516 kHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='086 −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='6 dB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='1 µW TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Temperatures of the various stages inside the dilution refrigerator and corresponding attenuations between the respective stages to estimate the number of thermal cavity photons, when the solenoid switch at the nominal 1 K stage is disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' stage i 1 2 3 4 5 6 Ti (K) 296 42 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='025 Ai,i+1 (dB) pump – – – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='5 2 probe 2 22 2 12 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='5 out 2 2 2 2 30 port 2 into the cavity using the lowest probe powers with −45 dB attenuation is then calculated as (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Table I) nmin trig = P −45 dB max ∆t ℏωc A2(1 − R2) ≈ 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' For the pulse train measurements in the reduced coop- erativity regime we use the same procedure to calculate the number of photons per 100 ns pulse entering through the pump-line (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Table II), npC<1 trig = PMW∆t ℏωc A1(1 − R1) ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='1 × 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' Lastly, we calculate an estimate for the number of thermal photons in the cavity, when the solenoid switch at the 1 K stage is open to decouple the higher temperature stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' We use the values of Table III and evaluate according to ni = n(Ti) + Ai -1,i ni -1 , n(T) = 1 exp(ℏωc/kBT) − 1 , (S16) going down the stages for all MW lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The dominant contribution are thermal photons from the 1 K stage of the pump-line, which result in a value of n ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='3 photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' T1 MEASUREMENTS USING THE VECTOR NETWORK ANALYZER In the main text we discuss a fast relaxation of the stored spin inversion with a characteristic timescale of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='6 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' This observed fast decay is contrasted by a slow relaxation from an approximately thermally mixed state, S5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='095 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='105 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='115 Frequency (GHz) 50 40 30 20 Transmission (dB) @ (a) scrambled detuned on res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' 0 100 200 300 400 Time (sec) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='2 0 @ (MHz) T1 = 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='3 sec (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (a) VNA transmission measurement of the hybrid system in its ground state on resonance (blue), with the detun- ing loop off (spins detuned, yellow) and measured using high input power to scramble the spins (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' (b) Dispersive shift χ over time, extracted from Lorentzian fits to the transmission data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The black fit line corresponds to a simple exponential decay law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The initial state for measuring this slow relaxation is created by repeatedly sweeping across the resonance with the VNA using a high input power for 30 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' This way, we scramble the spins, creating a state that resembles a thermal ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' For a large ensemble detuning δ ≫ gcoll the dispersive shift χ, represented graphically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' S4(a), allows a direct way to determine the long T1 time of the spins with the result χ(t) = g2 coll δ ⟨Sz(t)⟩ , (S17) as similarly employed in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' NITROGEN VACANCY CENTER SPINS AND DIAMOND SAMPLE The spin ensemble used in this work consists of negatively charged nitrogen vacancy centers in diamond (NV), which are made up of a substitutional nitrogen atom with an adjacent lattice vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' This paramagnetic impurity has an electron spin S = 1 and can be described by the Hamiltonian H = ℏDS2 z + µBS, with the zero field splitting D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content='88 GHz and µ = 28 MHz/mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The diamond symmetry results in four possible orien- tations of the NV centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' In the present experiment the external magnetic field orientation was chosen to tune all four sub-ensembles into resonance with the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The roughly cube shaped diamond samples with side length d ∼ 200 µm were cut from a larger sample by Delaware Diamond Knives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' The larger sample was cre- ated similarly to the one characterized in detail in [23], referred to as “N1” therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' It was made by irradiating a commercially available high-pressure high-temperature diamond with an initial nitrogen concentration of 200 ppm and naturally abundant 13C isotopes with our in- house neutron source (TRIGA Mark II reactor) for lattice vacancy creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} +page_content=' It was irradiated with a fluence of 5 × 1017cm−2 for 50 h and annealed at 900 °C for 3 h, resulting in a NV density of 40 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E2T4oBgHgl3EQfwQjN/content/2301.04100v1.pdf'} diff --git a/b9FKT4oBgHgl3EQfpS4j/content/tmp_files/2301.11869v1.pdf.txt b/b9FKT4oBgHgl3EQfpS4j/content/tmp_files/2301.11869v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a867bac476a87c0b0a53e9aca4b3764d9a8e1b7 --- /dev/null +++ b/b9FKT4oBgHgl3EQfpS4j/content/tmp_files/2301.11869v1.pdf.txt @@ -0,0 +1,1248 @@ +Observation of brane parity order in programmable optical lattices +David Wei,1, 2 Daniel Adler,1, 2 Kritsana Srakaew,1, 2 Suchita Agrawal,1, 2 +Pascal Weckesser,1, 2 Immanuel Bloch,1, 2, 3 and Johannes Zeiher1, 2 +1Max-Planck-Institut f¨ur Quantenoptik, 85748 Garching, Germany +2Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany +3Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at, 80799 Munich, Germany +(Dated: January 30, 2023) +The Mott-insulating phase of the two-dimensional (2d) Bose-Hubbard model is expected to be +characterized by a non-local brane parity order. Parity order captures the presence of microscopic +particle-hole fluctuations and entanglement, whose properties depend on the underlying lattice ge- +ometry. We realize 2d Bose-Hubbard models in dynamically tunable lattice geometries, using neutral +atoms in a novel passively phase-stable tunable optical lattice in combination with programmable +site-blocking potentials. We benchmark the performance of our system by single-particle quantum +walks in the square, triangular, kagome and Lieb lattice. In the strongly correlated regime, we mi- +croscopically characterize the geometry dependence of the quantum fluctuations and experimentally +validate the brane parity as a proxy for the non-local order parameter signaling the superfluid-to- +Mott insulating phase transition. +I. +INTRODUCTION +According to seminal work by Landau, second-order +phase transitions are signaled by a change of a local or- +der parameter. +However, some phase transitions defy +this classification in terms of a local order parameter, and +require, as a generalization, non-local order parameters +to describe their underlying structure [1, 2]. The Hal- +dane insulator constitutes a celebrated example for such +a phase, in which a string correlator captures the un- +derlying non-local hidden order [3, 4], which has recently +also been realized experimentally [5, 6]. Interestingly, the +Mott-insulating (MI) phase of the Bose-Hubbard (BH) +model also features non-local order, which accounts for +quantum fluctuations in the form of bound particle-hole +pairs [7–9]. In one-dimensional (1d) BH chains, the MI +order has been revealed by a parity order parameter of +the on-site occupation [7, 8, 10]. In two dimensions (2d) +the brane parity was proposed as a generalization of par- +ity order for square lattices [11, 12]. However, up to now, +experiments directly measuring the brane parity in any +2d lattice geometry are lacking, as well as its experimen- +tal validation as an order parameter for the MI phase in +2d. A strategy to achieve the latter is provided by mean- +field theory, which predicts that the location of the phase +transition should scale with the coordination number and +thus the underlying lattice geometry. This scaling was +explicitly probed by measuring the local order parame- +ter in the SF phase [13]. Observing such scaling also in +the brane parity provides an indication for the suitability +of the brane parity as 2d non-local order parameter. +Neutral atoms in optical lattices provide a pristine +testbed to realize low-dimensional Hubbard models [14] +and offer techniques for the detection of local observ- +ables using quantum gas microscopes [15, 16]. Optical +lattices arise through the interference pattern of laser +beams, whose layout is carefully chosen for a specific tar- +get geometry and has led to the realization of a variety +of lattices [17–21]. +While optical lattices benefit from +their inherent homogeneity and stability, the static na- +ture of a given beam layout restricts systems to fixed +spatial geometries and makes dynamical changes within +a single experimental run challenging. In contrast, ar- +rays of optical tweezers can be generated in almost freely +programmable geometries [22] and have allowed for stud- +ies of a variety of many-body spin models. +A num- +ber of approaches have been brought forward to allow +for similar programmability for itinerant atoms based on +realizing small systems of tunnel-coupled optical tweez- +ers [23, 24] or dynamically controllable lattices [17]. How- +ever, tweezer arrays in the itinerant regime are difficult +to scale to large system sizes due to inhomogeneities and +concomitantly a large calibration overhead, and the dy- +namical control over lattices typically involves complex +active phase stabilization techniques [17, 25]. +Here, we report on the realization of 2d Bose-Hubbard +models in passively phase-stable optical lattices with +square or triangular base geometry, which we combine +with local site-blocking beams to realize programmable +unit cells. We demonstrate this novel degree of flexible +control by implementing square, triangular, kagome and +Lieb lattices in one experimental setup, and benchmark +their quality through single-particle quantum walks. In- +creasing the atomic density, we microscopically probe the +strongly interacting regime through non-local quantum +fluctuations and show their dependence on the underly- +ing lattice structure. Our measurements provide a quan- +titative characterization of the phase transition point in +these lattices and experimentally establish the brane par- +ity [8, 11, 12] as a meaningful non-local observable to +characterize the SF-MI phase transition in 2d models. +II. +PROGRAMMABLE LATTICES +In our approach to realizing tunable lattice geome- +tries, we superpose a bow-tie lattice [26] (L2) with a +arXiv:2301.11869v1 [cond-mat.quant-gas] 27 Jan 2023 + +2 +−1 +0 +1 +−1 +0 +1 +Position y/¸ +−1 +0 +1 +−1 +0 +Potential (arb.) +0.0 +0.5 +1.0 +Position x/¸ +0.0 +0.5 +1.0 +Á2 +Á1 +¢x +−5 +0 +5 +Position x/¸ +−5 +0 +5 +y/¸ +−5 +0 +5 +x/¸ +−5 +0 +5 +y/¸ +MI (square) +MI (triangular) +SF +L2 +L1 +PBS +Á1 +Á2 +x +y +(c) +(a) +(b) +(f) +(d) +(g) +(h) +(i) +(e) +FIG. 1. +(a) Experimental setup realizing passively phase-stable tunable lattices. The lattice 2 beam (L2, blue) is out-of-plane +polarized and forms a bow-tie lattice, realizing a square lattice potential (b). The in-plane polarized lattice 1 beam (L1, orange) +can be added with a well-defined superlattice phase ∆ϕ = 2π∆x/(λ/2) (see lattice potentials sketched in right inset). For the +in-phase case, ∆ϕ = 0, this realizes an effective triangular lattice geometry (c). (Left inset) The combined lattice is passively +phase-stable due to the retro-reflection mirror serving as a common phase reference. Temporal fluctuations of path lengths lead +to translations along either common paths or along translationally invariant directions. The arrows indicate the movement of +the interference pattern generated by the respective lattice upon changes in the phase φ1,2. (d,e) Single-site resolved image of +a Mott insulator in the square (d) and Lieb lattice (e). (f) Lattices with more complex unit cells, e.g. as shown in (e), can be +dynamically generated by projecting repulsive local potentials through the objective, blocking out distinct lattice sites. (g-i) +The MI phase hosts doublon-hole pairs (red shading), observable as correlated parities (blue: positive, grey: negative). The +brane parity serves as a proxy for a non-local order parameter and is given by the product of the on-site parities evaluated +over an analysis area (black frame). In the MI phase (g,h) its value is positive (for finite areas) and depends on the number of +doublon-hole pairs cut by the analysis boundary. In the SF phase (i) parities are nearly uncorrelated, leading to a substantially +smaller brane parity. +mutually non-interfering retro-reflected 1d lattice (L1), +see Fig. 1(a). +The two lattices are not only intrinsi- +cally phase-stable, but also relative to each other as they +are both phase-referenced to a common retro-reflection +mirror[27]. The relative phase between the lattice poten- +tial minima, which we refer to as “superlattice phase” +∆ϕ, can be adjusted by introducing a slight detuning +between the lattice frequencies (taking into account the +distance between atoms and retro-reflecting mirror). By +additionally varying the power ratio between the lattice +beams V1/V2, the ground band behavior can be tuned +between square, triangular, honeycomb and 1d lattice, +without the need for active phase locking[27]. On top of +these base lattices, we employ a digital micromirror de- +vice (DMD) to project single-site-resolved beams through +the microscope objective, see Fig. 1(f). This procedure +results in a programmable repulsive potential landscape, +blocking atomic occupation on specific lattice sites, which +allows for the realization of an even larger class of derived +lattice potentials, see Fig. 1(d,e). At the same time, with +light only applied to blocked out sites, this scheme min- +imizes cross-talk, reducing undesired local disorder. The +phase stability between these microscopic blocking beams +and the base lattice is ensured by active feed-forward to +correct for slow thermal drifts [28]. +III. +LATTICE CHARACTERIZATION +In our experiment, we worked with about 200 87Rb +atoms in the |F = 1, mF = −1⟩ ground state, trapped in +lattices at a wavelength of λ = 1064 nm and with DMD +block-out beams operating at 670 nm. For the data pre- +sented here, we optimized the superlattice phase for the +triangular lattice condition ∆ϕ = 0, and extracted a +phase stability of σ∆ϕ = 0.01(1)π using L1 amplitude +modulation spectroscopy[27]. Starting with a 2d super- +fluid trapped in a single layer of a vertical 1d lattice, we +adiabatically ramped up the horizontal lattices (L2, L1), +such that the atoms formed a unity-filled Mott insulator +with a typical filling of 0.97. After performing measure- +ments in the desired lattice configuration, we ramped off +L1 and performed single-site resolved fluorescence imag- +ing in L2. +Due to pair-wise losses during fluorescence +imaging, the resulting single-shot images reveal the local +atom number parity [16]. +To demonstrate the flexible control over the lattice ge- +ometry and benchmark the corresponding properties of +the ground band, we perform single-particle quantum +walks [29–31] in the respective 2d lattices. To achieve +this, we flip the hyperfine state of a single atom us- +ing our local microwave addressing technique based on +a DMD [28, 29]. +After pushing out all but the spin- +flipped atom, we quench the lattices to a depth where + +3 +2Jt/ħ = 0.0 +exp. +sim. +0.0 +0.5 +1.0 +2Jt/ħ = 0.0 +exp. +sim. +0.0 +0.5 +1.0 +2Jt/ħ = 1.8 +0.00 +0.05 +0.10 +2Jt/ħ = 0.9 +0.0 +0.1 +0.2 +2Jt/ħ = 3.3 +0.00 +0.02 +0.04 +2Jt/ħ = 1.7 +0.00 +0.05 +0.10 +2Jt/ħ = 0.0 +exp. +sim. +0.0 +0.5 +1.0 +2Jt/ħ = 0.0 +exp. +sim. +0.0 +0.5 +1.0 +2Jt/ħ = 1.8 +0.0 +0.1 +2Jt/ħ = 0.9 +0.0 +0.2 +2Jt/ħ = 3.3 +0.0 +0.2 +0.4 +2Jt/ħ = 1.7 +0.00 +0.05 +0.10 +(a) +(b) +(c) +(d) +FIG. 2. +Atomic densities due to quantum walks in various +lattice geometries. After preparing a single localized atom in +the center of the lattice (red site in insets), we measure the +ballistic dynamics of the wavefunction at various times (top +to bottom). The square (a) and triangular (b) lattices are +realized in the ground band of our superlattice. The Lieb (c) +and kagome (d) lattices are generated by locally projecting +repulsive light on certain sites (grey sites in insets). The in- +terference fringes visible in the experimental data (left) agree +well with simulations (right), indicating coherent evolution in +a homogeneous and stable lattice. Note that some color map +ranges have been adjusted to facilitate displaying the large +dynamic range. +the particle is allowed to tunnel. As the wavefunction +spreads coherently, we expect the evolving site-resolved +probability distribution to display interference patterns +characteristic for the specific lattice. +The density dynamics averaged over 250 experimen- +tal repetitions and its hopping symmetry axes is dis- +played in Fig. 2, showing excellent agreement with sim- +ulations. +In the square lattice at 10.0(3)E(752) +r +depth, +where E(a/nm) +r += h2/8ma2 denotes the recoil energy of +the respective lattice with spacing a, the two dimen- +sions decouple and we observe the characteristic ballisti- +cally expanding wavefront with a fitted hopping energy of +J = h × 31(1) Hz along the horizontal and Jv = 0.92(1)J +along the vertical direction, see Fig. 2(a). +This agrees +well with the hopping rates obtained from band struc- +ture calculations using the lattice depth independently +calibrated by amplitude modulation spectroscopy. The +small observed anisotropy is well reproduced in our simu- +lations when considering the difference in the lattice spac- +ings as L2 intersects slightly non-orthogonally at an an- +gle of 90.7(1)°. For the triangular lattice, the depths are +tuned to an isotropic coupling ratio, following the rela- +tion V1/E(532) +r +≈ 4+V2/E(752) +r +. The associated quantum +walk measurements shown in Fig. 2(b) were performed +at V2 = 4.0(1)E(752) +r +with J = h × 21(1) Hz and exhibit +circularly symmetric expansion with a fitted residual di- +agonal anisotropy of Jd = 1.05(2)J. In general, the tun- +ability of the ratio V1/V2 enables us to deliberately vary +the diagonal anisotropy, interpolating between a square +and a 1d lattice along the diagonal[27]. +To characterize the emergent programmable lattices +in presence of microscopic site-blocking potentials of +Vb = h × 300(90) Hz, we measure quantum walks at the +same base-lattice parameters as above. For the block-out +potential presented in Fig. 2(c,d), the expected lattice ge- +ometries are the Lieb or kagome lattices for the square or +triangular base lattices, respectively. We again find ex- +cellent agreement with simulations, and observe that the +atom population remains on the non-blocked sites with +99(1) % probability, while cross-talk-induced disorder is +small. +IV. +DOUBLON-HOLE FLUCTUATIONS +After characterizing the single-particle tight-binding +bands and the stability of the generated lattices through +the quantum walks, we proceed to studying the interact- +ing regime in the unity-filling Bose-Hubbard model real- +ized on the various lattice geometries. While the ground +state in the atomic limit (J/U ≪ 1) corresponds to a +unity-filled product state, quantum fluctuations in the +form of doublon-hole pairs emerge on top of the product +state at finite tunnel couplings [7, 9]. In a perturbative +picture, regardless of the exact lattice geometry, every +bond in an isotropic lattice is expected to give rise to +equal nearest-neighbor ⟨i, j⟩ parity correlations of C = +⟨ˆs⟩−⟨ˆs⟩⟨ˆs⟩ ≈ 16J2/U 2, where ˆsj = eiπ(ˆnj−1) denotes the +local atom number parity. In the experiment, we started +with a 2d SF and then slowly ramped on the local block- +out potential in 150 ms to Vb = h × 450(120) Hz. Sub- +sequently, the horizontal lattices were adiabatically and +isotropically increased to the depth corresponding to the +desired J/U parameters in 200 ms, followed by a fast 1 ms +ramp to V2 = 90E(752) +r +, which froze all quantum fluctua- +tions. The interaction energies in this measurement were +in the range of U = h × 200 − 300 Hz. +In Fig. 3(a-c) +we compare the correlations from 200 experimental runs +evaluated over 9×9 sites along the straight and the diag- +onal neighbors for the square, triangular and Lieb lattice. +We clearly observe that diagonal correlations only arise +in the case of the triangular lattice. Furthermore, the +growth in correlations agrees with the perturbative de- + +4 +0.02 +0.04 +0.06 +Hopping energy J/U +0.00 +0.02 +0.04 +Parity correlations C +0.02 +0.04 +0.06 +Hopping energy J/U +0.00 +0.01 +0.02 +0.03 +Parity correlations C +0.05 +0.10 +Hopping energy J/U +0.00 +0.02 +0.04 +0.06 +Parity correlations C +0.05 +0.10 +Hopping energy J/U +0.2 +0.4 +0.6 +0.8 +Parity variance s2 +0.000 +0.025 +0.000 +0.025 +(a) +(b) +(c) +(d) +FIG. 3. +Doublon-hole fluctuations in the square (a) and +the triangular lattice (b). The straight-neighbor parity cor- +relations (blue) grow with (J/U)2 in both cases (line). The +correlations of the diagonal neighbor (orange) however only +grow in the case of the triangular lattice. +The color plots +(top) show the 2d parity correlations C of the neighboring +sites. +The colored edges in the left-most plot indicate the +value shown in the main plot. +(c) The fluctuations in the +Lieb lattice are averaged over both hub and rim sites and +show a behaviour similar to the square lattice case in the per- +turbative regime J ≪ U. (Insets) in (a-c) depict the lattice +geometry. (d) Fluctuations are driven by coupling to neigh- +boring sites, and thus the local coordination number. +The +coordination number of the Lieb lattice depends on the site +within the unit cell. Accordingly, the on-site variance on the +hub sites (green) grows twice as fast as the rim sites (grey), +see inset. Solid lines show perturbative calculations, with an +offset that accounts for the finite filling of 0.97. Error bars +denote the s.d. from a bootstrap analysis. +pendence within its range of validity for all lattice geome- +tries along their respective bond directions, with devia- +tions originating from hopping anisotropies, calibration +uncertainties and finite temperatures. When approach- +ing the phase transition, the pairs rapidly deconfine, re- +sulting in the observed reduction of neighboring correla- +tions [7]. +In the case of the tripartite Lieb lattice, there exist +two types of sublattices with differing local coordination +number z: the hub sites with z = 4 and the rim sites +with z = 2. This geometry gives rise to a flat central +band, whose Bloch wavefunctions exclusively populate +the rim sites [32], which suggests that the influence of +the flat band might manifest as spatially distinct behav- +ior on the two sublattice types. In particular, in the SF +phase the superfluid density is expected to be higher on +the hub sites [33, 34], and may be viewed as a tendency +to depopulate the flat band. To capture the effects of +this spatial inhomogeneity, we analyze the on-site vari- +ance s2 = ⟨ˆs⟩ − ⟨ˆs⟩2 averaged over either sublattice type, +0.05 +0.10 +Hopping energy ratio J/U +0.0 +0.2 +0.4 +0.6 +Brane parity OP +square +triangular +Lieb +0.0 +0.2 +0.4 +Rescaled ratio zJ/U +0.0 +0.5 +OP +2 +4 +6 +Analysis size L +0.1 +1.0 +0.3 +OP +(a) +(b) +(c) +FIG. 4. +(a) Brane parity across the SF-MI phase transi- +tion for various lattice geometries analyzed over 4 × 4 sites. +The measurements in the triangular (red), square (purple) +and Lieb (green) lattice all show a change from zero to fi- +nite values of the integer brane parity. The critical (J/U)c +agrees with the phase transition point obtained from quantum +Monte-Carlo simulations (solid lines), indicating its suitability +as non-local order parameter for the-Mott insulating phase. +(b) Rescaling the hopping energy with the respective (aver- +aged) coordination number z of the lattice, we find a collapse +of the data, showing that the phase transition scales with z. +(c) Dependence of the integer brane parity with the analysis +area containing L × L sites in the MI phase at J/U = 0.029 +(triangular, red), 0.029 (square, purple), and 0.033 (Lieb, +green). Solid lines denote an exponential fit consistent with +perimeter-law scaling. +The Lieb lattice only contains even +data points due to its 2 × 2-site unit cell. Error bars denote +the s.d. from a bootstrap analysis. +see Fig. 3(d). We can indeed observe that the variance +differs between the two types of sites when approaching +the phase transition, with the hub sites displaying higher +fluctuations. In the MI phase, the on-site fluctuations +correspond to the formation of doublon-hole pairs with +the site’s z neighbors and grow with J/U as described +by perturbation theory. In the SF phase, we would simi- +larly expect the sublattices to show distinct atom number +fluctuations due to the inhomogeneous superfluid density. +However, as the parity is bounded, the parity variance is +also bounded and at large J/U the difference in the par- +ity variance decreases again. +V. +BRANE PARITY +The correlation properties of the doublon-hole pairs +can furthermore be used to construct a non-local order +parameter characterizing the Mott-insulating phase: The +value of the product of all parities within a region of +interest, ˆOP = � +i∈L×L ˆsi, is determined by the num- +ber of correlated doublon-hole pairs cut by its bound- +ary. +Within the MI phase and for a finite brane, see +Fig. 1(g,h), this amounts to a positive value of OP = ⟨ ˆO⟩, +which decreases when approaching the phase transition +due to the increased number of cut pairs and thus a +higher probability for a negative OP . +When reaching +the SF phase, see Fig. 1(i), the proliferation of uncor- + +5 +related density fluctuations renders OP vanishing. +In +the 1d Bose-Hubbard model, the string parity has been +demonstrated to serve as a non-local order parameter, +being finite in the MI phase and vanishing in the SF +phase [7, 8, 10]. In 2d the integer brane parity would +vanish also in the MI phase in the thermodynamic limit, +and it was shown that using fractional instead of integer +parities retains a non-vanishing order parameter [11, 12]. +However, as OP decreases with a perimeter law [8] and +it remains finite for any finite analysis area, the inte- +ger brane parity is still useful as a proxy for the true +MI order parameter, and can capture the critical (J/U)c +within experimental uncertainties. In Fig. 4(a), we plot +the integer brane parity evaluated over a 4 × 4 area as +a function of J/U for the triangular, square and Lieb +lattice. For data in the MI phase, varying the analysis +area furthermore shows scaling consistent with a perime- +ter law, log OP ∼ −L, see Fig. 4(c). In all geometries, +the location of the phase transition is clearly represented +as a departure of the brane parity from zero. The experi- +mentally obtained critical values (J/U)c ≈ 0.04 and 0.06 +for the triangular and square lattice, respectively, agree +well with quantum Monte-Carlo simulations [35, 36]. We +furthermore find a collapse of OP for the different lattices +when rescaling the hopping energy with the coordination +number of the lattice, see Fig. 4(b). This scaling is con- +sistent with predictions by mean field theory and mea- +surements of the superfluid order parameter [13], thus +providing further validation for the use of the brane par- +ity as a proxy for the non-local order parameter. This +scaling appears to also approximately hold true for the +Lieb lattice at (J/U)c ≈ 0.09 when using the arithmetic +mean of its local constituents as an effective coordination +number. The systematic deviations for the Lieb lattice +towards higher parity values near the phase transition +could hint at a stabilizing effect of the flat band on the +MI phase, which is not captured by the applied simple +rescaling with coordination number—a point that needs +further investigation by theory and experiment. +VI. +CONCLUSION +Employing a quantum gas microscope, we have demon- +strated a novel passively phase-stable approach to real- +izing 2d Hubbard systems in programmable lattice ge- +ometries. +In various of these lattices, our microscopic +measurements have experimentally established the inte- +ger brane parity as a non-local observable suitable to +characterize the 2d SF-MI phase transition. +Imaging +within an out-of-phase superlattice would furthermore al- +low for resolving doublons [17, 37] and could enable the +detection of the fractional parity order for both bosonic +and fermionic systems [11, 12]. Finally, our microscopic +programmability of on-site potentials enables the explo- +ration of further lattice-dependent many-body phenom- +ena, ranging from the engineering of novel Hamiltoni- +ans on top of flat bands hosting exotic phases [32, 38] +to studying transport through interfaces between regions +with differing lattice geometry. +Note added: During the completion of this manuscript, +we became aware of related work studying fermionic +many-body systems in actively phase-stabilized tunable +lattice geometries [25]. +ACKNOWLEDGMENTS +We thank Jae-yoon Choi and Dan Stamper-Kurn +for valuable discussions. +We acknowledge funding by +the Max Planck Society (MPG), the European Union +(PASQuanS grant no. +817482), and the Deutsche +Forschungsgemeinschaft (DFG, German Research Foun- +dation) under Germany’s Excellence Strategy–EXC- +2111–390814868. +K.S. and S.A. acknowledge funding +from the International Max Planck Research School (IM- +PRS) for Quantum Science and Technology. +[1] M. den Nijs and K. 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Bissbort, Dynamical Effects and Disorder in Ultracold +Bosonic Matter, Ph.D. thesis, Johann Wolfgang Goethe- +Universit¨at, Frankfurt (2012). + +Supplemental Material for: +Observation of brane parity order in programmable optical lattices +David Wei,1, 2 Daniel Adler,1, 2 Kritsana Srakaew,1, 2 Suchita Agrawal,1, 2 +Pascal Weckesser,1, 2 Immanuel Bloch,1, 2, 3 and Johannes Zeiher1, 2 +1Max-Planck-Institut f¨ur Quantenoptik, 85748 Garching, Germany +2Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany +3Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at, 80799 Munich, Germany +(Dated: January 30, 2023) +I. +LATTICE PROPERTIES +A. +Lattice phase stability +In the tunable base lattices implemented in [17, 25], +where two independent but mutually interfering retro- +reflected laser beams are crossed, active phase stabiliza- +tion is required due to the “time phase” difference be- +tween the two beams, α, being an unrestricted degree of +freedom. For beams with wave number k = 2π/λ with +a combined field given by A ∼ eiky + e−iky + eiαeikx + +eiαe−ikx, the intensity becomes |A|2 ∝ cos 2x + cos 2y + +4 cos α cos x cos y and thus realizes an interference struc- +ture that depends on α. As bow-tie lattices (as used in +our setup) fold the same beam into the orthogonal axis, +the time phase difference is inherently fixed to α = 0. In +the following we furthermore show that this lattice is also +structurally phase stable with respect to variations in the +“spatial phases” when superposing with an additional 1d +lattice. +The layout of our lattice beams is shown in Fig. S1 with +the two axes kx,y = k(cos θ, ∓ sin θ). The square lattice +generated by lattice 2 and the 1d lattice generated by +a1 +a2 +b1 +b2 +Lattice 2 +c1 +c2 +Lattice 1 +µ +u +v +Á2 +Á1 +(a) +(b) +(c) +(d) +(e) +FIG. S1. +(a) Lattice beam layout denoting intersection +half-angle θ, field amplitudes of the incident beam passes +{ai, bi, ci}, and phase delays {φi} introduced by propagation. +The superlattice phase ∆ϕ is depicted in Fig. S2(a). At an +intersection angle of 2θ = 90°, various lattice geometries can +be realized, including: in the absence of lattice 1 a square +lattice (b), in its presence a honeycomb lattice for ∆ϕ = π +(c), a triangular lattice for ∆ϕ = 0 (d), and a 1d lattice in +the limit of deep lattice 1 (e). +lattice 1 have respective field amplitudes of +A2 = a1ei(ky·r) + b1ei(kx·r+φ2) ++ b2ei(−kx·r+φ2+2φ1) + a2ei(−ky·r+2φ2+2φ1) +and +A1 = c1ei(kx·r) + c2ei(−kx·r+2φ1+∆ϕ), +where ∆ϕ indicates the superlattice phase. This yields +an overall light intensity of +I = |A1|2 + |A2|2 += (a2 +1 + a2 +2 + b2 +1 + b2 +2 + c2 +1 + c2 +2) ++ 2c1c2 cos[2k(u − u0) cos θ − 2k(v − v0) sin θ − ∆ϕ] ++ 2a1a2 cos[2k(u − u0) cos θ + 2k(v − v0) sin θ] ++ 2b1b2 cos[2k(u − u0) cos θ − 2k(v − v0) sin θ] ++ 2(a1b1 + a2b2) cos[2k(v − v0) sin θ] ++ 2(a1b2 + a2b1) cos[2k(u − u0) cos θ] +(S1) +where we have defined 2ku0 cos θ += +φ2 + 2φ1 and +2kv0 sin θ = φ2. Thus, the lattice potential only depends +on a translated position (u − u0, v − v0), confirming that +the lattice is structurally passively phase stable. +B. +Bose-Hubbard parameters +For the Bose-Hubbard model +ˆH = − +� +⟨i,j⟩ +Jijˆc† +i ˆcj + U +2 +� +i +ˆni(ˆni − 1) + +� +i +Viˆni, (S2) +we calculate the hopping energy Jij and interaction en- +ergy U from band structure calculations of our opti- +cal lattice potential V (u, v) ∝ −I(u, v), with I given +in Eq. (S1). The calibration of the on-site potential Vi +is described in Sec. III. Since the square lattice potential +is not separable, we perform a full 2d band structure cal- +culation following [39]. On the one hand, this calculation +yields the band gaps used for the lattice depth calibration +(see Sec. II). On the other hand, we obtain the ground +state Wannier wavefunctions wj(u, v) on lattice site j, +which we use to determine the hopping energy between +sites i and j by evaluating +Jij = +� +du dv w∗ +i (u, v) +� +− ℏ2 +2m∇2 + V (u, v) +� +wj(u, v) + +8 +−0.5 +0.0 +0.5 +Position x/¸ +Lat. potential +¢x +−0.05 +0.00 +0.05 +Superlattice phase ¢'/π +0.1 +0.2 +Excitation prob. +(a) +(b) +FIG. S2. +(a) The superlattice phase, ∆ϕ = 2π∆x/(λ/2), +can be precisely calibrated by amplitude-modulating lattice 1 +(orange solid) near the band gap frequency of a much deeper +lattice 2 (blue solid). Single-band excitations require dipo- +lar modulation (black dashed), which is minimal when the +lattices are in phase (∆ϕ = 0). +The vertical dashed lines +represent the potential minima (and thus phase) of the re- +spective lattices. (b) Single-band amplitude modulation spec- +troscopy probing the resonant band excitation probability at +the respective superlattice phase. +The solid line shows a +fit from which we extract a superlattice phase stability of +σ∆ϕ = 0.01(1)π, which was confirmed in a long-time mea- +surement. +and the Hubbard interaction energy +U = 4πℏ2as +m +� +du dv dz |w(u, v)wz(z)|4 +where as is the s-wave scattering length. The Wannier +function for the vertical direction wz(z) is independently +obtained from a 1d band structure calculation due to the +separability of the lattice potential along this direction. +For the lattice geometries with site block-out, we con- +sider the influence of the block-out potentials within the +tight-binding model since the band gaps of ≳ h × 3 kHz +are much larger than the block-out potentials of ≲ h × +450 Hz. +II. +SINGLE-PARTICLE MEASUREMENTS +A. +Modulation spectroscopy +We calibrate the individual lattice depths by perform- +ing amplitude modulation spectroscopy and find two d- +band resonances from which we determine the lattice +depth with ∼ 2 % uncertainty. +For the superlattice phase measurements shown in +Fig. S2, we amplitude-modulate lattice 1 within a deep +lattice 2 potential near the upper p-band resonance. Due +to the weak single-particle drive in an isolated system, +we analyze the response assuming a two-level model +with coupling Ω and modulation-frequency detuning ∆. +This model yields a mean excited-state population of +Pe(Ω, ∆) = 2/(4 + δ2 + +� +δ2(4 + δ2)), with δ = ∆/Ω. +Close to the superlattice in-phase condition, ∆ϕ = 0, the +coupling is proportional to the superlattice phase (here: +Ω/∆ϕ ≈ 660 Hz h/π), which we calculate from the band +structure results. Considering the long and weak drive, +we assume that Gaussian fluctuations in the superlattice +phase (∝ σΩ) and in the lattice depth (∝ σ∆) dominate +the shape of the resonance. Thus, the excitation proba- +bility on resonance is given by the two-fold convolution +over the fluctuations, yielding +P e(Ω) ∼ 1 − +� +fN (ω,σ2)(x)ex2erfc|x|dx, +(S3) +where erfc denotes the complimentary error function, +and fN (ω,σ2) the probability distribution of a normal +distribution with center ω2 = 2Ω2/σ2 +Ω and variance +σ2 = σ2 +Ω/2σ2 +∆. +In the experiment, we tuned the superlattice phase by +varying the frequency difference ∆f between the lattices +by an acousto-optic modulator, which yields a tuning +slope of ∆ϕ/∆f ≈ π/250 MHz for the distance between +atoms and retro-reflecting mirror of ∼ 300 mm. At a lat- +tice 2 depth of V2 = 185(5)E(752) +r +, where all dynamics +in the lattice is frozen and where we can separate the +lattice in its local potential wells, we modulated lattice +1 at V1 = 5.0(2)E(532) +r +with a modulation depth of 0.25. +As a spectroscopic signature, we measured the fraction of +atoms remaining in the ground band after modulation by +adiabatically lowering the lattice depth to V2 ≈ 18E(752) +r +, +leading to the loss of atoms populating higher bands, see +Fig. S2(b). +By fitting the functional shape of Eq. (S3) +to our experimental data and converting from coupling +strength to the superlattice phase, this model allows us +to extract a standard deviation of the superlattice phase +of σ∆ϕ = 0.01(1)π. Repeating this measurement weeks +later gave a similar excitation probability, demonstrating +the long-term stability of this lattice scheme. +B. +Quantum walks +The quantum walk measurements shown in Fig. 2 were +performed by preparing a single atom and lowering the +lattice 2 depth from 25E(752) +r +(for square geometries) and +15E(752) +r +(for triangular geometries) to the depth used for +the dynamics measurements in 2.5 ms to avoid band exci- +tations. After the time evolution, we froze the dynamics +by ramping up lattice 2 to 25E(752) +r +in 0.8 ms. We posts- +elected the data for a single detected atom and fitted the +resulting time-dependent densities to numerical simula- +tions of the respective tight-binding lattice. For the base +lattice geometries, we fitted the hopping energy along +each bond direction, as well as a time offset t0 to account +for the finite ramp times, yielding 2Jt0 = 0.60(1) and +0.32(5) for the square and the triangular geometries, re- +spectively. The parameters of the base lattice fits were +used for the Lieb and kagome lattice simulations and +agreed with a direct fit to the data. +By varying the depth ratio V1/V2 between the lat- +tices, we can furthermore tune the hopping energy ra- +tio between the straight bonds and the diagonal bonds, +i.e., the geometry between a square lattice for V1 ≪ V2 + +9 +2Jt/ħ = 1.0 +exp. +sim. +0.00 +0.05 +0.10 +2Jt/ħ = 2.0 +exp. +sim. +0.00 +0.05 +0.10 +(a) +(b) +FIG. S3. +Densities due to quantum walks in anisotropic +triangular lattices with fitted hopping energy ratios between +diagonal and horizontal neighbors of Jd/J = 1.69(3) (a) and +0.79(2) (b). (Insets) show the site connectivity with stronger +couplings highlighted in orange. +and a 1d lattice for V1 ≫ V2. +In measurements sim- +ilar to the ones shown in Fig. 2 at lattice depths of +V2 = 3.9E(752) +r +and V1 ∈ {5.9, 9.4}E(532) +r +, we performed +quantum walks subject to intermediate anisotropic hop- +ping ratios, see Fig. S3. Fitting to numerical simulations +yields Jd/J = 1.69(3) and 0.79(2), which similarly shows +good agreement between simulations and calibrations, +demonstrating control of the anisotropy. +III. +MANY-BODY MEASUREMENTS +A. +On-site potential calibration +Our vertical lattice creates a spatially inhomogeneous +in-plane confinement potential. To estimate its poten- +tial depth, we increase the atom number loaded into the +system until a doubly-filled Mott insulator forms in the +center. The outline of the atomic cloud then gives us the +equipotential line at a potential depth of the Hubbard +interaction energy U. Due to fluctuations in the atom +number, the major source of uncertainty for this calibra- +tion method stems from determining the outline of the +cloud. +Using this information, we calibrated the projected +DMD potential for blocking out the lattice sites. +We +adiabatically ramped the square lattice into the atomic +limit with the projected potential switched on. +While +keeping the atom number such that the outline of the +atomic cloud remained near the U-equipotential line, we +varied the projected light power. When reaching a pro- +jected potential of U, we expect the population on the +central blocked-out sites to vanish. +We therefore cali- +brated the DMD potential by mapping the light power +where the average filling of the central blocked-out sites +reached ≲ 0.03 to a potential shift of ∼ U. +B. +Lieb sublattice inhomogeneity +In order to validate the observation that the parity +variance differs between the hub and rim sublattice sites +of the Lieb lattice as shown in Fig. 3(d), we plot the vari- +ance difference in Fig. S4. Comparing to the same analy- +sis performed in the square and triangular lattice, we can +0.00 +0.05 +0.10 +0.15 +0.20 +Hopping energy J/U +−0.05 +0.00 +0.05 +0.10 +0.15 +Parity variance s2 +hub − s2 +rim +FIG. S4. +Parity variance on the hub sites s2 +hub subtracted +by the variance on the rim sites s2 +rim for the triangular (red), +square (purple) and Lieb (green) lattices. The local variance +differs significantly only for the Lieb lattice. The solid line +indicates perturbative on-site fluctuations from doublon-hole +pairs in the MI phase. The dashed line indicates inhomoge- +neous mean-field calculations at µ/U = 0.5 in the SF phase. +see that only the Lieb lattice shows a significant deviation +from zero. +We would expect the sublattice-dependent occupation +fluctuations to grow further into the SF phase, however, +the data show a peak already around the phase transition +point. +We attribute this observation to the fact that, +in contrast to the atom number, the parity is bounded, +which limits the observable fluctuations. This behavior +is also qualitatively reproduced by inhomogeneous mean +field calculations [33], which similarly show a reduction +in the parity variance difference with increasing J/U. +C. +Finite-size scaling of the integer brane parity +To maximize the signal-to-noise ratio of the integer +brane parity extracted from experimental data, we first +crop the images to a 7 × 7-site area in the center of the +atomic cloud. We then evaluate the brane parity for all +possible L×L-site analysis areas within the original 7×7 +sites and average over the results. Note that in the case +of the Lieb lattice, we flip the sign of OP for analysis +areas with an odd number of total lattice sites. In this +section, we will discuss how the choice of L influences +the value of OP as well as the extracted phase transition +point. +In the MI phase, we show that the integer brane parity +OP is subject to a perimeter-law scaling, log OP ∼ −L, +see Fig. 4(c). Evaluated at different parameter regimes of +J/U, with increasing L we additionally observe a slight +trend towards lower values than expected for a perime- +ter law. This behaviour can be partially attributed to +finite-temperature effects, which lead to the formation of +uncorrelated individual holes. Uncorrelated holes would +lead to an area-law scaling, log OP ∼ −L2, and thus a +downward trend that becomes more dominant with in- +creasingly large analysis sizes (due to the perimeter-area +scaling) and with decreasing J/U (due to the reduced +probability of finding correlated pairs). Another reason + +10 +0.025 +0.050 +0.075 +Hopping energy J/U +0.0 +0.5 +1.0 +OP +0.05 +0.10 +Hopping energy J/U +0.0 +0.5 +1.0 +OP +0.05 +0.10 +0.15 +Hopping energy J/U +0.0 +0.5 +1.0 +OP +2 +3 +4 +5 +Analysis size L +0.00 +0.05 +0.10 +(J/U)0 +(a) +(b) +(c) +(d) +FIG. S5. +(a-c) Brane parity in the triangular (red), square +(purple) and Lieb (green) lattice for increasing analysis sizes +(light to dark color), ranging from L = 2−5 for the triangular +and square lattice, and L = 2, 4 for the Lieb lattice. The lines +indicate linear fits to the sloped J/U-parameter regime. We +extract a simplified estimate for the critical point, (J/U)0, as +the point where the fit vanishes. (d) With increasing anal- +ysis size L, the extracted phase transition point converges. +The horizontal lines indicate the critical value predicted by +quantum Monte Carlo simulations. +involves the inhomogeneous confining potential from the +lattice beams, leading to a coexistence of different phases +in the system depending on the local chemical poten- +tial [16]. As a consequence, we expect a bias towards a +superfluid when including regions of smaller local chem- +ical potential towards the edges of the atomic cloud. As +analyzing with larger L has a higher sampling frequency +at the edges than with smaller L, the inhomogeneity ef- +fects are stronger for larger L. +In the SF phase one would in contrast expect log OP ∼ +−L log L scaling [8]. We do not directly observe such scal- +ing since the absolute OP values are much smaller and +lie within experimental noise already at L ∼ 4. However, +due to the difference in scaling compared with the MI +phase, we expect the integer brane parity to serve as a +more accurate proxy for the order parameter when mea- +sured on larger analysis areas L × L. In Fig. S5 we show +the J/U-dependence of OP for different L and extract a +simplified estimate for the phase transition point (J/U)0: +We linearly fit the sloped part of the data (disregarding +nonlinear behavior predicted in the immediate vicinity of +the phase transition [7, 8]) and assign the value at which +the fit vanishes as (J/U)0, for which we indeed observe +convergent behavior for increasing L. + diff --git a/b9FKT4oBgHgl3EQfpS4j/content/tmp_files/load_file.txt b/b9FKT4oBgHgl3EQfpS4j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..33661224e4f9723f128a3dda27353c6ebfaa308b --- /dev/null +++ b/b9FKT4oBgHgl3EQfpS4j/content/tmp_files/load_file.txt @@ -0,0 +1,853 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf,len=852 +page_content='Observation of brane parity order in programmable optical lattices David Wei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2 Daniel Adler,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2 Kritsana Srakaew,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2 Suchita Agrawal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2 Pascal Weckesser,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2 Immanuel Bloch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 3 and Johannes Zeiher1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2 1Max-Planck-Institut f¨ur Quantenoptik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Germany 2Munich Center for Quantum Science and Technology (MCQST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 80799 Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Germany 3Fakult¨at f¨ur Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Ludwig-Maximilians-Universit¨at,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 80799 Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Germany (Dated: January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2023) The Mott-insulating phase of the two-dimensional (2d) Bose-Hubbard model is expected to be characterized by a non-local brane parity order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Parity order captures the presence of microscopic particle-hole fluctuations and entanglement, whose properties depend on the underlying lattice ge- ometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We realize 2d Bose-Hubbard models in dynamically tunable lattice geometries, using neutral atoms in a novel passively phase-stable tunable optical lattice in combination with programmable site-blocking potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We benchmark the performance of our system by single-particle quantum walks in the square, triangular, kagome and Lieb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In the strongly correlated regime, we mi- croscopically characterize the geometry dependence of the quantum fluctuations and experimentally validate the brane parity as a proxy for the non-local order parameter signaling the superfluid-to- Mott insulating phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' INTRODUCTION According to seminal work by Landau, second-order phase transitions are signaled by a change of a local or- der parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' However, some phase transitions defy this classification in terms of a local order parameter, and require, as a generalization, non-local order parameters to describe their underlying structure [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The Hal- dane insulator constitutes a celebrated example for such a phase, in which a string correlator captures the un- derlying non-local hidden order [3, 4], which has recently also been realized experimentally [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Interestingly, the Mott-insulating (MI) phase of the Bose-Hubbard (BH) model also features non-local order, which accounts for quantum fluctuations in the form of bound particle-hole pairs [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In one-dimensional (1d) BH chains, the MI order has been revealed by a parity order parameter of the on-site occupation [7, 8, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In two dimensions (2d) the brane parity was proposed as a generalization of par- ity order for square lattices [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' However, up to now, experiments directly measuring the brane parity in any 2d lattice geometry are lacking, as well as its experimen- tal validation as an order parameter for the MI phase in 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' A strategy to achieve the latter is provided by mean- field theory, which predicts that the location of the phase transition should scale with the coordination number and thus the underlying lattice geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' This scaling was explicitly probed by measuring the local order parame- ter in the SF phase [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Observing such scaling also in the brane parity provides an indication for the suitability of the brane parity as 2d non-local order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Neutral atoms in optical lattices provide a pristine testbed to realize low-dimensional Hubbard models [14] and offer techniques for the detection of local observ- ables using quantum gas microscopes [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Optical lattices arise through the interference pattern of laser beams, whose layout is carefully chosen for a specific tar- get geometry and has led to the realization of a variety of lattices [17–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' While optical lattices benefit from their inherent homogeneity and stability, the static na- ture of a given beam layout restricts systems to fixed spatial geometries and makes dynamical changes within a single experimental run challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In contrast, ar- rays of optical tweezers can be generated in almost freely programmable geometries [22] and have allowed for stud- ies of a variety of many-body spin models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' A num- ber of approaches have been brought forward to allow for similar programmability for itinerant atoms based on realizing small systems of tunnel-coupled optical tweez- ers [23, 24] or dynamically controllable lattices [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' How- ever, tweezer arrays in the itinerant regime are difficult to scale to large system sizes due to inhomogeneities and concomitantly a large calibration overhead, and the dy- namical control over lattices typically involves complex active phase stabilization techniques [17, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Here, we report on the realization of 2d Bose-Hubbard models in passively phase-stable optical lattices with square or triangular base geometry, which we combine with local site-blocking beams to realize programmable unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We demonstrate this novel degree of flexible control by implementing square, triangular, kagome and Lieb lattices in one experimental setup, and benchmark their quality through single-particle quantum walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In- creasing the atomic density, we microscopically probe the strongly interacting regime through non-local quantum fluctuations and show their dependence on the underly- ing lattice structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Our measurements provide a quan- titative characterization of the phase transition point in these lattices and experimentally establish the brane par- ity [8, 11, 12] as a meaningful non-local observable to characterize the SF-MI phase transition in 2d models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' PROGRAMMABLE LATTICES In our approach to realizing tunable lattice geome- tries, we superpose a bow-tie lattice [26] (L2) with a arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='11869v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='quant-gas] 27 Jan 2023 2 −1 0 1 −1 0 1 Position y/¸ −1 0 1 −1 0 Potential (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 Position x/¸ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 Á2 Á1 ¢x −5 0 5 Position x/¸ −5 0 5 y/¸ −5 0 5 x/¸ −5 0 5 y/¸ MI (square) MI (triangular) SF L2 L1 PBS Á1 Á2 x y (c) (a) (b) (f) (d) (g) (h) (i) (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (a) Experimental setup realizing passively phase-stable tunable lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The lattice 2 beam (L2, blue) is out-of-plane polarized and forms a bow-tie lattice, realizing a square lattice potential (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The in-plane polarized lattice 1 beam (L1, orange) can be added with a well-defined superlattice phase ∆ϕ = 2π∆x/(λ/2) (see lattice potentials sketched in right inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' For the in-phase case, ∆ϕ = 0, this realizes an effective triangular lattice geometry (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (Left inset) The combined lattice is passively phase-stable due to the retro-reflection mirror serving as a common phase reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Temporal fluctuations of path lengths lead to translations along either common paths or along translationally invariant directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The arrows indicate the movement of the interference pattern generated by the respective lattice upon changes in the phase φ1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (d,e) Single-site resolved image of a Mott insulator in the square (d) and Lieb lattice (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (f) Lattices with more complex unit cells, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' as shown in (e), can be dynamically generated by projecting repulsive local potentials through the objective, blocking out distinct lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (g-i) The MI phase hosts doublon-hole pairs (red shading), observable as correlated parities (blue: positive, grey: negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The brane parity serves as a proxy for a non-local order parameter and is given by the product of the on-site parities evaluated over an analysis area (black frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In the MI phase (g,h) its value is positive (for finite areas) and depends on the number of doublon-hole pairs cut by the analysis boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In the SF phase (i) parities are nearly uncorrelated, leading to a substantially smaller brane parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' mutually non-interfering retro-reflected 1d lattice (L1), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The two lattices are not only intrinsi- cally phase-stable, but also relative to each other as they are both phase-referenced to a common retro-reflection mirror[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The relative phase between the lattice poten- tial minima, which we refer to as “superlattice phase” ∆ϕ, can be adjusted by introducing a slight detuning between the lattice frequencies (taking into account the distance between atoms and retro-reflecting mirror).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' By additionally varying the power ratio between the lattice beams V1/V2, the ground band behavior can be tuned between square, triangular, honeycomb and 1d lattice, without the need for active phase locking[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' On top of these base lattices, we employ a digital micromirror de- vice (DMD) to project single-site-resolved beams through the microscope objective, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 1(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' This procedure results in a programmable repulsive potential landscape, blocking atomic occupation on specific lattice sites, which allows for the realization of an even larger class of derived lattice potentials, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 1(d,e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' At the same time, with light only applied to blocked out sites, this scheme min- imizes cross-talk, reducing undesired local disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The phase stability between these microscopic blocking beams and the base lattice is ensured by active feed-forward to correct for slow thermal drifts [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' LATTICE CHARACTERIZATION In our experiment, we worked with about 200 87Rb atoms in the |F = 1, mF = −1⟩ ground state, trapped in lattices at a wavelength of λ = 1064 nm and with DMD block-out beams operating at 670 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' For the data pre- sented here, we optimized the superlattice phase for the triangular lattice condition ∆ϕ = 0, and extracted a phase stability of σ∆ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='01(1)π using L1 amplitude modulation spectroscopy[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Starting with a 2d super- fluid trapped in a single layer of a vertical 1d lattice, we adiabatically ramped up the horizontal lattices (L2, L1), such that the atoms formed a unity-filled Mott insulator with a typical filling of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' After performing measure- ments in the desired lattice configuration, we ramped off L1 and performed single-site resolved fluorescence imag- ing in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Due to pair-wise losses during fluorescence imaging, the resulting single-shot images reveal the local atom number parity [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' To demonstrate the flexible control over the lattice ge- ometry and benchmark the corresponding properties of the ground band, we perform single-particle quantum walks [29–31] in the respective 2d lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' To achieve this, we flip the hyperfine state of a single atom us- ing our local microwave addressing technique based on a DMD [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' After pushing out all but the spin- flipped atom, we quench the lattices to a depth where 3 2Jt/ħ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 2Jt/ħ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 2Jt/ħ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='10 2Jt/ħ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='2 2Jt/ħ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='04 2Jt/ħ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='10 2Jt/ħ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 2Jt/ħ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 2Jt/ħ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1 2Jt/ħ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='2 2Jt/ħ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='4 2Jt/ħ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='10 (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Atomic densities due to quantum walks in various lattice geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' After preparing a single localized atom in the center of the lattice (red site in insets), we measure the ballistic dynamics of the wavefunction at various times (top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The square (a) and triangular (b) lattices are realized in the ground band of our superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The Lieb (c) and kagome (d) lattices are generated by locally projecting repulsive light on certain sites (grey sites in insets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The in- terference fringes visible in the experimental data (left) agree well with simulations (right), indicating coherent evolution in a homogeneous and stable lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Note that some color map ranges have been adjusted to facilitate displaying the large dynamic range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' the particle is allowed to tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' As the wavefunction spreads coherently, we expect the evolving site-resolved probability distribution to display interference patterns characteristic for the specific lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The density dynamics averaged over 250 experimen- tal repetitions and its hopping symmetry axes is dis- played in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2, showing excellent agreement with sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In the square lattice at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0(3)E(752) r depth, where E(a/nm) r = h2/8ma2 denotes the recoil energy of the respective lattice with spacing a, the two dimen- sions decouple and we observe the characteristic ballisti- cally expanding wavefront with a fitted hopping energy of J = h × 31(1) Hz along the horizontal and Jv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='92(1)J along the vertical direction, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' This agrees well with the hopping rates obtained from band struc- ture calculations using the lattice depth independently calibrated by amplitude modulation spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The small observed anisotropy is well reproduced in our simu- lations when considering the difference in the lattice spac- ings as L2 intersects slightly non-orthogonally at an an- gle of 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='7(1)°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' For the triangular lattice, the depths are tuned to an isotropic coupling ratio, following the rela- tion V1/E(532) r ≈ 4+V2/E(752) r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The associated quantum walk measurements shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2(b) were performed at V2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0(1)E(752) r with J = h × 21(1) Hz and exhibit circularly symmetric expansion with a fitted residual di- agonal anisotropy of Jd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05(2)J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In general, the tun- ability of the ratio V1/V2 enables us to deliberately vary the diagonal anisotropy, interpolating between a square and a 1d lattice along the diagonal[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' To characterize the emergent programmable lattices in presence of microscopic site-blocking potentials of Vb = h × 300(90) Hz, we measure quantum walks at the same base-lattice parameters as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' For the block-out potential presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2(c,d), the expected lattice ge- ometries are the Lieb or kagome lattices for the square or triangular base lattices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We again find ex- cellent agreement with simulations, and observe that the atom population remains on the non-blocked sites with 99(1) % probability, while cross-talk-induced disorder is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' DOUBLON-HOLE FLUCTUATIONS After characterizing the single-particle tight-binding bands and the stability of the generated lattices through the quantum walks, we proceed to studying the interact- ing regime in the unity-filling Bose-Hubbard model real- ized on the various lattice geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' While the ground state in the atomic limit (J/U ≪ 1) corresponds to a unity-filled product state, quantum fluctuations in the form of doublon-hole pairs emerge on top of the product state at finite tunnel couplings [7, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In a perturbative picture, regardless of the exact lattice geometry, every bond in an isotropic lattice is expected to give rise to equal nearest-neighbor ⟨i, j⟩ parity correlations of C = ⟨ˆs⟩−⟨ˆs⟩⟨ˆs⟩ ≈ 16J2/U 2, where ˆsj = eiπ(ˆnj−1) denotes the local atom number parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In the experiment, we started with a 2d SF and then slowly ramped on the local block- out potential in 150 ms to Vb = h × 450(120) Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Sub- sequently, the horizontal lattices were adiabatically and isotropically increased to the depth corresponding to the desired J/U parameters in 200 ms, followed by a fast 1 ms ramp to V2 = 90E(752) r , which froze all quantum fluctua- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The interaction energies in this measurement were in the range of U = h × 200 − 300 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 3(a-c) we compare the correlations from 200 experimental runs evaluated over 9×9 sites along the straight and the diag- onal neighbors for the square, triangular and Lieb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We clearly observe that diagonal correlations only arise in the case of the triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Furthermore, the growth in correlations agrees with the perturbative de- 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='06 Hopping energy J/U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='04 Parity correlations C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='06 Hopping energy J/U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='03 Parity correlations C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='10 Hopping energy J/U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='06 Parity correlations C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='10 Hopping energy J/U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='8 Parity variance s2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='025 (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Doublon-hole fluctuations in the square (a) and the triangular lattice (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The straight-neighbor parity cor- relations (blue) grow with (J/U)2 in both cases (line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The correlations of the diagonal neighbor (orange) however only grow in the case of the triangular lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The color plots (top) show the 2d parity correlations C of the neighboring sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The colored edges in the left-most plot indicate the value shown in the main plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (c) The fluctuations in the Lieb lattice are averaged over both hub and rim sites and show a behaviour similar to the square lattice case in the per- turbative regime J ≪ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (Insets) in (a-c) depict the lattice geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (d) Fluctuations are driven by coupling to neigh- boring sites, and thus the local coordination number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The coordination number of the Lieb lattice depends on the site within the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Accordingly, the on-site variance on the hub sites (green) grows twice as fast as the rim sites (grey), see inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Solid lines show perturbative calculations, with an offset that accounts for the finite filling of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Error bars denote the s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' from a bootstrap analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' pendence within its range of validity for all lattice geome- tries along their respective bond directions, with devia- tions originating from hopping anisotropies, calibration uncertainties and finite temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' When approach- ing the phase transition, the pairs rapidly deconfine, re- sulting in the observed reduction of neighboring correla- tions [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In the case of the tripartite Lieb lattice, there exist two types of sublattices with differing local coordination number z: the hub sites with z = 4 and the rim sites with z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' This geometry gives rise to a flat central band, whose Bloch wavefunctions exclusively populate the rim sites [32], which suggests that the influence of the flat band might manifest as spatially distinct behav- ior on the two sublattice types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In particular, in the SF phase the superfluid density is expected to be higher on the hub sites [33, 34], and may be viewed as a tendency to depopulate the flat band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' To capture the effects of this spatial inhomogeneity, we analyze the on-site vari- ance s2 = ⟨ˆs⟩ − ⟨ˆs⟩2 averaged over either sublattice type, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='10 Hopping energy ratio J/U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='6 Brane parity OP square triangular Lieb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='4 Rescaled ratio zJ/U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='5 OP 2 4 6 Analysis size L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='3 OP (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (a) Brane parity across the SF-MI phase transi- tion for various lattice geometries analyzed over 4 × 4 sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The measurements in the triangular (red), square (purple) and Lieb (green) lattice all show a change from zero to fi- nite values of the integer brane parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The critical (J/U)c agrees with the phase transition point obtained from quantum Monte-Carlo simulations (solid lines), indicating its suitability as non-local order parameter for the-Mott insulating phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (b) Rescaling the hopping energy with the respective (aver- aged) coordination number z of the lattice, we find a collapse of the data, showing that the phase transition scales with z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (c) Dependence of the integer brane parity with the analysis area containing L × L sites in the MI phase at J/U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='029 (triangular, red), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='029 (square, purple), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='033 (Lieb, green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Solid lines denote an exponential fit consistent with perimeter-law scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The Lieb lattice only contains even data points due to its 2 × 2-site unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Error bars denote the s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' from a bootstrap analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We can indeed observe that the variance differs between the two types of sites when approaching the phase transition, with the hub sites displaying higher fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In the MI phase, the on-site fluctuations correspond to the formation of doublon-hole pairs with the site’s z neighbors and grow with J/U as described by perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In the SF phase, we would simi- larly expect the sublattices to show distinct atom number fluctuations due to the inhomogeneous superfluid density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' However, as the parity is bounded, the parity variance is also bounded and at large J/U the difference in the par- ity variance decreases again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' BRANE PARITY The correlation properties of the doublon-hole pairs can furthermore be used to construct a non-local order parameter characterizing the Mott-insulating phase: The value of the product of all parities within a region of interest, ˆOP = � i∈L×L ˆsi, is determined by the num- ber of correlated doublon-hole pairs cut by its bound- ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Within the MI phase and for a finite brane, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 1(g,h), this amounts to a positive value of OP = ⟨ ˆO⟩, which decreases when approaching the phase transition due to the increased number of cut pairs and thus a higher probability for a negative OP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' When reaching the SF phase, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 1(i), the proliferation of uncor- 5 related density fluctuations renders OP vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In the 1d Bose-Hubbard model, the string parity has been demonstrated to serve as a non-local order parameter, being finite in the MI phase and vanishing in the SF phase [7, 8, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In 2d the integer brane parity would vanish also in the MI phase in the thermodynamic limit, and it was shown that using fractional instead of integer parities retains a non-vanishing order parameter [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' However, as OP decreases with a perimeter law [8] and it remains finite for any finite analysis area, the inte- ger brane parity is still useful as a proxy for the true MI order parameter, and can capture the critical (J/U)c within experimental uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 4(a), we plot the integer brane parity evaluated over a 4 × 4 area as a function of J/U for the triangular, square and Lieb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' For data in the MI phase, varying the analysis area furthermore shows scaling consistent with a perime- ter law, log OP ∼ −L, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In all geometries, the location of the phase transition is clearly represented as a departure of the brane parity from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The experi- mentally obtained critical values (J/U)c ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='04 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='06 for the triangular and square lattice, respectively, agree well with quantum Monte-Carlo simulations [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We furthermore find a collapse of OP for the different lattices when rescaling the hopping energy with the coordination number of the lattice, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' This scaling is con- sistent with predictions by mean field theory and mea- surements of the superfluid order parameter [13], thus providing further validation for the use of the brane par- ity as a proxy for the non-local order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' This scaling appears to also approximately hold true for the Lieb lattice at (J/U)c ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='09 when using the arithmetic mean of its local constituents as an effective coordination number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The systematic deviations for the Lieb lattice towards higher parity values near the phase transition could hint at a stabilizing effect of the flat band on the MI phase, which is not captured by the applied simple rescaling with coordination number—a point that needs further investigation by theory and experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' CONCLUSION Employing a quantum gas microscope, we have demon- strated a novel passively phase-stable approach to real- izing 2d Hubbard systems in programmable lattice ge- ometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In various of these lattices, our microscopic measurements have experimentally established the inte- ger brane parity as a non-local observable suitable to characterize the 2d SF-MI phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Imaging within an out-of-phase superlattice would furthermore al- low for resolving doublons [17, 37] and could enable the detection of the fractional parity order for both bosonic and fermionic systems [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Finally, our microscopic programmability of on-site potentials enables the explo- ration of further lattice-dependent many-body phenom- ena, ranging from the engineering of novel Hamiltoni- ans on top of flat bands hosting exotic phases [32, 38] to studying transport through interfaces between regions with differing lattice geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Note added: During the completion of this manuscript, we became aware of related work studying fermionic many-body systems in actively phase-stabilized tunable lattice geometries [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Jae-yoon Choi and Dan Stamper-Kurn for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We acknowledge funding by the Max Planck Society (MPG), the European Union (PASQuanS grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 817482), and the Deutsche Forschungsgemeinschaft (DFG, German Research Foun- dation) under Germany’s Excellence Strategy–EXC- 2111–390814868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' acknowledge funding from the International Max Planck Research School (IM- PRS) for Quantum Science and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' [1] M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Holthaus, Refer- ence data for phase diagrams of triangular and hexagonal bosonic lattices, EPL 91, 10004 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Greif, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Uehlinger, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Jotzu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Tarruell, and T.' metadata={'source': 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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 117, 163001 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' [39] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Bissbort, Dynamical Effects and Disorder in Ultracold Bosonic Matter, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' thesis, Johann Wolfgang Goethe- Universit¨at, Frankfurt (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Supplemental Material for: Observation of brane parity order in programmable optical lattices David Wei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2 Daniel Adler,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2 Kritsana Srakaew,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2 Suchita Agrawal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2 Pascal Weckesser,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2 Immanuel Bloch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 3 and Johannes Zeiher1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2 1Max-Planck-Institut f¨ur Quantenoptik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Germany 2Munich Center for Quantum Science and Technology (MCQST),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 80799 Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Germany 3Fakult¨at f¨ur Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Ludwig-Maximilians-Universit¨at,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 80799 Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Germany (Dated: January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2023) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' LATTICE PROPERTIES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Lattice phase stability In the tunable base lattices implemented in [17, 25], where two independent but mutually interfering retro- reflected laser beams are crossed, active phase stabiliza- tion is required due to the “time phase” difference be- tween the two beams, α, being an unrestricted degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' For beams with wave number k = 2π/λ with a combined field given by A ∼ eiky + e−iky + eiαeikx + eiαe−ikx, the intensity becomes |A|2 ∝ cos 2x + cos 2y + 4 cos α cos x cos y and thus realizes an interference struc- ture that depends on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' As bow-tie lattices (as used in our setup) fold the same beam into the orthogonal axis, the time phase difference is inherently fixed to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In the following we furthermore show that this lattice is also structurally phase stable with respect to variations in the “spatial phases” when superposing with an additional 1d lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The layout of our lattice beams is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' S1 with the two axes kx,y = k(cos θ, ∓ sin θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The square lattice generated by lattice 2 and the 1d lattice generated by a1 a2 b1 b2 Lattice 2 c1 c2 Lattice 1 µ u v Á2 Á1 (a) (b) (c) (d) (e) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (a) Lattice beam layout denoting intersection half-angle θ, field amplitudes of the incident beam passes {ai, bi, ci}, and phase delays {φi} introduced by propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The superlattice phase ∆ϕ is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' S2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' At an intersection angle of 2θ = 90°, various lattice geometries can be realized, including: in the absence of lattice 1 a square lattice (b), in its presence a honeycomb lattice for ∆ϕ = π (c), a triangular lattice for ∆ϕ = 0 (d), and a 1d lattice in the limit of deep lattice 1 (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' lattice 1 have respective field amplitudes of A2 = a1ei(ky·r) + b1ei(kx·r+φ2) + b2ei(−kx·r+φ2+2φ1) + a2ei(−ky·r+2φ2+2φ1) and A1 = c1ei(kx·r) + c2ei(−kx·r+2φ1+∆ϕ), where ∆ϕ indicates the superlattice phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' This yields an overall light intensity of I = |A1|2 + |A2|2 = (a2 1 + a2 2 + b2 1 + b2 2 + c2 1 + c2 2) + 2c1c2 cos[2k(u − u0) cos θ − 2k(v − v0) sin θ − ∆ϕ] + 2a1a2 cos[2k(u − u0) cos θ + 2k(v − v0) sin θ] + 2b1b2 cos[2k(u − u0) cos θ − 2k(v − v0) sin θ] + 2(a1b1 + a2b2) cos[2k(v − v0) sin θ] + 2(a1b2 + a2b1) cos[2k(u − u0) cos θ] (S1) where we have defined 2ku0 cos θ = φ2 + 2φ1 and 2kv0 sin θ = φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Thus, the lattice potential only depends on a translated position (u − u0, v − v0), confirming that the lattice is structurally passively phase stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Bose-Hubbard parameters For the Bose-Hubbard model ˆH = − � ⟨i,j⟩ Jijˆc† i ˆcj + U 2 � i ˆni(ˆni − 1) + � i Viˆni, (S2) we calculate the hopping energy Jij and interaction en- ergy U from band structure calculations of our opti- cal lattice potential V (u, v) ∝ −I(u, v), with I given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The calibration of the on-site potential Vi is described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Since the square lattice potential is not separable, we perform a full 2d band structure cal- culation following [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' On the one hand, this calculation yields the band gaps used for the lattice depth calibration (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' On the other hand, we obtain the ground state Wannier wavefunctions wj(u, v) on lattice site j, which we use to determine the hopping energy between sites i and j by evaluating Jij = � du dv w∗ i (u, v) � − ℏ2 2m∇2 + V (u, v) � wj(u, v) 8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='5 Position x/¸ Lat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' potential ¢x −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content="05 Superlattice phase ¢'/π 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='2 Excitation prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (a) The superlattice phase, ∆ϕ = 2π∆x/(λ/2), can be precisely calibrated by amplitude-modulating lattice 1 (orange solid) near the band gap frequency of a much deeper lattice 2 (blue solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Single-band excitations require dipo- lar modulation (black dashed), which is minimal when the lattices are in phase (∆ϕ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The vertical dashed lines represent the potential minima (and thus phase) of the re- spective lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (b) Single-band amplitude modulation spec- troscopy probing the resonant band excitation probability at the respective superlattice phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The solid line shows a fit from which we extract a superlattice phase stability of σ∆ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='01(1)π, which was confirmed in a long-time mea- surement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' and the Hubbard interaction energy U = 4πℏ2as m � du dv dz |w(u, v)wz(z)|4 where as is the s-wave scattering length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The Wannier function for the vertical direction wz(z) is independently obtained from a 1d band structure calculation due to the separability of the lattice potential along this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' For the lattice geometries with site block-out, we con- sider the influence of the block-out potentials within the tight-binding model since the band gaps of ≳ h × 3 kHz are much larger than the block-out potentials of ≲ h × 450 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' SINGLE-PARTICLE MEASUREMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Modulation spectroscopy We calibrate the individual lattice depths by perform- ing amplitude modulation spectroscopy and find two d- band resonances from which we determine the lattice depth with ∼ 2 % uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' For the superlattice phase measurements shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' S2, we amplitude-modulate lattice 1 within a deep lattice 2 potential near the upper p-band resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Due to the weak single-particle drive in an isolated system, we analyze the response assuming a two-level model with coupling Ω and modulation-frequency detuning ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' This model yields a mean excited-state population of Pe(Ω, ∆) = 2/(4 + δ2 + � δ2(4 + δ2)), with δ = ∆/Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Close to the superlattice in-phase condition, ∆ϕ = 0, the coupling is proportional to the superlattice phase (here: Ω/∆ϕ ≈ 660 Hz h/π), which we calculate from the band structure results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Considering the long and weak drive, we assume that Gaussian fluctuations in the superlattice phase (∝ σΩ) and in the lattice depth (∝ σ∆) dominate the shape of the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Thus, the excitation proba- bility on resonance is given by the two-fold convolution over the fluctuations, yielding P e(Ω) ∼ 1 − � fN (ω,σ2)(x)ex2erfc|x|dx, (S3) where erfc denotes the complimentary error function, and fN (ω,σ2) the probability distribution of a normal distribution with center ω2 = 2Ω2/σ2 Ω and variance σ2 = σ2 Ω/2σ2 ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In the experiment, we tuned the superlattice phase by varying the frequency difference ∆f between the lattices by an acousto-optic modulator, which yields a tuning slope of ∆ϕ/∆f ≈ π/250 MHz for the distance between atoms and retro-reflecting mirror of ∼ 300 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' At a lat- tice 2 depth of V2 = 185(5)E(752) r , where all dynamics in the lattice is frozen and where we can separate the lattice in its local potential wells, we modulated lattice 1 at V1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0(2)E(532) r with a modulation depth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' As a spectroscopic signature, we measured the fraction of atoms remaining in the ground band after modulation by adiabatically lowering the lattice depth to V2 ≈ 18E(752) r , leading to the loss of atoms populating higher bands, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' S2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' By fitting the functional shape of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (S3) to our experimental data and converting from coupling strength to the superlattice phase, this model allows us to extract a standard deviation of the superlattice phase of σ∆ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='01(1)π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Repeating this measurement weeks later gave a similar excitation probability, demonstrating the long-term stability of this lattice scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Quantum walks The quantum walk measurements shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2 were performed by preparing a single atom and lowering the lattice 2 depth from 25E(752) r (for square geometries) and 15E(752) r (for triangular geometries) to the depth used for the dynamics measurements in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='5 ms to avoid band exci- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' After the time evolution, we froze the dynamics by ramping up lattice 2 to 25E(752) r in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='8 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We posts- elected the data for a single detected atom and fitted the resulting time-dependent densities to numerical simula- tions of the respective tight-binding lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' For the base lattice geometries, we fitted the hopping energy along each bond direction, as well as a time offset t0 to account for the finite ramp times, yielding 2Jt0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='60(1) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='32(5) for the square and the triangular geometries, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The parameters of the base lattice fits were used for the Lieb and kagome lattice simulations and agreed with a direct fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' By varying the depth ratio V1/V2 between the lat- tices, we can furthermore tune the hopping energy ra- tio between the straight bonds and the diagonal bonds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=', the geometry between a square lattice for V1 ≪ V2 9 2Jt/ħ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='10 2Jt/ħ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='10 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Densities due to quantum walks in anisotropic triangular lattices with fitted hopping energy ratios between diagonal and horizontal neighbors of Jd/J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='69(3) (a) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='79(2) (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (Insets) show the site connectivity with stronger couplings highlighted in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' and a 1d lattice for V1 ≫ V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In measurements sim- ilar to the ones shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 2 at lattice depths of V2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='9E(752) r and V1 ∈ {5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='9, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='4}E(532) r , we performed quantum walks subject to intermediate anisotropic hop- ping ratios, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Fitting to numerical simulations yields Jd/J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='69(3) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='79(2), which similarly shows good agreement between simulations and calibrations, demonstrating control of the anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' MANY-BODY MEASUREMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' On-site potential calibration Our vertical lattice creates a spatially inhomogeneous in-plane confinement potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' To estimate its poten- tial depth, we increase the atom number loaded into the system until a doubly-filled Mott insulator forms in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The outline of the atomic cloud then gives us the equipotential line at a potential depth of the Hubbard interaction energy U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Due to fluctuations in the atom number, the major source of uncertainty for this calibra- tion method stems from determining the outline of the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Using this information, we calibrated the projected DMD potential for blocking out the lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We adiabatically ramped the square lattice into the atomic limit with the projected potential switched on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' While keeping the atom number such that the outline of the atomic cloud remained near the U-equipotential line, we varied the projected light power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' When reaching a pro- jected potential of U, we expect the population on the central blocked-out sites to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We therefore cali- brated the DMD potential by mapping the light power where the average filling of the central blocked-out sites reached ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='03 to a potential shift of ∼ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Lieb sublattice inhomogeneity In order to validate the observation that the parity variance differs between the hub and rim sublattice sites of the Lieb lattice as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 3(d), we plot the vari- ance difference in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Comparing to the same analy- sis performed in the square and triangular lattice, we can 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='20 Hopping energy J/U −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='15 Parity variance s2 hub − s2 rim FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Parity variance on the hub sites s2 hub subtracted by the variance on the rim sites s2 rim for the triangular (red), square (purple) and Lieb (green) lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The local variance differs significantly only for the Lieb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The solid line indicates perturbative on-site fluctuations from doublon-hole pairs in the MI phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The dashed line indicates inhomoge- neous mean-field calculations at µ/U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='5 in the SF phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' see that only the Lieb lattice shows a significant deviation from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We would expect the sublattice-dependent occupation fluctuations to grow further into the SF phase, however, the data show a peak already around the phase transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We attribute this observation to the fact that, in contrast to the atom number, the parity is bounded, which limits the observable fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' This behavior is also qualitatively reproduced by inhomogeneous mean field calculations [33], which similarly show a reduction in the parity variance difference with increasing J/U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Finite-size scaling of the integer brane parity To maximize the signal-to-noise ratio of the integer brane parity extracted from experimental data, we first crop the images to a 7 × 7-site area in the center of the atomic cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We then evaluate the brane parity for all possible L×L-site analysis areas within the original 7×7 sites and average over the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Note that in the case of the Lieb lattice, we flip the sign of OP for analysis areas with an odd number of total lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In this section, we will discuss how the choice of L influences the value of OP as well as the extracted phase transition point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In the MI phase, we show that the integer brane parity OP is subject to a perimeter-law scaling, log OP ∼ −L, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Evaluated at different parameter regimes of J/U, with increasing L we additionally observe a slight trend towards lower values than expected for a perime- ter law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' This behaviour can be partially attributed to finite-temperature effects, which lead to the formation of uncorrelated individual holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Uncorrelated holes would lead to an area-law scaling, log OP ∼ −L2, and thus a downward trend that becomes more dominant with in- creasingly large analysis sizes (due to the perimeter-area scaling) and with decreasing J/U (due to the reduced probability of finding correlated pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' Another reason 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='075 Hopping energy J/U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 OP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='10 Hopping energy J/U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 OP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='15 Hopping energy J/U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='0 OP 2 3 4 5 Analysis size L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content='10 (J/U)0 (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (a-c) Brane parity in the triangular (red), square (purple) and Lieb (green) lattice for increasing analysis sizes (light to dark color), ranging from L = 2−5 for the triangular and square lattice, and L = 2, 4 for the Lieb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The lines indicate linear fits to the sloped J/U-parameter regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We extract a simplified estimate for the critical point, (J/U)0, as the point where the fit vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' (d) With increasing anal- ysis size L, the extracted phase transition point converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' The horizontal lines indicate the critical value predicted by quantum Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' involves the inhomogeneous confining potential from the lattice beams, leading to a coexistence of different phases in the system depending on the local chemical poten- tial [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' As a consequence, we expect a bias towards a superfluid when including regions of smaller local chem- ical potential towards the edges of the atomic cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' As analyzing with larger L has a higher sampling frequency at the edges than with smaller L, the inhomogeneity ef- fects are stronger for larger L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In the SF phase one would in contrast expect log OP ∼ −L log L scaling [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' We do not directly observe such scal- ing since the absolute OP values are much smaller and lie within experimental noise already at L ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' However, due to the difference in scaling compared with the MI phase, we expect the integer brane parity to serve as a more accurate proxy for the order parameter when mea- sured on larger analysis areas L × L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} +page_content=' S5 we show the J/U-dependence of OP for different L and extract a simplified estimate for the phase transition point (J/U)0: We linearly fit the sloped part of the data (disregarding nonlinear behavior predicted in the immediate vicinity of the phase transition [7, 8]) and assign the value at which the fit vanishes as (J/U)0, for which we indeed observe convergent behavior for increasing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FKT4oBgHgl3EQfpS4j/content/2301.11869v1.pdf'} diff --git a/cdFAT4oBgHgl3EQf5x5t/content/tmp_files/2301.08735v1.pdf.txt b/cdFAT4oBgHgl3EQf5x5t/content/tmp_files/2301.08735v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a83a219971a9e1fa394f5926ebe067e5b8e71389 --- /dev/null +++ b/cdFAT4oBgHgl3EQf5x5t/content/tmp_files/2301.08735v1.pdf.txt @@ -0,0 +1,2183 @@ +TU-1177 +Thermal production of cold “hot dark matter” around eV +Wen Yin1 +1Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan +A very simple production mechanism of feebly interacting dark matter (DM) that rarely annihi- +lates is thermal production, which predicts the DM mass around eV. This has been widely known +as the hot DM scenario. Despite there are several observational hints from background lights sug- +gesting a DM in this mass range, the hot DM scenario has been considered strongly in tension with +the structure formation of our Universe because the free-streaming length of the DM produced from +thermal reactions was thought to be too long. In this paper, I show that the previous conclusions are +not always true depending on the reaction for bosonic DM because of the Bose-enhanced reaction +at very low momentum. By using the simple 1 ↔ 2 decay/inverse decay process to produce the +DM, I demonstrate that the eV range bosonic DM can be thermally produced coldly from a hot +plasma by performing a model-independent analysis applicable to axion, hidden photon, and other +bosonic DM candidates. Therefore, the bosonic DM in the eV mass range may still be special and +theoretically well-motivated. +I. +INTRODUCTION +The origin of the dark matter (DM) of our Universe +has been one of the leading mysteries of particle theory, +cosmology, and astronomy for around a century [1]. A +few decades ago, thermally produced feebly-interacting +DM in the eV mass range was popularly considered, with +a leading candidate of the standard model (SM) neu- +trino (see, e.g., [2]). Indeed, if the feebly-interacting DM +once reaches the thermal equilibrium with the thermal +bath in the early Universe, the number density of the +DM is around that of the photon relic, and the matter- +radiation equality, which is known to be around eV tem- +perature, happens at the cosmic temperature around the +DM mass. Then the DM mass is predicted around eV. +This scenario is well known as the hot DM, which has +been considered highly in tension with the structure for- +mation. To be consistent, we need an entropy dilution +making DM heavier than a few keV [3, 4]. Moreover, if +the DM is a fermion, the Pauli exclusion principle for the +DM in galaxies, i.e., the Tremaine-Gunn bound, excludes +the mass below ∼ 100 eV [5–7]. If the DM is not fully +thermalized in the early Universe, e.g., it is produced +from a freeze-in mechanism [8], the free-streaming bound +still restricts the mass above several keVs and excludes +the eV mass range [9, 10]. In any case, DM interacting +with thermal plasma na¨ıvely acquires momentum around +the cosmic temperature, and if the DM is lighter than a +keV, the free-streaming length will be too long. It seems +that any DM from thermal production with eV mass +range is a no-go. On the other hand, recently, there have +been hints from the observations of anisotropic cosmic +infrared background and TeV gamma-ray spectrum, in- +dependently suggesting an axion-like particle (ALP) DM +around the eV mass range [11–14] (see also [15–17]).1 In +1 In contrast, the anisotropic cosmic infrared background data and +the TeV gamma-ray spectrum suggest that the LORRI excess +[18] cannot be simply explained by the decay of cold DM [19, +20](See also [14, 21]). +the future, there will be various experiments confirming +the eV range DM, like the direct detection [22], indi- +rect detection [23], line-intensity mapping [24] (see also +some experiments for a generic ALP including this mass +range , e.g., solar axion helioscope [25–28] and photon +collider [29]). In this paper, I study if the aforementioned +no-go theorem for the eV range DM is true. I will show by +using a concrete example that the cold eV-range bosonic +DM can be produced via the thermal interaction with hot +plasma by taking into account the Bose-enhancement ef- +fect. +As we mentioned, the DM, much lighter than keV, is +very likely to be a bosonic one due to the Tremaine-Gunn +bound. +A known successful scenario that predicts the +eV DM is the ALP miracle scenario [30, 31], where the +ALP DM is also the inflaton, driving the cosmic infla- +tion. The potential of the ALP is assumed to have an +upside-down symmetry, via which the mass, as well as +the self-couplings, of the ALP in the vacuum, is related +to that during the hilltop inflation. The DM is a rem- +nant of inflaton from a predicted incomplete reheating. +Interestingly, the eV mass range is predicted from the +conditions for explaining the DM abundance, and the +cosmic-microwave background normalization and spec- +tral index for the power spectrum of the scalar density +perturbation.2 +In this paper, we study another simple production +mechanism, predicting the eV mass range: the thermal +production that was thought to be excluded in the early +studies. I show by considering a two-body decay/inverse +decay process, +χ1 ↔ χ2φ +(1) +2 Alternatively, there are also various simple DM production mech- +anisms in standard cosmology that are consistent (but not pre- +dict) the eV mass range, like the DM production via inflation- +ary fluctuation [32–36], the light DM production via inflaton +decay [37, 38], for ALP with modified potentials [39–41]. +arXiv:2301.08735v1 [hep-ph] 20 Jan 2023 + +2 +of a thermal distributed mother particle, χ1, with a mass, +M1(≪ T), into two daughter particles, χ2 and φ, in- +cluding a light bosonic particle, φ, has a burst popula- +tion era of the low-momentum mode pbusrt +φ +∼ M 2 +1 /T of +φ. +Here, T is the cosmic temperature at which χ1 is +thermalized. The burst production of φ is triggered by +the reaction in a timescale ∆tignition ∼ +� +T 3 +M 3 +1 Γrest +χ1→χ2φ +�−1 +with Γrest +χ1→χ2φ being the proper decay rate of χ1 → χ2φ. +Immediately, the Bose-enhanced production of φ pop- +ulates the momentum modes around pburst +φ +until the φ +number density reaches about the number density of χ1. +Thus low momentum modes of φ are produced with a +number density around T 3. In the expanding Universe +with the Hubble parameter, H, the condition for this +to happen is ∆t−1 +ignition ≫ H. +The momentum of the +cold component of φ redshifts to be below pburst +φ +which +blue shifts in time. If the usual thermalization rate of +χ2 or φ, ∆t−1 +decay ∼ Γrest +χ1→χ2φM1/T, is smaller than H at +the burst production, it cannot interact with χ1, χ2 any- +more through Eq. (1) due to kinematics, and the burst- +produced φ free-streams until it becomes non-relativistic. +Thus if the mass of φ is around eV, we get the cold com- +ponent abundance of φ consistent with the measured DM +abundance. Since the condition mostly relies on kinemat- +ics and statistics, this mechanism easily applies to pro- +duce generic bosonic DM, such as axion, hidden photon, +and CP-even scalar (a candidate is CP-even ALP [42–44] +with dark sector PQ fermions [42]), etc. +The main difference from the previous approaches of +freeze-in or thermal production of heavier DM is that I +use the unintegrated Boltzmann equation for the evolu- +tion of the distribution functions of φ and χ2 by includ- +ing Bose-enhancement and Pauli-blocking effect as well +as the mother particle mass effect. The important as- +sumption is that χ2, φ are both not thermalized when +the burst production happens since I focus on light DM, +which is typically considered to have a feeble interaction. +The rest of this paper is organized as follows. In the +next Sec.II, I will review the Boltzmann equation. +In +Sec.III, I use a simplified setup neglecting the expansion +of the Universe to explain analytically and numerically +my mechanism, the burst production of φ. +In Sec.IV, +I will remove several assumptions made in Sec.III, and +apply the mechanism to the DM production. I also dis- +cuss the conditions that the mechanism is not spoiled +by other effects in more generic setups. The last section +Sec.V is devoted to discussion and conclusions, in which +I will also comment on the application of the mechanism +to produce the DM around keV. +II. +BOLTZMANN EQUATION +Let us study the production of φ via (1) by employ- +ing the standard (unintegrated) Boltzmann equation in +expanding Universe (see e.g. [45]), +∂fi[pi, t] +∂t +− piH ∂fi[pi, t] +∂pi += Ci[pi, t], +(2) +with i = χ1, χ2, φ and Ci being the collision term of i; fi +is the distribution function of i; H = ˙a/a is the Hubble +parameter with a being the scale factor; As aforemen- +tioned, I do not specify whether φ is a vector or a scalar +field (or a more generic field with integer spins) but I +only assume that φ is a boson while χ1,2 may be either +fermions or bosons; I assumed the rotational invariance +for the equations, and pi = |⃗pi|. +The collision term for, e.g., i = φ of the 1 ↔ 2 process +(1) has the form of +Cφ = +1 +2Eφgφ +� � +dΠχ1 dΠχ2 +(2π)4δ4(pχ1 − pφ − pχ2) × |Mχ1→χ2φ|2 +× S (fχ1[pχ1], fχ2[pχ2], fφ[pφ]) . +(3) +with gi being the internal degrees of freedom including +spins, Mχ1→χ2φ the amplitude, dΠi = +d3pi +2Ei(2π)3 is phase +space integral, the sum is performed over all internal de- +grees of freedom of the initial and final states, +S ≡ fχ1[pχ1](1 ± fχ2[pχ2])(1 + fφ[pφ]) +− (1 ± fχ1[pχ1])fφ[pφ]fχ2[pχ2] +(4) += {fχ1(pχ1)(±fχ2(pχ2) + fφ(pφ) + 1) − fχ2(pχ2)fφ(pφ)} +(5) +includes the Bose-enhancement and Pauli-exclusion ef- +fects. ++ and − correspond to the cases that χ1,2 are +bosons and fermions, respectively. +a. +Simplified form with only (1) reaction +In this pa- +per, I treat the reaction by Eq. (1) seriously and make +some approximations for the other possible reactions. By +using the comoving momentum ˆpi = pia, the Boltzmann +equation with collision term (3) reduces to the simplified +form +d ˆfφ[ˆpφ] +ˆp2 +φ +2π2 +dt += gχ1 +gφ +� ˆp+ +χ1 +ˆp− +χ1 +dˆpχ1 +ˆp2 +χ1 +2π2 ˆfχ1(ˆpχ1) +∂Γrest +χ1→χ2φ +γ∂ˆpφ +S[ ˆfi] +ˆfχ1(ˆpχ1) +. +(6) +Here ˆp± +χ1 and dΓrest +χ1→χ2φ/dˆpφ depend on the kinemat- +ics, which will be explained later, γ = Eχ1/M1 (with- +out a hat) the Lorentz factor, and ˆfi(ˆpi) ≡ fi(pi). This +equation can be understood by multiplying dˆpφ on both +sides. Then the number density of φ in the momentum +range ˆpφ ∼ ˆpφ + dˆpφ is produced by the decays minus +inverse decays of χ1 in the whole kinematically-allowed +phase space. The reaction rate is accompanied by the +Lorentz factor, and the Bose-enhancement and Pauli- +blocking factors, S/ ˆfχ1, which also includes the inverse +decay effect. +The equations for the other particles can be similarly + +3 +obtained, e.g., for χ2, we have +dfχ2[ˆpχ2] +ˆp2 +χ2 +2π2 +dt += gχ1 +gχ2 +� ˆp+ +χ1 +ˆp− +χ1 +dˆpχ1 +ˆp2 +χ1 +2π2 fχ1(ˆpχ1) +∂Γrest +χ1→χ2φ +γ∂ˆpχ2 +S[ ˆfi] +fχ1(ˆpχ1). +(7) +The collision term via (1) conserves the difference of +the comoving number density of χ2 minus that of φ, +(nχ2 − nφ)a3 = const, +(8) +as long as we do not have other fast interactions to +change the comoving number of φ or χ2. We also have +− d +dt(nχ1a3) = +d +dt(nφa3) = +d +dt(nχ2a3) in the case χ1 does +not have other fast interaction than (1). +b. +Kinematics +To discuss kinematics, let us first +estimate the energy distribution in the boosted frame +of χ1 moving along the z-axis with the Lorentz factor +γ = Eχ1/M1. The momentum of the injected φ, which +has the momentum prest +φ += M 2 +1 −M 2 +2 +M 2 +1 +M1 +2 +with an angle θχ1 +to the z-axis in the rest frame, is boosted as well +pφ = (γ + γβ cos θχ1) × η M1 +2 += η +2 (Eχ1 + pχ1 cos θχ1) +(9) +where β = +� +γ2 − 1/γ. I include the effect of the mass +of χ2, M2, in +η ≡ M 2 +1 − M 2 +2 +M 2 +1 +(10) +for generality and later convenience. However, I neglect +the small φ mass, mφ, which is only taken into account +when we estimate the DM energy density in the next +section. Note that the lowest value of pφ is ηM 2 +1 /4pχ1 +when pχ1 ≫ M1, and cos θχ1 = −1, i.e., φ is injected +backward. +Then I get +∂Γrest +χ1→χ2φ +∂ˆpφ += +Γrest +χ1→χ2φ +ηˆpχ1 +. +(11) +Due to the rotational invariance, we do not have any +preferred direction in the rest frame. p± +χ1 can be obtained +from Eq. (9) with cos θχ1 in the range [−1, 1]. +Before ending this section, I would like to note that +given Γrest +χ1→χ2φ M1, η, gi and statistics, the equations are +irrelevant to intrinsic interactions. This is the reason why +I did not specify a model by using an explicit Lagrangian +so far (for one explicit Lagrangian, see Sec.IV D). In other +words, the mechanism explained by the equations in this +section should apply to large classes of models in which +φ is a bosonic particle. +III. +A BURST PRODUCTION OF BOSONIC +PARTICLE +In this section, I will study the particle production of φ +carefully by using the Boltzmann equation in flat space +introduced in the previous section. For clarity, I use a +simplified setup to describe the mechanism, and I will +remove and discuss the simplifications in the next section, +where the mechanism is applied to DM production in +cosmology. The simplifications are listed as follows, +Flat Universe: I will neglect the expansion of the Uni- +verse, i.e., a = 1, and I use Eqs. (6) and (7) by +removing the hat. I will recover the effects of the +expanding Universe in the following section. +Hierarchical timescales of other interactions: I +assume for simplicity that χ1 is in the thermal +equilibrium with the Bose-Einstein (Fermi-Dirac) +distribution +fχ1 ≈ f eq +χ1 ≡ +� +eEχ1/T ∓ 1 +�−1 +, +(12) +where − and + are for the bosonic and fermionic +χ1,2, respectively. Here and hereafter, I use the su- +perscript “eq” to denote the quantity in the thermal +equilibrium. fχ2, and fφ are treated as variables +that evolve via Eqs. (6) and (7). This is a realis- +tic condition if the reaction timescale for the other +interactions for χ1 (χ2, φ) is much faster (slower) +than the reactions induced by the process (1). The +fast reaction is assumed to keep χ1 always in the +thermal equilibrium. I will come back to argue the +case this is not satisfied in Sec.IV C. +Initial conditions: The initial conditions are taken as +fχ2,φ[pi] = 0 at t = ti +(13) +for any pi. +I will comment on what happens by +other initial conditions at the end of Sec.III D. +Relativistic plasma: I will focus on the case +T ≫ M1 ̸= 0. +(14) +In Sec.IV, this assumption is removed in the expan- +sion Universe. +A. +First stage: Ignition +Now we are ready to discuss particle production. Let +us focus on the mode of pφ ≪ M1. Then we get from +Eq. (9), +p− +χ1 ≈ η M 2 +1 +4pφ +, +p+ +χ1 = ∞ +(15) +Thus in Eqs. (6) and (7) only the higher momentum +modes of χ1 can produce the lower momentum mode of +pφ. In particular, by noting that the dominant mode of +the thermal distributed χ1 has pχ1 ∼ T(≫ M1), fφ(pφ) +(not fφp2 +φ) with momentum +pφ ∼ pburst +φ +≡ η M 2 +1 +2T +(16) + +4 +is popularly produced. From the energy-momentum con- +servation with pburst +φ +≪ T, +pχ1, pχ2 ∼ T +(17) +in the reaction. This first stage is characterized by con- +ditions close to the initial one, +fφ[pφ ∼ pburst +φ +] ≲ 1 and fχ2[pχ2 ∼ T] ≪ 1, +(18) +and the other momentum modes are also suppressed. +Let us follow the evolution of fφ[pφ ∼ pburst +φ +]. By not- +ing S/fχ1 ≃ 1, a timescale that fφ(pφ ∼ pburst +φ +) reaches +unity is derived from Eqs. (6), (7) and (18) as +∆t−1 +ignition ∼ gχ1 +gφ +4T 3 +η3M 3 +1 +Γrest +χ1→χ2φ. +(19) +We note that at this timescale, χ1 rarely decays because +the thermally averaged decay rate is +∆t−1 +decay ∼ Γrest +χ1→χ2φ +M1 +T . +(20) +Only a fraction of +∆tignition +∆tdecay +∼ gφη3 +gχ1 +M 4 +1 +4T 4 (≪ 1) , +(21) +of χ1 decays. +Although the slow decay with a small +branching fraction of pburst +φ +/T decays into φ with mo- +mentum pφ ∼ pburst +φ +, it can fill the occupation number in +the low momenta modes within a short period because +of the small phase space volume ∼ gφ +� +pburst +φ +�3 +. At the +end of this stage characterized by fφ(pφ ∼ pburst +φ +) ∼ 1, or +t−ti ∼ ∆tignition we have a small occupation number for +pφ,χ ∼ T, fφ[pφ ∼ T], fχ2[pχ2 ∼ T] ≪ 1, because of (21). +The numerical result3 for this stage is shown in red +shaded region in Fig.1, where I plot the solutions in +the [pφ/T vs fφ], [pφ/T vs (pφ/T)3fφ], [pχ2/T vs fχ2] +planes in the three panels from top to bottom, with tak- +ing M1/T = 1/10, η = 1, ∆tignition = ∆tdecay/2500 ≪ +∆tdecay, ti = 0, χ1, χ2 as Dirac fermions, and φ as singlet +scalar with gχ1,2 = 4, gφ = 1. In the top panel, the mo- +mentum modes around pφ/T ∼ O(0.001) ∼ ηM 2 +1 /(2T 2) +grow to unity with t ∼ ∆tiginition. The timescale is much +shorter than ∆tdecay. +B. +Second stage: Burst +What happens afterward is a violent production of φ. +This stage is characterized by +fφ[pφ ∼ pburst +φ +] ≳ 1, fχ2(pχ2 ∼ T) ≪ 1, +(22) +3 Throughout the paper, the Boltzmann equation is solved on the +lattices of the momenta, {log ˆpφ, log ˆpχ2}, in relevant ranges by +using Mathematica. +with which conditions, we have +S +fχ1 +���� +pφ∼pburst +φ +∼ fφ[pφ]. +(23) +From Eq. (6), we derive +˙fφ[pφ ∼ pburst +φ +] ∼ gχ1 +gφ +4T 3 +η3M 3 +1 +Γrest +χ1→χ2φfχ1[pχ1 ∼ T]fφ[pφ]. +(24) +By using the time-independent +(12), fφ[pφ ∼ pburst +φ +] +has exponential growth, thanks to the Bose enhance- +ment. +The growth rate is +gχ1 +gφ +4T 3 +η3M 3 +1 Γrest +χ1→χ2φfχ1 +∼ +gχ1 +gφ +4T 3 +η3M 3 +1 Γrest +χ1→χ2φ ∼ ∆t−1 +ignition where I used fχ1[pχ1 ∼ +T] ∼ 1. Thus we get +log +� +fφ[pφ ∼ pburst +φ +] +� +∼ +t +∆tignition +. +(25) +Therefore a burst production of φ in the low momentum +modes around pburst +φ +of φ happens in a timescale not too +different from ∆tiginition. +This stage can be found in the blue-shaded region in +Fig.1, which is indeed characterized by the exponential +growth of fφ(pφ ∼ M 2 +1 /T). +Note that t are changed +with an interval 2∆tignition for the plots here (rather than +the exponential changes of the time for the plots in the +previous stage). +The growth rate is indeed ∼ O(1) × +1/∆tignition. +C. +Final stage: Saturation +The second stage is terminated due to the back reac- +tion from the χ2 particles, which are simultaneously pro- +duced via the bose-enhanced φ production. The relevant +χ2 momentum in the reaction χ1(pχ1 ∼ T) → φ(pφ ∼ +pburst +φ +)χ2 is pχ2 ∼ T. +Although the phase space vol- +ume of χ2 is much larger than that of φ modes around +pφ ∼ pburst +φ +, the exponential production of particles +makes fχ2(pχ2 ∼ T) soon reaches a quasi-equilibrium. +The back reaction from χ2 stops a further burst produc- +tion of φ. This equilibrium can be estimated by using +S ≃ 0 with fφ(pφ ∼ pburst +φ +) ≫ 1, which leads to +fχ2(pχ2 ∼ T) ≃ fχ1(pχ1 ∼ T) +(26) +With (12), the number density of χ2 at this stage is +nχ2 ∼ gχ2 +� +pχ2∼T +d3pχ2 +2π2 fχ2 ∼ gχ2 +T 3 +π2 . +(27) +From Eq. (8) and Eq. (13) we arrive at +nφ[pφ ∼ pburst +φ +] = nburst +φ +∼ gχ2 +T 3 +π2 . +(28) +This form is similar to that from thermal distribution, ∼ +gφT 3/π2, but it is different because the internal degrees of + +5 +freedom, gχ2, is for χ2 and, importantly, it is composed +of the low momentum modes, pφ ∼ pburst +φ +≪ T. I also +showed that up to the saturation, the time is only passed +by a few ∆tignition, therefore, we get the timescale for the +burst production process to complete within +∆tburst ∼ O(1)∆tignition +(29) +In the following analytical estimation, I neglect the short +duration of the second stage, and I will use ∆tignition to +approximate the timescale to reach the final stage. +The stage discussed here is shown by the plots in +the blue-shaded region. +They overlap strongly be- +cause the system reaches a quasi-equilibrium. In addi- +tion, we numerically checked that nχ2[> 0.5T] ≈ nφ[< +0.01T] ≈ 0.36T 3 at t = 15∆tignition. Here, ni[> pcutoff] ≡ +gi +� ∞ +pcutoff +dpip2 +i +2π2 fi(pi), ni[< pcutoff] ≡ gi +� pcutoff +0 +dpip2 +i +2π2 fi(pi). +As a consequence, we have confirmed that the thermal +reactions can produce φ modes around pburst +φ +≪ T vio- +lently until the number density reaches ∼ gχ2T 3/π2. +D. +Slow thermalization after burst, and initial +condition dependence +In Fig.1, I also displayed the distribution functions +with t ≫ tignition in the dashed lines. +In the middle +panel, the number density (∝ the areas below the lines) +around pburst +φ +are suppressed. Although in the next sec- +tion, I will consider the parameter region that the physics +at this timescale is seriously changed due to the expan- +sion of the Universe, let us discuss the evolution in the +flat Universe for the understanding of the stability of the +quasi-equilibrium reached by the final stage of the burst +production. +What is happening is the usual thermalization via the +decay/inverse decay at the timescale of t ∼ ∆tdecay. +At this timescale, an O(1) fraction of plasma of χ1 +of energy O(T) decays into χ2 and φ with energies of +O(T/2). Thus fχ2(pχ2 ∼ T/2) and fφ(pφ ∼ T/2) tend +to increase. This process did not reach an equilibrium +so far because the burst production does not produce +fφ(pφ ∼ T/2). This happens much after the burst pro- +duction. From the large hierarchy of the timescale (21) +with T ≫ M1, the inverse decay via the burst process, +φ(pφ ∼ pburst +φ +)χ2(pχ2 ∼ T) → χ1(pχ2 ∼ T) happens +immediately compensating the usual decay, i.e., it de- +creases nφ(pφ ∼ pburst +φ +) immediately to keep the quasi- +equilibrium (26). +The phenomena discussed here can be seen from +the +dashed +lines +with +t += +{0.1, 1, 10}∆tdecay +in +Fig. +1. +Strictly +speaking, +the +decrease +happens +from larger pφ which corresponds to larger M 2 +1 /pχ1,χ2, +which corresponds to the faster boosted decay rate, +Γχ1→φχ2M1/Eχ1. +With the exponential hierarchy of +timescale (21), fφ(pφ +∼ M 2 +1 /4p0 +≪ pburst +φ +) at the +moment t ∼ ∆tdecay has the production due to igni- +tion with a timescale suppressed by the Boltzmann fac- +10-4 +0.001 +0.010 +0.100 +1 +10 +10-10 +10-5 +1 +105 +1010 +Ignition +Burst +10-4 +0.001 +0.010 +0.100 +1 +10 +10-10 +10-5 +1 +105 +1010 +10-3Δtignition +10-2Δtignition +10-1Δtignition +Δtignition +3Δtignition +5Δtignition +7Δtignition +9Δtignition +11Δtignition +13Δtignition +15Δtignition +17Δtignition +10-1Δtdecay +Δtdecay +10Δtdecay +10-4 +0.001 +0.010 +0.100 +1 +10 +10-10 +10-5 +1 +105 +1010 +10-3Δtignition +10-2Δtignition +10-1Δtignition +Δtignition +3Δtignition +5Δtignition +7Δtignition +9Δtignition +11Δtignition +13Δtignition +15Δtignition +17Δtignition +10-1Δtdecay +Δtdecay +10Δtdecay +Saturation +pϕ/T +fϕ +10-4 +0.001 +0.010 +0.100 +1 +10 +10-9 +10-6 +0.001 +1 +pϕ/T +(pϕ/T)3fϕ +0.001 +1 +10-9 +10-6 +0.001 +1 +pχ2/T +fχ2 +Fig. 1. The numerical solutions of the Boltzmann equation +for fφ, fχ2 due to the process Eq. (1) is shown in fφ[t] − pφ/T +plane at several t with ti = 0 [top panel]. The Hubble ex- +pansion is neglected We take M1/T = 1/10, M2 = 0, and +χ1,2 to be Dirac fermions with gχ1,2 = 4, and φ a scalar +boson with gφ = 1. The first stage, the ignition, of the +burst production, is shaded in red with four plots for t = +{10−3, 10−2, 10−1, 1}∆tignition from bottom to top. The sec- +ond stage, the burst, corresponds to the purple-shaded regime +with four plots of t = {3, 5, · · · 9}∆tignition from bottom to +top. The final stage, saturation, is found in the narrow blue +shaded regime with four plots of t = {11, 13, · · · 17}∆tignition. +We also show for comparison that the plots with t = ∆tdecay +in dashed lines. Here ∆tignition = 2500∆tdecay for the param- +eter set. In the middle panel, the solutions for (pφ/T)3 fφ is +also shown. In the bottom panel, we display the solutions in +fχ2 − pφ/T plane in the same setup and the time choices. +tor of fχ1(p0 ≫ T) ∼ exp (−p0/T) in the reaction +fχ1(p0) → fχ2(pχ2 ∼ p0)fφ(pφ ∼ M 2 +1 /4p0). This is the +reason why the deeper IR modes are still populated at + +6 +t ∼ ∆tdecay in the top and middle panels of Fig. 1. Since +the usual thermalization for the deeper UV modes of +χ1, χ2, corresponding to the deeper IR modes of φ, is +suppressed by the Lorentz factor, and Boltzmann sup- +pression ∼ exp (−2p0T) for fχ1(pχ1 ∼ 2p0) → fχ2(pχ2 ∼ +p0), fφ(pφ ∼ p0) for the usual thermalization process, +eliminating the deeper IR mode of φ requires a longer +timescale than ∆tdecay.4 +The lesson we have learned here is that if the usual +thermalization of χ2 at pχ2 ∼ T occurs, the burst- +produced φ in the IR modes is more-or-less eliminated to +maintain the quasi-equilibrium (26). This phenomenon +may also happen for the thermalization of χ2 via the +other interactions if they exist. +So far, we have discussed the case that fχ2 = fφ = 0 +as the initial condition. Let me also comment on the nu- +merical results for other initial conditions. I have checked +that if we take fχ2 ∝ f eq +χ2, and fχ2 ≳ fχ1 initially, the +burst φ production does not happen. On the contrary, +it is also checked that even if φ initially has a thermal +distribution, we have the burst φ production if fχ2 is +smaller than fχ1. They can be well understood by using +the quasi-equilibrium (26) and the number conserving +feature Eq. (8) of the burst production via (1). +IV. +COSMOLOGY OF DM BURST +PRODUCTION +Let us apply the burst production mechanism of the +bosonic particles studied in the previous Sec.III to pro- +duce light DM because the burst-produced particles have +pburst +φ +≪ T. To discuss the burst production in a more +realistic case, let me redefine +pburst +φ +≡ pburst +φ +[T(t)] = ηM 2 +1 +2T(t) +(30) +here and hereafter. It has the same form as the previous +section’s pburst +φ +, but I introduced the time dependence in +T[t] in pburst +φ +, taking account of the Hubble expansion or +thermalization of χ1. In Sec.IV A, I remove the assump- +tion of the flat Universe and assume the temperature +T[t] ∝ a[t]−1 to show that the burst-produced bosons re- +main afterward. Then we estimate the DM abundance +and discuss some model-independent constraints. Since +the production era of the DM is at the highest tempera- +ture of T[t] in the regime T[t] ∝ a[t]−1, this production +depends on the UV scenario of the radiation-dominated +Universe. In Secs. IV B and IV C, I will consider the sce- +narios that the DM produced during the reheating and +through the thermalization of χ1, respectively. In Sec. +4 With this kind of suppressed IR modes, we can produce heav- +ier DM than eV range, coldly, by explaining the measured DM +abundance. +IV D I will discuss the model-building for this produc- +tion mechanism. +A. +DM burst production in radiation-dominated +Universe +To produce the DM, we need to guarantee that the +number density of φ via the burst production is not elim- +inated in the later history of the Universe. This is nat- +urally achieved due to the expansion of the Universe, in +which the momenta of free-particle redshifts, while the +mass Mi and Γrest +χ1→χ2φ do not. +In this section, let us +further consider the case in the radiation-dominant Uni- +verse by assuming that the burst production timescale or +∆t−1 +iginition is much faster than H, at +t = ti = tprod +(31) +which is our initial time for the discussion. The temper- +ature is +T[tprod] = Tprod. +(32) +I further consider that ∆t−1 +decay is much smaller than the +Hubble parameter at t = tprod. Then the burst produc- +tion occurs because a Hubble time has many ∆tignition, +and to discuss the burst, we can neglect the Hubble ex- +pansion resulting in the essentially same setup as dis- +cussed in the previous section. Afterwards the thermal +distribution of χ1 has a time-dependent temperature scal- +ing as +T[t] = Tprod +a[tprod] +a[t] +. +(33) +I assumed there is no entropy production or dilution to +increase or decrease Ta. Thus the typical momentum for +the burst production blueshifts, i.e., pburst +φ +∝ a, but par- +ticle momentum redshifts, pφ ∝ a−1. Within one Hubble +time, the momentum of produced light DM before at +tprod soon becomes smaller than pburst +φ +[t], i.e., +pburst +φ +(tprod)a[tprod] +a[t] +< pburst +φ +(t) = pburst +φ +(tprod) +a[t] +a[tprod]. +(34) +with a[t] > a[tprod] due to the redshift and blueshift. +Since the production/destruction rate of the modes with +pφ ≪ pburst +φ +via Eq. (1) is Boltzmann suppressed by +fχ1 +∼ e−ηM 2 +1 /(2pφT ) (see Eq. (6)), the DM produc- +tion/destruction for the mode produced at t = tprod will +be kept intact later thanks to the expansion of Universe. +The production of modes around pφ ∼ pburst +φ +[t] later +is also suppressed since fχ2(pχ2 ∼ T) reaches the quasi- +equilibrium fχ1(pχ1 ∼ T) ∼ fχ2(pχ2 ∼ T) at the first +short moment t ≈ tprod. +This is because pχ1, pχ2 ∝ +a−1, T ∝ a−1 later. +In other words, once χ2 has the +number density ∼ gχ2T 3/π2, which is close to the up- +per bound from the thermal production, Eq. (8) also sets + +7 +the upper bound of the number density of φ. Therefore +once nφ ∼ nburst +φ +∼ gχ2T 3 +prod/π2 is fulfilled due to the +burst production, the reaction is afterward suppressed. +A numerical simulation for a similar setup is shown in +Fig. 2 by assuming a phase of reheating followed by the +radiation-dominated Universe (see for a detailed explana- +tion of the figure in Sec.IV B). We see the burst-produced +component indeed is frozen at a much later time at which +the pφ ∼ T modes are mostly thermalized. This is a very +different point from the case in flat Universe Sec.III D. +To sum up, the condition for the burst production to +occur in the radiation dominated Universe is +∆t−1 +ignition ≫ H ≫ ∆t−1 +decay at T = Tprod +(35) +The first inequality is that the Hubble expansion can +be neglected compared with the timescale for the burst +production. The second inequality is for suppressing the +usual thermalization eliminating the burst-produced φ. +This is because with ∆t−1 +ignition, ∆t−1 +decay ≫ H, the setup +by neglecting the Hubble expansion will be essentially +the same as in Sec.III D (with additional interaction for +φ, χ2, the additionally induced thermalization of χ2 and +φ should probably be also smaller than the Hubble rate +[see Sec.IV C]). If this is satisfied the DM is produced +with (28) with T = Tprod. +One notices with the assumption T ∝ a−1 and H ∝ +a−2, the condition (35) is more likely to satisfy in an +early time. In other words, the production should be UV +scenario dependent. Since in the following subsections, +we will focus on some natural scenarios that the discus- +sion here is applicable by properly choosing Tprod, let us +continue our discussion. +a. +DM abundance and the mass range +Once the con- +dition (35) is satisfied, later, the ratio of the burst- +produced DM number density to plasma entropy density +conserves until today. The cold component of the abun- +dance can be estimated from +Ωφ = mφ +nburst +φ +s +����� +T =Tprod +s0 +ρc +. +(36) +Here ρc, s0 are the present critical density and the en- +tropy density, respectively, and nburst +φ +∼ gχ2T 3 +prod/π2 (see +Eq. (28)),s = g⋆,s 2π2 +45 T 3 +prod. g⋆,s is the relativistic degrees +of freedom for entropy, (g⋆ will be used as the degrees +for energy density). g⋆,s should include (7/8)f(gχ1 +gχ2) +with f = 1(0) for fermionic (bosonic) χ1,2, because they +are relativistic soon after the burst. +In the following, +I assume that the comoving entropy carried by χ1,2 is +released to the lighter SM particles before the neutrino +decoupling. In addition, χ1,2 are supposed not to domi- +nate the Universe during the thermal history (no further +entropy production other than ∼ gχ1T 3, gχ2T 3). By re- +quiring the φ abundance Ωφ equal to the measured DM +density [1] ΩDM ≈ 0.26, I, therefore, get +mφ = 50eV 4 +gχ2 +g⋆,s[Tprod] +100 +. +(37) +Since g⋆,s include gχ2, at the gχ2 → ∞ limit, this reduces +to the lower bound of the mass of φ, while we may also +have an upper bound by restricting g⋆,s ≲ O(100): +2 eV ≲ mφ ≲ O(100) eV +(38) +is the generic prediction.5 +b. +Constraints from structure formation +To have +successful +structure +formation, +we +need +the +free- +streaming length of the DM to be sufficiently suppressed. +The free-streaming length of the cold component can be +estimated by using, +LFS = a0 +� teq +dtvφ[t]. +(39) +Here vφ is the typical physical velocity of φ DM, +a0 is the present scale factor, +teq is the time at +matter-radiation equality. +By approximating vφ +∼ +pburst +φ +(tprod)a[tprod]/a +� +(pburst +φ +(tprod)a[tprod]/a) +2+m2 +φ +, the bound LFS < 0.06Mpc +[3, 4] (see a similar mapping for heavy DM from inflaton +decay [46]) leads to +√η M1 +Tprod +≲ 0.02 +�gs⋆[Tprod] +100 +�1/6 �mφ +eV . +(40) +The required hierarchy is not too large.6 +c. +Suppression +of +the +hot +components +Strictly +speaking, other than the burst-produced component, pro- +duction of φ may occur via the usual decay/inverse decay +process. This implies we may have a mixed DM after the +decoupling of φ from the thermal plasma +ntot +φ [t] = nburst +φ +[t] + nth +φ [t] +(41) +where the first term denotes the part from the burst +produced component, which is dominated by the mo- +mentum mode pburst +φ +∼ ηM 2 +1 a[tprod]/(2Tproda[t]) ≪ T[t], +while the latter one is from the ordinary thermal pro- +duction via Eq. (1). We have neglected the latter com- +ponent so far in discussing the free-streaming length. In- +deed, the hot component nth +φ of φ should be suppressed +to be below O(1 − 10)% level depending on the mass +range [47]. Indeed in the numerical simulation for Fig. 2, +I get nth +φ /ntot +φ +≡ +nφ[<0.1T ] +nφ[<0.1T ]+nφ[>0.1T ]|t=5×105tR ∼ 30% for +χ1,2 being the Dirac fermion case (top panel), and domi- +nant hot components for χ1,2 being singlet scalars (mid- +dle panel). They may be in tension with the constraint. +5 If there is a large amount of entropy production after the pro- +duction by, e.g., χ1, χ2 dominating the Universe, the DM can be +heavier, which is not taken into account here. +6 Therefore, I will not discuss the model-dependent issues, e.g., +the coherent scattering and the perturbatively of the Boltzmann +equation, that are important when the occupation number is very +large. + +8 +Here let me point out three kinds of parameter regions +with suppressed hot DM components. +First, we can suppress the hot DM component by con- +sidering η ≲ 1, i.e., the mass of χ2 is non-negligible. +The contribution can be analytically estimated by us- +ing nth +φ ∼ η3T 3/π2, because the momentum of φ can be +at most produced up to ηT (see Eq. (9)). We expect a +near thermal distribution at pφ ≲ ηT while the spectrum +is suppressed at pφ ≳ ηT compared to the thermal one. +For instance, with η = 1/2, in the bottom panel of Fig. 2, +the thermal component is indeed suppressed. We checked +nth +φ /ntot +φ +∼ 3%. In this case, pburst +φ +is also suppressed (see +the Figure c.f. Eq. (16)). +The second possibility is to consider the small mφ +range by taking gχ1/g⋆,s large in Eq. (37). The effects +have already been seen by comparing the top and mid- +dle panels in Fig. 2. If the mass is below 5 − 10 eV with +Tprod ≳ 100 GeV, we can have a successful cold DM sce- +nario together with the hot DM similar to the scenar- +ios [30, 31]. This scenario may be naturally justified if χi +are charged under some non-abelian group.7 +The last possibility is that ∆t−1 +decay never becomes +faster than H until T ∼ M1. This may be considered +as the “freeze-in” scenario in which the usual interac- +tion rate with the plasma for φ is always smaller than +the Hubble expansion. +Then the usual thermal pro- +duction of fφ(pφ ∼ T) is always suppressed, and thus +nth +φ /ntot +φ +is suppressed. One numerical example is shown +in Fig.3, where the hot component is suppressed to be +nth +φ /ntot +φ +∼ 5% (See Sec.IV B for detail of the figure). +I also comment that the hot DM bound disfavors the +simple scenario χ2 = φ. This is because fφ[pφ ∼ T] = +fχ2[pφ ∼ T] is also obtained via the burst production (as +seen from the middle panel of Fig.2). Any of the above +possibilities may not be useful for this case. +B. +DM burst production during reheating +One realistic possibility that the setup for the numer- +ical simulation can apply is the DM production at the +end of reheating. Given that χ1 is always thermalized, +the burst production was found to be UV-dependent in +the radiation-dominated Universe, which motivates me +also to study the behavior during the reheating phase. +To be more concrete, let us focus on the case ∆t−1 +ignition +is faster than the Hubble expansion rate at the end of +7 In the scenarios solving the quality problem by using a non- +abelian gauge group, multiple PQ Higgs bosons/singlet fermions +with similar masses may be predicted [48–51]. Also, the scenario +generically predicts fermions/bosons with large internal degrees. +There may also be additional neutral bosons in the scenarios en- +hancing the sphaleron rate for baryogenesis with lower reheating +temperature than electroweak scale [52] and the model having +QCD axion DM around eV with larger decay constant than the +conventional one [53]. +10-4 +0.001 +0.010 +0.100 +1 +10 +10-5 +10-4 +0.001 +0.010 +0.100 +1 +10 +̂pϕ/TR +( ̂p3 +ϕ/T3 +R) ̂fϕ +t = tR 10tR +102tR +5 × 103tR +103tR +tR/3 +Thermal distribution +η = 1, χ1,2 :Dirac fermions. +10-4 +0.001 +0.010 +0.100 +1 +10 +10-5 +10-4 +0.001 +0.010 +0.100 +1 +10 +̂pϕ/TR +( ̂p3 +ϕ/T3 +R) ̂fϕ +η = 1, χ1,2 :singlet bosons +10-5 +10-4 +0.001 +0.010 +0.100 +1 +10 +10-5 +10-4 +0.001 +0.010 +0.100 +1 +10 +̂pϕ/TR +( ̂p3 +ϕ/T3 +R) ̂fϕ +η = 1/2, χ1,2 :Dirac fermions. +Fig. 2. The numerical simulation of DM distribution func- +tion ˆp3 +φT −3 +R +ˆfφ in expanding Universe by varying ˆpφ/TR with +t = {1/3, 1, 10, 102, 103, 5 × 103}tR. The initial cosmic time +is ti = 1/10tR at which ˆfχ2 = ˆfφ = 0 is set. +Γχ1→χ2φ = +10−3t−1 +R , M1 = TR/10. We take gχ1,2 = 4 with χ1,2 being +fermions in the top panel. The dotted line shows the thermal +distribution for φ. +ˆpi ≡ pia[t]/aR. In the middle and bot- +tom panels, cases with η = 1, and 1/2 with gχ1,2 = 1 and 4 +and χ1,2 being singlet scalars and fermions are shown, respec- +tively. Other parameters/variables are the same as in the top +panel. In all panels, I consider φ as a scalar with gφ = 1. +reheating t = tR, +∆t−1 +ignition|t=tR ≫ H(t = tR). +(42) +t = tR is defined by H = +� +ρtot/3M 2 +pl = Γreh where Γreh +is the decay rate of inflaton, moduli or other particle that +is responsible for reheating, ρtot the total energy density +of the Universe, Mpl ≈ 2.4×1018 GeV the reduced Planck +scale. The cosmic temperature at this moment is defined +as T = TR. + +9 +̂pϕ/TR +( ̂p3 +ϕ/T3 +R) ̂fϕ +10-6 +10-4 +0.01 +1 +10-12 +10-10 +10-8 +10-6 +10-4 +0.01 +1 +tR/2 +t = tR +10tR +102tR +103tR +104tR +105tR +106tR +107tR +108tR +Thermal distribution +"Freeze-in" +Fig. 3. +The numerical simulation for the “freeze-in” sce- +nario that the usual thermalization rate of φ is always slower +than the Hubble rate. +Setups are the same as the top +panel in Fig.2. +I take ti += 1/3tR, M1 += T at t = +2.5 × 105tR, at which Γrest +χ1→χ2φ = 0.01H. The plots are for +t = {1/2, 1, 10, 102, 103, 104, 105, 106, 107, 108}tR from bottom +to top. +During the reheating, the radiation is contiuously pro- +duced via ρr ∼ ρtotΓreh/H. +As conventionally, I as- +sume the matter-dominated Universe during the reheat- +ing, H ∝ a−3/2, ρtot ∝ a−3, which gradually decays into +radiation. Thus T ∝ ρ1/4 +r +∝ a−3/8, i.e., the temperature +of the plasma due to the reheating decreases slower than +a−1. The ignition rate scales as +∆t−1 +ignition|t 1) during the +thermalization. In any case, slightly before tth,1, the ig- +nition rate is still faster than the Hubble parameter sat- +isfying Eq. (48). +In the time regime, (∆tignition)−1 +≫ +H, during +the thermalization of χ1 Eq. (26) is reached with +fχ1 < f eq +χ1.10 +Since nburst +φ +∼ gχ2 +� +pχ2∼T d3pχ2fχ2 ∼ +gχ2 +� +pχ1∼T d3pχ1fχ1 ∝ Γth +H T 3 ∝ a−3+qth,1, the comoving +number density of nburst +φ +a3 increases in time. At t > tth,1, +9 If this is not satisfied, but satisfied in the early stage of thermal- +ization, t ≪ tth,1 when fχ1 < feq +χ1, we will have the suppressed +nburst +φ +< gχ2T 3 +prod/π2 with Tprod being the cosmic tempera- +ture at H = ∆t−1 +ignition. Thus the DM mass to explain the DM +abundance is enhanced. +This is the case qth,1 < 1, because +t−1 +ignition ∝ a−3+qth,1 decreases faster than H ∝ a−2 does. +10 Strictly speaking, when Γth,1 is slower than H, that is slower +than ∆t−1 +ignition, the back reaction to fχ,1 due to the interaction +(1) exists. +It decreases fχ1 at t < tth,1 compared to the dis- +cussed case neglecting this backreaction. The decrease is in a +way satisfying the comoving number conservation −∆(nχ1a3) = +∆(nχ2a3) = ∆(nφa3) via Eq. (1). +Taking account of this ef- +fect should not change our conclusions because, in the end, we +will get χ1 thermalized, with fχ2(pχ2 ∼ T) ∼ fχ1(pχ1 ∼ T) ∼ +feq +χ1(pχ1 ∼ T). During the whole process Eq. (8) is guaranteed. +it is frozen out as discussed in Sec.IV A. Therefore the +dominant production happens at the cosmic tempera- +ture T = Tth,1 which is the cosmic temperature that χ1 +is fully thermalized, fχ1 ≈ f eq +χ1. At this moment, we get +fχ2 ∼ f eq +χ1. Thus +Tprod ∼ Tth,1. +(50) +I numerically checked a similar behavior in the setup +more-or-less close to this scenario with a simple modi- +fication, Eq. (12)→ fχ1 = tanh(Γth,1/H) +� +eEχ1/T ∓ 1 +�−1 +with a ∝ t1/2. +Lastly, I comment on the thermalization rate of Γth,2 +and Γth,φ. +Γth,2 for the momentum mode around T +should be slower than the H at least at T = Tth,1 since, +otherwise, the pχ2 ∼ T modes of χ2 reach the equilib- +rium, and thus the resulting burst-produced φ is sup- +pressed (c.f. Eq. (8) is only for the reaction of Eq. (1), and +Sec.III D). Similarly, Γth,φ for the low momentum mode +should be smaller than H at the production. In addition, +after the burst production, Γth,φ for the low momentum +mode may also be required to be smaller than the Hubble +parameter because otherwise, the produced φ is washed +out. This is a usual assumption for the light DM. The +consistency of the argument is checked by introducing the +terms −Γth,φ( ˆfφ[ˆpφ] − ˆf eq +φ [ˆpφ]), and − Γth,2( ˆfχ2[ˆpχ2] − +ˆf eq +χ2[ˆpχ2]) in the Boltzmann equations (6) and (7), re- +spectively. That said, I emphasize that the arguments +may have exceptions due to the momentum dependence +of the reaction and Bose enhancement. A more detailed +model-dependent analysis by performing the Boltzmann +equation will be desired. +D. +Model-building –case of ALP coupled to +right-handed neutrinos– +Let me roughly discuss possible models for the burst +production of φ DM. By assuming that φ is an SM gauge +singlet, χ1,2 should have the representation for the gauge +group. In the case, χ1,2 has a non-trivial representation +of the SM gauge group, the requirement Γth,2 ≪ H can +be only satisfied in the high-temperature regime, for in- +stance, TR ≳ 1013−14 GeV SU(2) gauge interactions are +decoupled (e.g. [62]). In this case, we can use charged +heavy beyond SM (BSM) particles to play the roles of χ1 +in the burst production of φ. We will not consider this +possibility but focus on the case that χ1,2 are also gauge +singlets. The theoretical candidates may be the BSM sin- +glet scalars, and fermions in various BSM scenarios (see +also the footnote. 7), e.g., some supersymmetric partners, +or right-handed neutrinos (RHN), the latter of which I +will explain in more detail. +The RHNs may exist to explain the smallness of the +active neutrino masses by the seesaw mechanism [63–66] +(see also [67] and a UV completion for charge quanti- +zation predicting the neutrino mass scale [68]) and to +produce baryon asymmetry via leptogenesis [69–73] (see + +11 +also [62, 74, 75] in the effective theory with lepton flavor +oscillation). The Lagrangian is given as +LN = i ¯Ni∂µγµNi−(1 +2Mij ¯ +N c +i Nj+yN,iα ¯Lα ˜H ˆPRNi+h.c.), +(51) +where Lα is a left-handed lepton field in the chiral repre- +sentation, ˜H is the Higgs doublet field, i, (α) runs from +1 to n (e, µ, τ), and we take Mij = Miδij and Mi to be +real without loss of generality. I do not restrict to the +case n = 3 or 2 but take n generic. +The thermalization of the RHN in the mass range of +interest can be estimated as +Γth +N,i ≃ γN +� +α +|yN,iα|2T +(52) +with γN ≃ 0.01 being the numerical result from Refs. [76, +77] which includes 2 ↔ 2 and 1 ↔ 2 processes with +SM particles as well as the Landau-Pomeranchuk-Migdal +effects [78, 79]. +By comparing Γth +N,i with the Hub- +ble parameter at the radiation dominated era, H ≃ +� +g⋆π2T 4/90M 2 +pl, one obtains the temperature that N +is thermalized +Tth,i ≃ 7 TeV +�� +α |yN,iα| +10−6 +�2 +. +(53) +The right-handed neutrino, Ni, can also couple to the +light bosonic DM, especially the ALP, with a derivative +coupling like +Lint +eff ⊃ +� +i≥j +CNiNj +∂µφ +2fφ +¯ +Niγ5γµNj. +(54) +Moving to a mass basis by field redefinitions to remove +φ in the derivative, we obtain, +Lmass = i +� +i≥j +CNiNj(Mi + Mj) φ +2fφ +¯ +N c +i γ5Nj. +(55) +This interaction introduces the decay of +N1 → aNi̸=1 +(56) +where N1 is the heaviest RHN. The ignition rate can be +estimated as +(∆tiginition)−1 ∼ +� +i̸=1 +C2 +N1,Ni +T 3 +4πf 2 +φ +. +(57) +After this timescale, the ALP burst production occurs, +stimulating all reactions (56) via the Bose enhancement if +the ALP couplings in the mass basis are not exponentially +small. We can easily find that the ignition rate is faster +than the Hubble expansion rate at T if +fφ ≲ 2 × 109 GeV +�� +i̸=1 +C2 +N1,Ni +� +T +100 GeV. +(58) +From this, for the ALP with fφ ∼ 106−8 GeV that is +relevant to EBL hints and future reaches mentioned in +the introduction, the process is very efficient. If N1 has +the highest thermalization temperature after reheating, +TR ≫ Tth,1 ≫ Tth,i̸=1 (→ Sec.III D) +(59) +this becomes the setup of Sec.III D with Tprod = Tth,1, +while if +Tth,1 ≫ TR ≫ Tth,i̸=1 (→ Sec.IV B) +(60) +it becomes the setup of Sec.IV B with Tprod = TR. +In any case, Ni̸=1 are gradually thermalized after the +burst production via the reaction to all channels N1 → +aN2, · · · aNn. We can estimate the DM abundance with +Eq. (37) by taking gχ2 = 2 × (n − 1). Via the active- +neutrino Yukawa interactions, the comoving entropy car- +ried by RHNs gets back to the SM much after the φ burst +production. +The hot component of φ from the decay and inverse +decay can be suppressed with the mild degeneracy of +M1 ∼ Mi̸=1, which can lead to slightly larger yN,ij to ex- +plain the neutrino mass via the seesaw mechanism. Also, +the hot component can be suppressed by considering rela- +tively large fφ for the suppressed ∆t−1 +decay, or simply have +DM light as discussed in Sec.IV A.11 +Since ALP is usually defined as an axion coupled to +a pair of photons, I also check the thermalization of the +ALP via the photon coupling. Thermalization rate for +the 2 → 2 process involving an ALP and photon, e.g., +e¯e → γa, is roughly estimated Γth ∼ α3T 3/f 2 +φ, which +is several orders of magnitude smaller than the ignition +rate. Here α ∼ 1/128 is the fine-structure constant.12 +The thermalization does not occur for T ≲ T (γ) +th,φ ∼ +0.3 TeV +fφ +107 GeV. As long as this thermalization is inef- +ficient at the burst production period at T ∼ Tprod < +T (γ) +th,φ, the cold DM production happens. Even if the ther- +mal relic of φ exists at T = Tprod the burst production +occurs (see the last part of Sec.III D). Interestingly, the +hot component produced initially via the ALP-photon +coupling is suppressed with η < 1 due to the inverse de- +cay φNi̸=1 → N1 at T ∼ M1, as numerically checked in +11 It is straightforward to apply the model to a hidden photon DM +whose gauge coupling is not too large. Thanks to the equiva- +lence theorem, we can consider φ as the longitudinal mode of +the photon with certain UV completions. Model-independently, +the ignition rate for N1 decaying into the longitudinal mode and +Ni̸=1 does not change. The N1 decay into the transverse mode is +neglected because of the small gauge coupling. However, the dis- +cussion in the following, including the thermalization via photon +coupling and decays into neutrinos, are only for the ALP. +12 This may be replaced by the one for SM SU(2)L or U(1)Y cou- +pling in the symmetric phase, which decreases the upper bound +of T. With only U(1)Y coupling, the discussion does not change +much. + +12 +the last panel of Fig.2.13 Later, the ALP produced via +the burst is kept intact because the dissipation rate via +the interactions is suppressed by the tiny momentum as +usual [84] (see also another estimation by treating the +ALP as an oscillating field [85]). +A prediction of this scenario is that the ALP also +decays into active neutrinos. +The mass range can be +reached by future cosmic neutrino background searches +like PTOLEMY [86–88]. +V. +CONCLUSIONS AND DISCUSSION +In this paper, I have shown that the thermal produc- +tion of dark matter (DM) with a mass around eV may +not result in the DM being as hot as has been consid- +ered. The coldness of the produced DM depends on the +details of the reaction that produces the DM, given that +the TG bound favors the DM as a bosonic field in the +mass range of eV. In a very short period, the bosonic DM +may be burst-produced in much lower momentum modes +than the cosmic temperature due to a Bose enhancement. +Since the DM with the burst production naturally has the +number density around that of the thermalized mother +particles due to the back-reaction, the mass is predicted +around eV, the mass range of the conventional hot DM. +The cold component of the DM can remain until today +if the cosmic expansion makes the burst reaction freeze +out. One successful example has been discussed by fo- +cusing on the simple 1 ↔ 2 decay/inverse decay reaction +without adopting the conventional approximations: (1) +all the particles except for the DM are treated in ther- +mal distribution and (2) neglect of the Bose-enhancement +and Pauli-blocking factors. The resulting DM abundance +predicts the mass in the range of +mDM = O (1 − 100) eV. +(61) +In summary, I claim that the eV mass range for the DM +may still be special, and it should be theoretically well +motivated. +So far, I have used the simplest reaction to demonstrate +my claim. There may be other examples of the light and +cold bosonic DM production from hot plasma, e.g., via +generic many to many scatterings, Bremsstrahlung emis- +sions of hidden photons, etc., which are worth further +studies. +I also comment that in the whole discussion, I con- +sidered the possibility that various timescales have hier- +archies, e.g., Eq. (35). Without the hierarchy, we may +have less cold DM number and thus heavier DM mass, +e.g., a few keV, for the abundance. Some examples were +explained in footnotes 4 and 9. +The DM in this sce- +nario is colder than that of the usual thermally produced +DM with the same mass. 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D 100, +015050 (2019), arXiv:1812.11154 [hep-ph]. + diff --git a/cdFAT4oBgHgl3EQf5x5t/content/tmp_files/load_file.txt b/cdFAT4oBgHgl3EQf5x5t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..920429bbf3a4c93e9f9a74e758f6532793b6a4aa --- /dev/null +++ b/cdFAT4oBgHgl3EQf5x5t/content/tmp_files/load_file.txt @@ -0,0 +1,1151 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf,len=1150 +page_content='TU-1177 Thermal production of cold “hot dark matter” around eV Wen Yin1 1Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan A very simple production mechanism of feebly interacting dark matter (DM) that rarely annihi- lates is thermal production, which predicts the DM mass around eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This has been widely known as the hot DM scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Despite there are several observational hints from background lights sug- gesting a DM in this mass range, the hot DM scenario has been considered strongly in tension with the structure formation of our Universe because the free-streaming length of the DM produced from thermal reactions was thought to be too long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In this paper, I show that the previous conclusions are not always true depending on the reaction for bosonic DM because of the Bose-enhanced reaction at very low momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' By using the simple 1 ↔ 2 decay/inverse decay process to produce the DM, I demonstrate that the eV range bosonic DM can be thermally produced coldly from a hot plasma by performing a model-independent analysis applicable to axion, hidden photon, and other bosonic DM candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Therefore, the bosonic DM in the eV mass range may still be special and theoretically well-motivated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' INTRODUCTION The origin of the dark matter (DM) of our Universe has been one of the leading mysteries of particle theory, cosmology, and astronomy for around a century [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' A few decades ago, thermally produced feebly-interacting DM in the eV mass range was popularly considered, with a leading candidate of the standard model (SM) neu- trino (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Indeed, if the feebly-interacting DM once reaches the thermal equilibrium with the thermal bath in the early Universe, the number density of the DM is around that of the photon relic, and the matter- radiation equality, which is known to be around eV tem- perature, happens at the cosmic temperature around the DM mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Then the DM mass is predicted around eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This scenario is well known as the hot DM, which has been considered highly in tension with the structure for- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' To be consistent, we need an entropy dilution making DM heavier than a few keV [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Moreover, if the DM is a fermion, the Pauli exclusion principle for the DM in galaxies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', the Tremaine-Gunn bound, excludes the mass below ∼ 100 eV [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' If the DM is not fully thermalized in the early Universe, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', it is produced from a freeze-in mechanism [8], the free-streaming bound still restricts the mass above several keVs and excludes the eV mass range [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In any case, DM interacting with thermal plasma na¨ıvely acquires momentum around the cosmic temperature, and if the DM is lighter than a keV, the free-streaming length will be too long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' It seems that any DM from thermal production with eV mass range is a no-go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' On the other hand, recently, there have been hints from the observations of anisotropic cosmic infrared background and TeV gamma-ray spectrum, in- dependently suggesting an axion-like particle (ALP) DM around the eV mass range [11–14] (see also [15–17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='1 In 1 In contrast, the anisotropic cosmic infrared background data and the TeV gamma-ray spectrum suggest that the LORRI excess [18] cannot be simply explained by the decay of cold DM [19, 20](See also [14, 21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' the future, there will be various experiments confirming the eV range DM, like the direct detection [22], indi- rect detection [23], line-intensity mapping [24] (see also some experiments for a generic ALP including this mass range , e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', solar axion helioscope [25–28] and photon collider [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In this paper, I study if the aforementioned no-go theorem for the eV range DM is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' I will show by using a concrete example that the cold eV-range bosonic DM can be produced via the thermal interaction with hot plasma by taking into account the Bose-enhancement ef- fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' As we mentioned, the DM, much lighter than keV, is very likely to be a bosonic one due to the Tremaine-Gunn bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' A known successful scenario that predicts the eV DM is the ALP miracle scenario [30, 31], where the ALP DM is also the inflaton, driving the cosmic infla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The potential of the ALP is assumed to have an upside-down symmetry, via which the mass, as well as the self-couplings, of the ALP in the vacuum, is related to that during the hilltop inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The DM is a rem- nant of inflaton from a predicted incomplete reheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Interestingly, the eV mass range is predicted from the conditions for explaining the DM abundance, and the cosmic-microwave background normalization and spec- tral index for the power spectrum of the scalar density perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='2 In this paper, we study another simple production mechanism, predicting the eV mass range: the thermal production that was thought to be excluded in the early studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' I show by considering a two-body decay/inverse decay process, χ1 ↔ χ2φ (1) 2 Alternatively, there are also various simple DM production mech- anisms in standard cosmology that are consistent (but not pre- dict) the eV mass range, like the DM production via inflation- ary fluctuation [32–36], the light DM production via inflaton decay [37, 38], for ALP with modified potentials [39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='08735v1 [hep-ph] 20 Jan 2023 2 of a thermal distributed mother particle, χ1, with a mass, M1(≪ T), into two daughter particles, χ2 and φ, in- cluding a light bosonic particle, φ, has a burst popula- tion era of the low-momentum mode pbusrt φ ∼ M 2 1 /T of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Here, T is the cosmic temperature at which χ1 is thermalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The burst production of φ is triggered by the reaction in a timescale ∆tignition ∼ � T 3 M 3 1 Γrest χ1→χ2φ �−1 with Γrest χ1→χ2φ being the proper decay rate of χ1 → χ2φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Immediately, the Bose-enhanced production of φ pop- ulates the momentum modes around pburst φ until the φ number density reaches about the number density of χ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Thus low momentum modes of φ are produced with a number density around T 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In the expanding Universe with the Hubble parameter, H, the condition for this to happen is ∆t−1 ignition ≫ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The momentum of the cold component of φ redshifts to be below pburst φ which blue shifts in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' If the usual thermalization rate of χ2 or φ, ∆t−1 decay ∼ Γrest χ1→χ2φM1/T, is smaller than H at the burst production, it cannot interact with χ1, χ2 any- more through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (1) due to kinematics, and the burst- produced φ free-streams until it becomes non-relativistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Thus if the mass of φ is around eV, we get the cold com- ponent abundance of φ consistent with the measured DM abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Since the condition mostly relies on kinemat- ics and statistics, this mechanism easily applies to pro- duce generic bosonic DM, such as axion, hidden photon, and CP-even scalar (a candidate is CP-even ALP [42–44] with dark sector PQ fermions [42]), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The main difference from the previous approaches of freeze-in or thermal production of heavier DM is that I use the unintegrated Boltzmann equation for the evolu- tion of the distribution functions of φ and χ2 by includ- ing Bose-enhancement and Pauli-blocking effect as well as the mother particle mass effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The important as- sumption is that χ2, φ are both not thermalized when the burst production happens since I focus on light DM, which is typically considered to have a feeble interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In the next Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='II, I will review the Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='III, I use a simplified setup neglecting the expansion of the Universe to explain analytically and numerically my mechanism, the burst production of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='IV, I will remove several assumptions made in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='III, and apply the mechanism to the DM production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' I also dis- cuss the conditions that the mechanism is not spoiled by other effects in more generic setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The last section Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='V is devoted to discussion and conclusions, in which I will also comment on the application of the mechanism to produce the DM around keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' BOLTZMANN EQUATION Let us study the production of φ via (1) by employ- ing the standard (unintegrated) Boltzmann equation in expanding Universe (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' [45]), ∂fi[pi, t] ∂t − piH ∂fi[pi, t] ∂pi = Ci[pi, t], (2) with i = χ1, χ2, φ and Ci being the collision term of i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' fi is the distribution function of i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' H = ˙a/a is the Hubble parameter with a being the scale factor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' As aforemen- tioned, I do not specify whether φ is a vector or a scalar field (or a more generic field with integer spins) but I only assume that φ is a boson while χ1,2 may be either fermions or bosons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' I assumed the rotational invariance for the equations, and pi = |⃗pi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The collision term for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', i = φ of the 1 ↔ 2 process (1) has the form of Cφ = 1 2Eφgφ � � dΠχ1 dΠχ2 (2π)4δ4(pχ1 − pφ − pχ2) × |Mχ1→χ2φ|2 × S (fχ1[pχ1], fχ2[pχ2], fφ[pφ]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (3) with gi being the internal degrees of freedom including spins, Mχ1→χ2φ the amplitude, dΠi = d3pi 2Ei(2π)3 is phase space integral, the sum is performed over all internal de- grees of freedom of the initial and final states, S ≡ fχ1[pχ1](1 ± fχ2[pχ2])(1 + fφ[pφ]) − (1 ± fχ1[pχ1])fφ[pφ]fχ2[pχ2] (4) = {fχ1(pχ1)(±fχ2(pχ2) + fφ(pφ) + 1) − fχ2(pχ2)fφ(pφ)} (5) includes the Bose-enhancement and Pauli-exclusion ef- fects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' + and − correspond to the cases that χ1,2 are bosons and fermions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Simplified form with only (1) reaction In this pa- per, I treat the reaction by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (1) seriously and make some approximations for the other possible reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' By using the comoving momentum ˆpi = pia, the Boltzmann equation with collision term (3) reduces to the simplified form d ˆfφ[ˆpφ] ˆp2 φ 2π2 dt = gχ1 gφ � ˆp+ χ1 ˆp− χ1 dˆpχ1 ˆp2 χ1 2π2 ˆfχ1(ˆpχ1) ∂Γrest χ1→χ2φ γ∂ˆpφ S[ ˆfi] ˆfχ1(ˆpχ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (6) Here ˆp± χ1 and dΓrest χ1→χ2φ/dˆpφ depend on the kinemat- ics, which will be explained later, γ = Eχ1/M1 (with- out a hat) the Lorentz factor, and ˆfi(ˆpi) ≡ fi(pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This equation can be understood by multiplying dˆpφ on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Then the number density of φ in the momentum range ˆpφ ∼ ˆpφ + dˆpφ is produced by the decays minus inverse decays of χ1 in the whole kinematically-allowed phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The reaction rate is accompanied by the Lorentz factor, and the Bose-enhancement and Pauli- blocking factors, S/ ˆfχ1, which also includes the inverse decay effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The equations for the other particles can be similarly 3 obtained, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', for χ2, we have dfχ2[ˆpχ2] ˆp2 χ2 2π2 dt = gχ1 gχ2 � ˆp+ χ1 ˆp− χ1 dˆpχ1 ˆp2 χ1 2π2 fχ1(ˆpχ1) ∂Γrest χ1→χ2φ γ∂ˆpχ2 S[ ˆfi] fχ1(ˆpχ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (7) The collision term via (1) conserves the difference of the comoving number density of χ2 minus that of φ, (nχ2 − nφ)a3 = const, (8) as long as we do not have other fast interactions to change the comoving number of φ or χ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' We also have − d dt(nχ1a3) = d dt(nφa3) = d dt(nχ2a3) in the case χ1 does not have other fast interaction than (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Kinematics To discuss kinematics, let us first estimate the energy distribution in the boosted frame of χ1 moving along the z-axis with the Lorentz factor γ = Eχ1/M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The momentum of the injected φ, which has the momentum prest φ = M 2 1 −M 2 2 M 2 1 M1 2 with an angle θχ1 to the z-axis in the rest frame, is boosted as well pφ = (γ + γβ cos θχ1) × η M1 2 = η 2 (Eχ1 + pχ1 cos θχ1) (9) where β = � γ2 − 1/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' I include the effect of the mass of χ2, M2, in η ≡ M 2 1 − M 2 2 M 2 1 (10) for generality and later convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' However, I neglect the small φ mass, mφ, which is only taken into account when we estimate the DM energy density in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Note that the lowest value of pφ is ηM 2 1 /4pχ1 when pχ1 ≫ M1, and cos θχ1 = −1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', φ is injected backward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Then I get ∂Γrest χ1→χ2φ ∂ˆpφ = Γrest χ1→χ2φ ηˆpχ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (11) Due to the rotational invariance, we do not have any preferred direction in the rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' p± χ1 can be obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (9) with cos θχ1 in the range [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Before ending this section, I would like to note that given Γrest χ1→χ2φ M1, η, gi and statistics, the equations are irrelevant to intrinsic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This is the reason why I did not specify a model by using an explicit Lagrangian so far (for one explicit Lagrangian, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='IV D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In other words, the mechanism explained by the equations in this section should apply to large classes of models in which φ is a bosonic particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' A BURST PRODUCTION OF BOSONIC PARTICLE In this section, I will study the particle production of φ carefully by using the Boltzmann equation in flat space introduced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' For clarity, I use a simplified setup to describe the mechanism, and I will remove and discuss the simplifications in the next section, where the mechanism is applied to DM production in cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The simplifications are listed as follows, Flat Universe: I will neglect the expansion of the Uni- verse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', a = 1, and I use Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (6) and (7) by removing the hat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' I will recover the effects of the expanding Universe in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Hierarchical timescales of other interactions: I assume for simplicity that χ1 is in the thermal equilibrium with the Bose-Einstein (Fermi-Dirac) distribution fχ1 ≈ f eq χ1 ≡ � eEχ1/T ∓ 1 �−1 , (12) where − and + are for the bosonic and fermionic χ1,2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Here and hereafter, I use the su- perscript “eq” to denote the quantity in the thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' fχ2, and fφ are treated as variables that evolve via Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This is a realis- tic condition if the reaction timescale for the other interactions for χ1 (χ2, φ) is much faster (slower) than the reactions induced by the process (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The fast reaction is assumed to keep χ1 always in the thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' I will come back to argue the case this is not satisfied in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='IV C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Initial conditions: The initial conditions are taken as fχ2,φ[pi] = 0 at t = ti (13) for any pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' I will comment on what happens by other initial conditions at the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='III D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Relativistic plasma: I will focus on the case T ≫ M1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (14) In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='IV, this assumption is removed in the expan- sion Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' First stage: Ignition Now we are ready to discuss particle production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Let us focus on the mode of pφ ≪ M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Then we get from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (9), p− χ1 ≈ η M 2 1 4pφ , p+ χ1 = ∞ (15) Thus in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (6) and (7) only the higher momentum modes of χ1 can produce the lower momentum mode of pφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In particular, by noting that the dominant mode of the thermal distributed χ1 has pχ1 ∼ T(≫ M1), fφ(pφ) (not fφp2 φ) with momentum pφ ∼ pburst φ ≡ η M 2 1 2T (16) 4 is popularly produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' From the energy-momentum con- servation with pburst φ ≪ T, pχ1, pχ2 ∼ T (17) in the reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This first stage is characterized by con- ditions close to the initial one, fφ[pφ ∼ pburst φ ] ≲ 1 and fχ2[pχ2 ∼ T] ≪ 1, (18) and the other momentum modes are also suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Let us follow the evolution of fφ[pφ ∼ pburst φ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' By not- ing S/fχ1 ≃ 1, a timescale that fφ(pφ ∼ pburst φ ) reaches unity is derived from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (6), (7) and (18) as ∆t−1 ignition ∼ gχ1 gφ 4T 3 η3M 3 1 Γrest χ1→χ2φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (19) We note that at this timescale, χ1 rarely decays because the thermally averaged decay rate is ∆t−1 decay ∼ Γrest χ1→χ2φ M1 T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (20) Only a fraction of ∆tignition ∆tdecay ∼ gφη3 gχ1 M 4 1 4T 4 (≪ 1) , (21) of χ1 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Although the slow decay with a small branching fraction of pburst φ /T decays into φ with mo- mentum pφ ∼ pburst φ , it can fill the occupation number in the low momenta modes within a short period because of the small phase space volume ∼ gφ � pburst φ �3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' At the end of this stage characterized by fφ(pφ ∼ pburst φ ) ∼ 1, or t−ti ∼ ∆tignition we have a small occupation number for pφ,χ ∼ T, fφ[pφ ∼ T], fχ2[pχ2 ∼ T] ≪ 1, because of (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The numerical result3 for this stage is shown in red shaded region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='1, where I plot the solutions in the [pφ/T vs fφ], [pφ/T vs (pφ/T)3fφ], [pχ2/T vs fχ2] planes in the three panels from top to bottom, with tak- ing M1/T = 1/10, η = 1, ∆tignition = ∆tdecay/2500 ≪ ∆tdecay, ti = 0, χ1, χ2 as Dirac fermions, and φ as singlet scalar with gχ1,2 = 4, gφ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In the top panel, the mo- mentum modes around pφ/T ∼ O(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='001) ∼ ηM 2 1 /(2T 2) grow to unity with t ∼ ∆tiginition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The timescale is much shorter than ∆tdecay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Second stage: Burst What happens afterward is a violent production of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This stage is characterized by fφ[pφ ∼ pburst φ ] ≳ 1, fχ2(pχ2 ∼ T) ≪ 1, (22) 3 Throughout the paper, the Boltzmann equation is solved on the lattices of the momenta, {log ˆpφ, log ˆpχ2}, in relevant ranges by using Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' with which conditions, we have S fχ1 ���� pφ∼pburst φ ∼ fφ[pφ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (23) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (6), we derive ˙fφ[pφ ∼ pburst φ ] ∼ gχ1 gφ 4T 3 η3M 3 1 Γrest χ1→χ2φfχ1[pχ1 ∼ T]fφ[pφ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (24) By using the time-independent (12), fφ[pφ ∼ pburst φ ] has exponential growth, thanks to the Bose enhance- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The growth rate is gχ1 gφ 4T 3 η3M 3 1 Γrest χ1→χ2φfχ1 ∼ gχ1 gφ 4T 3 η3M 3 1 Γrest χ1→χ2φ ∼ ∆t−1 ignition where I used fχ1[pχ1 ∼ T] ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Thus we get log � fφ[pφ ∼ pburst φ ] � ∼ t ∆tignition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (25) Therefore a burst production of φ in the low momentum modes around pburst φ of φ happens in a timescale not too different from ∆tiginition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This stage can be found in the blue-shaded region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='1, which is indeed characterized by the exponential growth of fφ(pφ ∼ M 2 1 /T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Note that t are changed with an interval 2∆tignition for the plots here (rather than the exponential changes of the time for the plots in the previous stage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The growth rate is indeed ∼ O(1) × 1/∆tignition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Final stage: Saturation The second stage is terminated due to the back reac- tion from the χ2 particles, which are simultaneously pro- duced via the bose-enhanced φ production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The relevant χ2 momentum in the reaction χ1(pχ1 ∼ T) → φ(pφ ∼ pburst φ )χ2 is pχ2 ∼ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Although the phase space vol- ume of χ2 is much larger than that of φ modes around pφ ∼ pburst φ , the exponential production of particles makes fχ2(pχ2 ∼ T) soon reaches a quasi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The back reaction from χ2 stops a further burst produc- tion of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This equilibrium can be estimated by using S ≃ 0 with fφ(pφ ∼ pburst φ ) ≫ 1, which leads to fχ2(pχ2 ∼ T) ≃ fχ1(pχ1 ∼ T) (26) With (12), the number density of χ2 at this stage is nχ2 ∼ gχ2 � pχ2∼T d3pχ2 2π2 fχ2 ∼ gχ2 T 3 π2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (27) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (8) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (13) we arrive at nφ[pφ ∼ pburst φ ] = nburst φ ∼ gχ2 T 3 π2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (28) This form is similar to that from thermal distribution, ∼ gφT 3/π2, but it is different because the internal degrees of 5 freedom, gχ2, is for χ2 and, importantly, it is composed of the low momentum modes, pφ ∼ pburst φ ≪ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' I also showed that up to the saturation, the time is only passed by a few ∆tignition, therefore, we get the timescale for the burst production process to complete within ∆tburst ∼ O(1)∆tignition (29) In the following analytical estimation, I neglect the short duration of the second stage, and I will use ∆tignition to approximate the timescale to reach the final stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The stage discussed here is shown by the plots in the blue-shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' They overlap strongly be- cause the system reaches a quasi-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In addi- tion, we numerically checked that nχ2[> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='5T] ≈ nφ[< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='01T] ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='36T 3 at t = 15∆tignition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Here, ni[> pcutoff] ≡ gi � ∞ pcutoff dpip2 i 2π2 fi(pi), ni[< pcutoff] ≡ gi � pcutoff 0 dpip2 i 2π2 fi(pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' As a consequence, we have confirmed that the thermal reactions can produce φ modes around pburst φ ≪ T vio- lently until the number density reaches ∼ gχ2T 3/π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Slow thermalization after burst, and initial condition dependence In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='1, I also displayed the distribution functions with t ≫ tignition in the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In the middle panel, the number density (∝ the areas below the lines) around pburst φ are suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Although in the next sec- tion, I will consider the parameter region that the physics at this timescale is seriously changed due to the expan- sion of the Universe, let us discuss the evolution in the flat Universe for the understanding of the stability of the quasi-equilibrium reached by the final stage of the burst production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' What is happening is the usual thermalization via the decay/inverse decay at the timescale of t ∼ ∆tdecay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' At this timescale, an O(1) fraction of plasma of χ1 of energy O(T) decays into χ2 and φ with energies of O(T/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Thus fχ2(pχ2 ∼ T/2) and fφ(pφ ∼ T/2) tend to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This process did not reach an equilibrium so far because the burst production does not produce fφ(pφ ∼ T/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This happens much after the burst pro- duction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' From the large hierarchy of the timescale (21) with T ≫ M1, the inverse decay via the burst process, φ(pφ ∼ pburst φ )χ2(pχ2 ∼ T) → χ1(pχ2 ∼ T) happens immediately compensating the usual decay, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', it de- creases nφ(pφ ∼ pburst φ ) immediately to keep the quasi- equilibrium (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The phenomena discussed here can be seen from the dashed lines with t = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='1, 1, 10}∆tdecay in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Strictly speaking, the decrease happens from larger pφ which corresponds to larger M 2 1 /pχ1,χ2, which corresponds to the faster boosted decay rate, Γχ1→φχ2M1/Eχ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' With the exponential hierarchy of timescale (21), fφ(pφ ∼ M 2 1 /4p0 ≪ pburst φ ) at the moment t ∼ ∆tdecay has the production due to igni- tion with a timescale suppressed by the Boltzmann fac- 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='100 1 10 10-10 10-5 1 105 1010 Ignition Burst 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='100 1 10 10-10 10-5 1 105 1010 10-3Δtignition 10-2Δtignition 10-1Δtignition Δtignition 3Δtignition 5Δtignition 7Δtignition 9Δtignition 11Δtignition 13Δtignition 15Δtignition 17Δtignition 10-1Δtdecay Δtdecay 10Δtdecay 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='100 1 10 10-10 10-5 1 105 1010 10-3Δtignition 10-2Δtignition 10-1Δtignition Δtignition 3Δtignition 5Δtignition 7Δtignition 9Δtignition 11Δtignition 13Δtignition 15Δtignition 17Δtignition 10-1Δtdecay Δtdecay 10Δtdecay Saturation pϕ/T fϕ 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='100 1 10 10-9 10-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='001 1 pϕ/T (pϕ/T)3fϕ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='001 1 10-9 10-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='001 1 pχ2/T fχ2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The numerical solutions of the Boltzmann equation for fφ, fχ2 due to the process Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (1) is shown in fφ[t] − pφ/T plane at several t with ti = 0 [top panel].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The Hubble ex- pansion is neglected We take M1/T = 1/10, M2 = 0, and χ1,2 to be Dirac fermions with gχ1,2 = 4, and φ a scalar boson with gφ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The first stage, the ignition, of the burst production, is shaded in red with four plots for t = {10−3, 10−2, 10−1, 1}∆tignition from bottom to top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The sec- ond stage, the burst, corresponds to the purple-shaded regime with four plots of t = {3, 5, · · · 9}∆tignition from bottom to top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The final stage, saturation, is found in the narrow blue shaded regime with four plots of t = {11, 13, · · · 17}∆tignition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' We also show for comparison that the plots with t = ∆tdecay in dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Here ∆tignition = 2500∆tdecay for the param- eter set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In the middle panel, the solutions for (pφ/T)3 fφ is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In the bottom panel, we display the solutions in fχ2 − pφ/T plane in the same setup and the time choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' tor of fχ1(p0 ≫ T) ∼ exp (−p0/T) in the reaction fχ1(p0) → fχ2(pχ2 ∼ p0)fφ(pφ ∼ M 2 1 /4p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This is the reason why the deeper IR modes are still populated at 6 t ∼ ∆tdecay in the top and middle panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Since the usual thermalization for the deeper UV modes of χ1, χ2, corresponding to the deeper IR modes of φ, is suppressed by the Lorentz factor, and Boltzmann sup- pression ∼ exp (−2p0T) for fχ1(pχ1 ∼ 2p0) → fχ2(pχ2 ∼ p0), fφ(pφ ∼ p0) for the usual thermalization process, eliminating the deeper IR mode of φ requires a longer timescale than ∆tdecay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='4 The lesson we have learned here is that if the usual thermalization of χ2 at pχ2 ∼ T occurs, the burst- produced φ in the IR modes is more-or-less eliminated to maintain the quasi-equilibrium (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This phenomenon may also happen for the thermalization of χ2 via the other interactions if they exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' So far, we have discussed the case that fχ2 = fφ = 0 as the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Let me also comment on the nu- merical results for other initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' I have checked that if we take fχ2 ∝ f eq χ2, and fχ2 ≳ fχ1 initially, the burst φ production does not happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' On the contrary, it is also checked that even if φ initially has a thermal distribution, we have the burst φ production if fχ2 is smaller than fχ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' They can be well understood by using the quasi-equilibrium (26) and the number conserving feature Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (8) of the burst production via (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' COSMOLOGY OF DM BURST PRODUCTION Let us apply the burst production mechanism of the bosonic particles studied in the previous Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='III to pro- duce light DM because the burst-produced particles have pburst φ ≪ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' To discuss the burst production in a more realistic case, let me redefine pburst φ ≡ pburst φ [T(t)] = ηM 2 1 2T(t) (30) here and hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' It has the same form as the previous section’s pburst φ , but I introduced the time dependence in T[t] in pburst φ , taking account of the Hubble expansion or thermalization of χ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='IV A, I remove the assump- tion of the flat Universe and assume the temperature T[t] ∝ a[t]−1 to show that the burst-produced bosons re- main afterward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Then we estimate the DM abundance and discuss some model-independent constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Since the production era of the DM is at the highest tempera- ture of T[t] in the regime T[t] ∝ a[t]−1, this production depends on the UV scenario of the radiation-dominated Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' IV B and IV C, I will consider the sce- narios that the DM produced during the reheating and through the thermalization of χ1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 4 With this kind of suppressed IR modes, we can produce heav- ier DM than eV range, coldly, by explaining the measured DM abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' IV D I will discuss the model-building for this produc- tion mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' DM burst production in radiation-dominated Universe To produce the DM, we need to guarantee that the number density of φ via the burst production is not elim- inated in the later history of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This is nat- urally achieved due to the expansion of the Universe, in which the momenta of free-particle redshifts, while the mass Mi and Γrest χ1→χ2φ do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In this section, let us further consider the case in the radiation-dominant Uni- verse by assuming that the burst production timescale or ∆t−1 iginition is much faster than H, at t = ti = tprod (31) which is our initial time for the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The temper- ature is T[tprod] = Tprod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (32) I further consider that ∆t−1 decay is much smaller than the Hubble parameter at t = tprod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Then the burst produc- tion occurs because a Hubble time has many ∆tignition, and to discuss the burst, we can neglect the Hubble ex- pansion resulting in the essentially same setup as dis- cussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Afterwards the thermal distribution of χ1 has a time-dependent temperature scal- ing as T[t] = Tprod a[tprod] a[t] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (33) I assumed there is no entropy production or dilution to increase or decrease Ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Thus the typical momentum for the burst production blueshifts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', pburst φ ∝ a, but par- ticle momentum redshifts, pφ ∝ a−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Within one Hubble time, the momentum of produced light DM before at tprod soon becomes smaller than pburst φ [t], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', pburst φ (tprod)a[tprod] a[t] < pburst φ (t) = pburst φ (tprod) a[t] a[tprod].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (34) with a[t] > a[tprod] due to the redshift and blueshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Since the production/destruction rate of the modes with pφ ≪ pburst φ via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (1) is Boltzmann suppressed by fχ1 ∼ e−ηM 2 1 /(2pφT ) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (6)), the DM produc- tion/destruction for the mode produced at t = tprod will be kept intact later thanks to the expansion of Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The production of modes around pφ ∼ pburst φ [t] later is also suppressed since fχ2(pχ2 ∼ T) reaches the quasi- equilibrium fχ1(pχ1 ∼ T) ∼ fχ2(pχ2 ∼ T) at the first short moment t ≈ tprod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This is because pχ1, pχ2 ∝ a−1, T ∝ a−1 later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In other words, once χ2 has the number density ∼ gχ2T 3/π2, which is close to the up- per bound from the thermal production, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (8) also sets 7 the upper bound of the number density of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Therefore once nφ ∼ nburst φ ∼ gχ2T 3 prod/π2 is fulfilled due to the burst production, the reaction is afterward suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' A numerical simulation for a similar setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 2 by assuming a phase of reheating followed by the radiation-dominated Universe (see for a detailed explana- tion of the figure in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='IV B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' We see the burst-produced component indeed is frozen at a much later time at which the pφ ∼ T modes are mostly thermalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This is a very different point from the case in flat Universe Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='III D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' To sum up, the condition for the burst production to occur in the radiation dominated Universe is ∆t−1 ignition ≫ H ≫ ∆t−1 decay at T = Tprod (35) The first inequality is that the Hubble expansion can be neglected compared with the timescale for the burst production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The second inequality is for suppressing the usual thermalization eliminating the burst-produced φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This is because with ∆t−1 ignition, ∆t−1 decay ≫ H, the setup by neglecting the Hubble expansion will be essentially the same as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='III D (with additional interaction for φ, χ2, the additionally induced thermalization of χ2 and φ should probably be also smaller than the Hubble rate [see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='IV C]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' If this is satisfied the DM is produced with (28) with T = Tprod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' One notices with the assumption T ∝ a−1 and H ∝ a−2, the condition (35) is more likely to satisfy in an early time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In other words, the production should be UV scenario dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Since in the following subsections, we will focus on some natural scenarios that the discus- sion here is applicable by properly choosing Tprod, let us continue our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' DM abundance and the mass range Once the con- dition (35) is satisfied, later, the ratio of the burst- produced DM number density to plasma entropy density conserves until today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The cold component of the abun- dance can be estimated from Ωφ = mφ nburst φ s ����� T =Tprod s0 ρc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (36) Here ρc, s0 are the present critical density and the en- tropy density, respectively, and nburst φ ∼ gχ2T 3 prod/π2 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (28)),s = g⋆,s 2π2 45 T 3 prod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' g⋆,s is the relativistic degrees of freedom for entropy, (g⋆ will be used as the degrees for energy density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' g⋆,s should include (7/8)f(gχ1 +gχ2) with f = 1(0) for fermionic (bosonic) χ1,2, because they are relativistic soon after the burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In the following, I assume that the comoving entropy carried by χ1,2 is released to the lighter SM particles before the neutrino decoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In addition, χ1,2 are supposed not to domi- nate the Universe during the thermal history (no further entropy production other than ∼ gχ1T 3, gχ2T 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' By re- quiring the φ abundance Ωφ equal to the measured DM density [1] ΩDM ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='26, I, therefore, get mφ = 50eV 4 gχ2 g⋆,s[Tprod] 100 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (37) Since g⋆,s include gχ2, at the gχ2 → ∞ limit, this reduces to the lower bound of the mass of φ, while we may also have an upper bound by restricting g⋆,s ≲ O(100): 2 eV ≲ mφ ≲ O(100) eV (38) is the generic prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='5 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Constraints from structure formation To have successful structure formation, we need the free- streaming length of the DM to be sufficiently suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The free-streaming length of the cold component can be estimated by using, LFS = a0 � teq dtvφ[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (39) Here vφ is the typical physical velocity of φ DM, a0 is the present scale factor, teq is the time at matter-radiation equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' By approximating vφ ∼ pburst φ (tprod)a[tprod]/a � (pburst φ (tprod)a[tprod]/a) 2+m2 φ , the bound LFS < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='06Mpc [3, 4] (see a similar mapping for heavy DM from inflaton decay [46]) leads to √η M1 Tprod ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='02 �gs⋆[Tprod] 100 �1/6 �mφ eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (40) The required hierarchy is not too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='6 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Suppression of the hot components Strictly speaking, other than the burst-produced component, pro- duction of φ may occur via the usual decay/inverse decay process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This implies we may have a mixed DM after the decoupling of φ from the thermal plasma ntot φ [t] = nburst φ [t] + nth φ [t] (41) where the first term denotes the part from the burst produced component, which is dominated by the mo- mentum mode pburst φ ∼ ηM 2 1 a[tprod]/(2Tproda[t]) ≪ T[t], while the latter one is from the ordinary thermal pro- duction via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' We have neglected the latter com- ponent so far in discussing the free-streaming length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In- deed, the hot component nth φ of φ should be suppressed to be below O(1 − 10)% level depending on the mass range [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Indeed in the numerical simulation for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 2, I get nth φ /ntot φ ≡ nφ[<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='1T ] nφ[<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='1T ]+nφ[>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='1T ]|t=5×105tR ∼ 30% for χ1,2 being the Dirac fermion case (top panel), and domi- nant hot components for χ1,2 being singlet scalars (mid- dle panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' They may be in tension with the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 5 If there is a large amount of entropy production after the pro- duction by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', χ1, χ2 dominating the Universe, the DM can be heavier, which is not taken into account here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 6 Therefore, I will not discuss the model-dependent issues, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', the coherent scattering and the perturbatively of the Boltzmann equation, that are important when the occupation number is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 8 Here let me point out three kinds of parameter regions with suppressed hot DM components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' First, we can suppress the hot DM component by con- sidering η ≲ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', the mass of χ2 is non-negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The contribution can be analytically estimated by us- ing nth φ ∼ η3T 3/π2, because the momentum of φ can be at most produced up to ηT (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' We expect a near thermal distribution at pφ ≲ ηT while the spectrum is suppressed at pφ ≳ ηT compared to the thermal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' For instance, with η = 1/2, in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 2, the thermal component is indeed suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' We checked nth φ /ntot φ ∼ 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In this case, pburst φ is also suppressed (see the Figure c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The second possibility is to consider the small mφ range by taking gχ1/g⋆,s large in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The effects have already been seen by comparing the top and mid- dle panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' If the mass is below 5 − 10 eV with Tprod ≳ 100 GeV, we can have a successful cold DM sce- nario together with the hot DM similar to the scenar- ios [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This scenario may be naturally justified if χi are charged under some non-abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='7 The last possibility is that ∆t−1 decay never becomes faster than H until T ∼ M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This may be considered as the “freeze-in” scenario in which the usual interac- tion rate with the plasma for φ is always smaller than the Hubble expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Then the usual thermal pro- duction of fφ(pφ ∼ T) is always suppressed, and thus nth φ /ntot φ is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' One numerical example is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='3, where the hot component is suppressed to be nth φ /ntot φ ∼ 5% (See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='IV B for detail of the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' I also comment that the hot DM bound disfavors the simple scenario χ2 = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' This is because fφ[pφ ∼ T] = fχ2[pφ ∼ T] is also obtained via the burst production (as seen from the middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Any of the above possibilities may not be useful for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' DM burst production during reheating One realistic possibility that the setup for the numer- ical simulation can apply is the DM production at the end of reheating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Given that χ1 is always thermalized, the burst production was found to be UV-dependent in the radiation-dominated Universe, which motivates me also to study the behavior during the reheating phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' To be more concrete, let us focus on the case ∆t−1 ignition is faster than the Hubble expansion rate at the end of 7 In the scenarios solving the quality problem by using a non- abelian gauge group, multiple PQ Higgs bosons/singlet fermions with similar masses may be predicted [48–51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Also, the scenario generically predicts fermions/bosons with large internal degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' There may also be additional neutral bosons in the scenarios en- hancing the sphaleron rate for baryogenesis with lower reheating temperature than electroweak scale [52] and the model having QCD axion DM around eV with larger decay constant than the conventional one [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='100 1 10 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='100 1 10 ̂pϕ/TR ( ̂p3 ϕ/T3 R) ̂fϕ t = tR 10tR 102tR 5 × 103tR 103tR tR/3 Thermal distribution η = 1, χ1,2 :Dirac fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='100 1 10 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='100 1 10 ̂pϕ/TR ( ̂p3 ϕ/T3 R) ̂fϕ η = 1, χ1,2 :singlet bosons 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='100 1 10 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='100 1 10 ̂pϕ/TR ( ̂p3 ϕ/T3 R) ̂fϕ η = 1/2, χ1,2 :Dirac fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The numerical simulation of DM distribution func- tion ˆp3 φT −3 R ˆfφ in expanding Universe by varying ˆpφ/TR with t = {1/3, 1, 10, 102, 103, 5 × 103}tR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The initial cosmic time is ti = 1/10tR at which ˆfχ2 = ˆfφ = 0 is set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Γχ1→χ2φ = 10−3t−1 R , M1 = TR/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' We take gχ1,2 = 4 with χ1,2 being fermions in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The dotted line shows the thermal distribution for φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' ˆpi ≡ pia[t]/aR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In the middle and bot- tom panels, cases with η = 1, and 1/2 with gχ1,2 = 1 and 4 and χ1,2 being singlet scalars and fermions are shown, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Other parameters/variables are the same as in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' In all panels, I consider φ as a scalar with gφ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' reheating t = tR, ∆t−1 ignition|t=tR ≫ H(t = tR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' (42) t = tR is defined by H = � ρtot/3M 2 pl = Γreh where Γreh is the decay rate of inflaton, moduli or other particle that is responsible for reheating, ρtot the total energy density of the Universe, Mpl ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='4×1018 GeV the reduced Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The cosmic temperature at this moment is defined as T = TR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 9 ̂pϕ/TR ( ̂p3 ϕ/T3 R) ̂fϕ 10-6 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='01 1 10-12 10-10 10-8 10-6 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='01 1 tR/2 t = tR 10tR 102tR 103tR 104tR 105tR 106tR 107tR 108tR Thermal distribution "Freeze-in" Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The numerical simulation for the “freeze-in” sce- nario that the usual thermalization rate of φ is always slower than the Hubble rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Setups are the same as the top panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' I take ti = 1/3tR, M1 = T at t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='5 × 105tR, at which Γrest χ1→χ2φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='01H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The plots are for t = {1/2, 1, 10, 102, 103, 104, 105, 106, 107, 108}tR from bottom to top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' During the reheating, the radiation is contiuously pro- duced via ρr ∼ ρtotΓreh/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' As conventionally, I as- sume the matter-dominated Universe during the reheat- ing, H ∝ a−3/2, ρtot ∝ a−3, which gradually decays into radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' Thus T ∝ ρ1/4 r ∝ a−3/8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=', the temperature of the plasma due to the reheating decreases slower than a−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFAT4oBgHgl3EQf5x5t/content/2301.08735v1.pdf'} +page_content=' The ignition rate scales as ∆t−1 ignition|t 0. +5.2 NGSIM Dataset Simulation +Our proposed planner was tested in a dense traffic scenario +based on reconstructed NGSIM dataset (Montanino and +Punzo, 2013). The NGSIM dataset contains the recorded +trajectories (sampling frequency of 10 Hz) of all vehicles +passing a segment of freeway I-80 for 15 minutes (Mon- +tanino and Punzo, 2013). The initial scene is generated by +randomly selecting the initial frame and two adjacent lanes +(i.e. lane 1 and 2) out of 5 total lanes. Then, three vehicles +with the shortest inter-vehicle distances in the initial scene +are selected as CAVs. The inter-vehicle distances must be +smaller than the user-defined threshold which is set to +9 meters in our simulation. If the threshold condition is +not satisfied, the scene is discarded and generated again +randomly. After selecting the CAVs, the desired velocity +of all CAVs and NPCs are set to the maximum velocity +of all vehicles in the initial scene: vmax +scene. The NPC vehi- +cles are simulated using the NPC controller as described +previously. CAVs and NPCs are simulated in closed loop +with kinematic bicycle model by applying the planner and +NPC controller at 20 Hz update rate using the parameters +in Table 2 for 25 seconds. Note that the NPC vehicles +in the scene have different initial speed and time-varying +acceleration, which renders it a more challenging scenario +for multi-CAV platoon to change lanes. +In 200 randomly generated dense traffic scenarios, the +baseline and our proposed lane change strategies were ap- +plied. The performance metrics are the number of success- +ful lane change completion of all three CAVs and the aver- +age lane change competition time in successful lane change +simulation runs. Table 3 reports the obtained results. Over +two-fold increase in number of lane change completions + +(a) +(b) +(c) +(d) +(a) +(b) +(c) +(d) +Fig. 4. Top: A multi-CAV platoon changes lanes using the proactive and cooperative lane change strategy in a dense +traffic scenario. Note that orange vehicle 1, vehicle 2, and vehicle 3 correspond to CAV 1, CAV 2, and CAV 3, +respectively. Scene (a) shows the initial scene of lane 2 and 3 at frame 561 from the NGSIM dataset. Scene (b)-(c) +show the facilitator’s (CAV 1) changing lanes and regulating the distance while CAV 2 and 3 get in the desired +formation. Scene (d) shows the completion of platoon-wise lane change. Bottom: The multi-CAV platoon vehicle’s +states and inputs are shown. The labeled sections on the time axis correspond to the scenes described above. +Table 3. Performance comparison of baseline +and proposed lane change strategies +Num. of LCP +complete +Avg. LCP +complete time +Baseline +27 +12.20 s +Proposed +57 +12.64 s +using our proposed strategy is mainly attributed to the +single vehicle lane change characteristics. It only requires +a single vehicle (facilitator) to change lanes to initiate the +coordinated maneuvers to enable multi-CAV lane change +whereas the baseline controller requires three-vehicles to +change lanes simultaneously. After changing lanes, the fa- +cilitator regulates the free space by proactively modifying +the environment to render the lane change feasible for +the other CAVs. Moreover, multi-CAV platoon completed +lane change using the proposed strategy in every traffic +scenario that the baseline strategy successfully completed +lane change. On average, the baseline strategy takes 12.20 +seconds to complete a multi-CAV platoon lane change +while the proposed strategy takes 12.64 seconds. Fig. 4 +shows snapshots of platoon-wise lane change using the +proposed strategy in a dense traffic scenario generated +from NGSIM dataset as well as time series of states and +inputs of the CAVs during the simulation. In this scenario, +platoon-wise lane change using the baseline strategy is not +feasible during the duration of the simulation. +6. CONCLUSION +In this paper, we presented the concept of a facilitator +and a proactive and cooperative multi-CAV platoon lane +change strategy. A distributed coordination method with +MPC path-planners with three modes and a higher-level +FSM to manage mode transitions were used to implement +the proposed lane change strategy. Leveraging vehicle con- +nectivity, the facilitator assists the multi-CAV platoon’s +objective by proactively modifying the environment and +enhances, as a result, the feasibility of the lane change +objective as shown in numerical simulations. Tested in +200 dense traffic scenarios randomly generated based on +the NGSIM dataset, the proposed lane change strategy +demonstrated over a two-fold increase in numbers of lane +change completion of a three-CAV platoon in comparison +to the passive and opportunistic lane change strategy. The +simulation results substantiate the utility and potential +of proactive and cooperative approach to multi-CAV lane +change in certain traffic conditions (i.e. dense traffic and +small speed difference between OL and TL). The future +work will consider vehicle interaction model and a fa- +cilitator selection algorithm to improve the performance. +Experimental results are also envisaged for the future. +REFERENCES +Andersson, J.A.E., Gillis, J., Horn, G., Rawlings, J.B., and +Diehl, M. (2018). CasADi – A software framework for +nonlinear optimization and optimal control. Mathemat- +ical Programming Computation. +Byrd, R.H., Hribar, M.E., and Nocedal, J. (1999). +An +interior point algorithm for large-scale nonlinear pro- +gramming. SIAM Journal on Optimization, 9(4), 877– +900. doi:10.1137/S1052623497325107. + +116 +97 +93 +87 +79 +59 +54 +55 +24 +11 +1 +11 +108 +90 +3 +2 +1 +117 +335 +51 +36116 +97 +931 +87 +79 +[59 +54 +55 +44 +24 +11 +1 +11 +108 +90 +321 +117 +335 +51 +36 +-79 +1 +59 +-- +3 +2 +11779 +59 +90 +11779 +1 +59 +32 +117Debada, E., Makarem, L., and Gillet, D. (2017). A virtual +vehicle based coordination framework for autonomous +vehicles in heterogeneous scenarios. In 2017 IEEE Inter- +national Conference on Vehicular Electronics and Safety +(ICVES), 51–56. doi:10.1109/ICVES.2017.7991900. +Firoozi, R., Zhang, X., and Borrelli, F. (2021). +For- +mation and reconfiguration of tight multi-lane pla- +toons. Control Engineering Practice, 108, 104714. doi: +10.1016/j.conengprac.2020.104714. +Guanetti, J., Kim, Y., and Borrelli, F. (2018). Control of +connected and automated vehicles: State of the art and +future challenges. Annual Reviews in Control, 45, 18–40. +doi:https://doi.org/10.1016/j.arcontrol.2018.04.011. +Jurgen, R. (2012). +V2V/V2I Communications for Im- +proved Road Safety and Efficiency, i–viii. +Kesting, A., Treiber, M., and Helbing, D. (2010). +En- +hanced intelligent driver model to access the im- +pact of driving strategies on traffic capacity. +Philo- +sophical transactions. Series A, Mathematical, phys- +ical, and engineering sciences, 368, 4585–605. +doi: +10.1098/rsta.2010.0084. +Kim, Y., Guanetti, J., and Borrelli, F. (2019). Robust eco +adaptive cruise control for cooperative vehicles. In 2019 +18th European Control Conference (ECC), 1214–1219. +doi:10.23919/ECC.2019.8796252. +Lam, +S. +and +Katupitiya, +J. +(2013). +Cooperative +autonomous platoon maneuvers on highways. +In +2013 IEEE/ASME International Conference on Ad- +vanced +Intelligent +Mechatronics, +1152–1157. +doi: +10.1109/AIM.2013.6584249. +Li, K., Wang, J., and Zheng, Y. (2022). +Coopera- +tive formation of autonomous vehicles in mixed traf- +fic flow: Beyond platooning. +IEEE Transactions +on Intelligent Transportation Systems, 1–16. +doi: +10.1109/TITS.2022.3146612. +Luo, Y., Yang, G., Xu, M., Qin, Z., and Li, K. (2019). +Cooperative lane-change maneuver for multiple auto- +mated vehicles on a highway. Automotive Innovation, +2(3), 157–168. doi:10.1007/s42154-019-00073-1. +Montanino, M. and Punzo, V. (2013). +Reconstructed +ngsim i80-1. cost action tu0903 - multitude. +Sun, X., Horowitz, R., and Tan, C.W. (2003). An efficient +lane change maneuver for platoons of vehicles in an +automated highway system. +Dynamic Systems and +Control, Volumes 1 and 2. +doi:10.1115/imece2003- +41845. + diff --git a/dNE3T4oBgHgl3EQfGgn0/content/tmp_files/load_file.txt b/dNE3T4oBgHgl3EQfGgn0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..094fd0b1d9d4296c87f09e6866c0698b8594b708 --- /dev/null +++ b/dNE3T4oBgHgl3EQfGgn0/content/tmp_files/load_file.txt @@ -0,0 +1,540 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf,len=539 +page_content='Facilitating Cooperative and Distributed Multi-Vehicle Lane Change Maneuvers Hansung Kim ∗ Francesco Borrelli ∗ ∗ Department of Mechanical Engineering, University of California-Berkeley, Berkeley, CA 94703 USA (e-mail: hansung@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='edu, fborrelli@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='edu) Abstract: A distributed coordination method for solving multi-vehicle lane changes for connected autonomous vehicles (CAVs) is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Existing approaches to multi-vehicle lane changes are passive and opportunistic as they are implemented only when the environment allows it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The novel approach of this paper relies on the role of a facilitator assigned to a CAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The facilitator interacts with and modifies the environment to enable lane changes of other CAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Distributed MPC path planners and a distributed coordination algorithm are used to control the facilitator and other CAVs in a proactive and cooperative way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' We demonstrate the effectiveness of the proposed approach through numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In particular, we show enhanced feasibility of a multi-CAV lane change in comparison to the simultaneous multi-CAV lane change approach in various traffic conditions generated by using a data-set from real-traffic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Keywords: Intelligent Autonomous Vehicles, Optimal Control, Connected Autonomous Vehicles, Multi-vehicle Lane Change, Cooperative Navigation, Facilitator 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' INTRODUCTION Connected vehicle technology including vehicle-to-vehicle (V2V), vehicle-to-cloud (V2C), and vehicle-to-infrastructure (V2I) communication reduces the risk of collisions and energy usage, enabled by the enhanced knowledge of the environment (Jurgen, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' By leveraging connectivity, connected autonomous vehicles (CAVs) can form platoons with short headway time and coordinate multi-vehicle maneuvers such as lane change (Guanetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Platooning increases traffic throughput by allowing small inter-vehicle distances, reducing motion delays in response to changing conditions, and enhances energy efficiency by minimizing undesirable braking using shared motion forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Because of these advantages, there has been an in- creasing interest in vehicle platooning and its cooperative systems to enhance safety, energy efficiency, robustness of CAV controllers in the last decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' For instance, Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' (2019) demonstrated energy savings of 12% and jerk reduction of 31% using a model predictive control architecture to incorporate preceding platoon vehicle’s state and acceleration forecast information received via V2V in a model predictive control (MPC) framework while guaranteeing collision avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' With advance- ments in vehicle connectivity technology, more complex coordinated and cooperative maneuvers such as platoon merging, splitting, reconfiguration, and lane change can be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In existing literature, cooperative platoon maneuver coordination is typically formulated in two ways: 1) a centralized control assuming either no uncooperative non-player character (NPC) vehicles are present in the environment or their state forecasts are exactly known (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Firoozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', 2021), 2) a distributed control assuming no NPC vehicles are present in the environment (Lam and Katupitiya, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' However, NPC vehicles, typically human- driven vehicles, are present in the environment and their motion forecasts are uncertain in a real world environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Moreover, platoon lane change maneuvers in Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' (2003), Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' (2019), and Firoozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' (2021) are executed simultaneously and are passive and opportunis- tic as the platoon adapt their policies to the changing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' When the environment is configured such that a platoon lane change is not possible, the platoon remains in its original lane until the environment evolves into a configuration where safe execution of a lane change is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Simultaneous platoon lane change has fewer opportunities to change lanes in certain traffic conditions than a single vehicle because larger free space in the target lane is required for multiple vehicles for a safe execution of a lane change maneuver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' These limitations underscore the increased complexity of lane change maneuvers when considering a multi-vehicle platoon interacting with sur- rounding vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In this paper, we propose a distributed coordination method to implement a novel proactive and cooperative multi-CAV platoon lane change strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In the proposed strategy, a CAV performs a role of a facilitator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The fa- cilitator actively interacts and modifies the environment to aid the multi-CAV platoon in executing a lane change while reducing the interaction between the multi-CAV platoon and NPC vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' A multi-CAV platoon refers to a single-lane platoon with a small number of inter- connected vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Our approach to distributed coordina- tion uses three modes (Lane Keeping, Lane Change, Gap Regulation) to describe the CAV maneuver modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Finite state machines which control the mode transitions of each CAVs are also used to coordinate CAV maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The mode transition logic depends on not only the state of arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='04316v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='RO] 11 Jan 2023 NPC 3 NPC 2 NPC 1 Facilitator NPC 3 NPC 1 NPC 3 NPC 1 TL OL TL OL TL OL NPC 2 NPC 2 CAV 3 CAV 2 (a) (b) (c) NPC 3 NPC 1 TL OL NPC 2 (d) CAV 1 Facilitator CAV 1 Facilitator CAV 1 Facilitator CAV 1 CAV 2 CAV 3 CAV 3 CAV 2 CAV 2 CAV 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Snapshots of the proactive and cooperative lane change strategy: (a) the facilitator change lane into the Target Lane (TL), (b) CAV 1 and CAV 2 accelerate while the facilitator decelerate to create free space for CAV 1 and CAV 2, (c) lane change for CAV 1 and CAV 2 is rendered feasible, (d) CAV 1 and CAV 2 completes a lane change the controlled CAV but also on those of other CAVs in the multi-CAV platoon by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In each mode, distinct constraints and cost functions in distributed MPC path- planners are designed to execute the desired maneuvers for multi-CAV coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' We demonstrate that the proposed lane change strategy enhances feasibility of multi-CAV platoon lane change compared to the passive and opportunistic multi-CAV platoon lane change strategy through numerical simula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In numerical simulation, a real-world NGSIM dataset (Montanino and Punzo, 2013) is used to generate realistic initial conditions for the set of CAVs and NPC under con- sideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The contributions are summarized as follows: (1) A novel proactive and cooperative multi-CAV platoon lane change strategy which relies on the role of a facilitator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' (2) A distributed coordination method consisting of a MPC based path-planner with three modes—in which different terminal constraints and safety constraints as well as cost functions are used—and a finite state machine that controls the mode transitions of the CAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' (3) A simulation study demonstrating enhanced feasi- bility of multi-CAV platoon lane change using the proposed strategy compared to opportunistic simul- taneous multi-CAV lane change strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Sec- tion 2 introduces the concept of a facilitator and its role in proactive and cooperative multi-CAV platoon lane change strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Section 3 discusses the vehicle dynamic model and NPC prediction model used in our MPC path- planner formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Section 4 discusses the distributed coordination method including the higher-level FSM and distributed MPC path-planner used to implement the proposed lane change strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The numerical simulation setup and results of multi-vehicle lane change in dense traf- fic is presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Finally, Section 6 concludes the paper and discusses future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' PROACTIVE AND COOPERATIVE MULTI-CAV PLATOON LANE CHANGE STRATEGY: THE CONCEPT OF FACILITATOR Consider an environment with a multi-CAV platoon of n- CAVs in the same lane as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Assuming that the multi-CAVs’ objective is to change into the Target Lane (TL) from the Original Lane (OL), the multi-CAV platoon must interact with surrounding vehicles when exe- cuting the maneuver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' For instance, the multi-CAV platoon (orange vehicles) of 3 CAVs in the environment shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 1(a) must consider NPC 2 and NPC 3’s positions and velocities to determine the feasibility of a lane change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Using the passive and opportunistic lane change strategy, the multi-CAV platoon may change lanes only when the surrounding vehicles’ positions and velocities are config- ured in a way such that lane change is simultaneously feasible for all CAVs in the multi-CAV platoon (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' if lane change is feasible for all but one vehicle such as CAV 3, the multi-CAV platoon does not change lanes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In certain traffic conditions, the multi-CAV platoon lane change may be infeasible for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' However, using our proposed lane change strategy, the facilitator (without loss of generality, CAV 1 will be assigned the facilitator role in this work) enables the lane change of the other CAVs in the multi-CAV platoon by proactively manipulating the environment in favor of the multi-CAV platoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' For instance, the facilitator changes lanes when feasible and regulates the free space between itself and the vehicle in front of it (NPC 2) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 1(b)-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' This allows CAV 2 and CAV 3 to change lanes into the free space regulated by the facilitator without considering NPC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Thus, the complexity of interaction between surrounding vehicles and the multi-CAV platoon is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In order to implement the proactive and cooperative lane change strategy, a sequence of coordinated maneuvers must be executed by each CAVs in the multi-CAV platoon as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 1(a)-(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' VEHICLE MODELS In this section, we first describe the kinematic bicycle model of nonlinear vehicle dynamics in Frenet frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Then, we describe the NPC prediction model used in our path-planner formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In this work, we assume the OL center line and curvature information are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 Vehicle Dynamics 𝑋 𝑌 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 𝑙𝑓 𝛿𝑓 𝑠 𝑒𝑦 𝑒𝜓 𝑆 Tangent Line Center Line Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The kinematic bicycle model in Frenet Frame The i-th CAV state variables are defined as zi(t) := [si(t), ei y(t), ei ψ(t), vi(t)]⊤ for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', n}, where si(t) and ei y(t) represent the longitudinal and lateral displace- ment of the vehicle with respect to the desired lane’s center line, respectively, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' n is the number of CAVs in the platoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' ei ψ(t) is the heading angle difference between that of the vehicle and the center line and vi is the longitudinal velocity of the vehicle at the center of gravity (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The control input vector is ui(t) = [ai(t), δi f(t)]⊤, where ai(t) is the longitudinal acceleration of the vehicle at the C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' and δi f(t) is the steering angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The vehicle dynamics are ˙si(t) = vi(t) cos ei ψ(t) 1−eiy(t) κ(t), ˙ey i(t) = vi(t) sin ei ψ(t), ˙eψ i(t) = vi(t) ( tan δi f (t) li f +lir − κ(t) cos ei ψ(t) 1−κ(t) eiy(t) ), ˙vi(t) = ai(t), (1) li f and li r are distance from the front axle to the C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' and distance from the rear axle to the C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' κ(t) is the curvature of the reference centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The continuous-time dynamics model is discretized using forward Euler method with sampling time ∆t as follows zi(k + 1) = f(zi(k), ui(k)) (2) where zi(k) denotes the state vector zi at time k∆t+t for k ∈ Z+ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='2 NPC Prediction Model During the prediction horizon N, we use a longitudi- nal uncertain prediction model to predict NPC vehicle’s motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The model assumes uncertainty on acceleration and that the NPC vehicle does not change lanes within the prediction horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Given the OL centerline and road curvature information: (κt, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' , κt+N), the resulting dis- cretized dynamics of the NPC vehicle is as follows sNP C k+1|t = sNP C k|t + ∆t vNP C k|t cos eNP C ψ,k|t 1−eNP C y,k|t κk , eNP C y,k+1|t = eNP C y (t), eNP C ψ,k+1|t = 0, vNP C k+1|t = vNP C k|t + ∆t wk, (3) for k = t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', t + N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' where eNP C y (t) is ey of the NPC vehicle at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' wk ∼ N(0, σ2) is a random variable representing acceleration at time k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The variance σ2 is the design parameter of the model and depends on the acceleration limitation of the NPC vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In our sim- ulation, we set σ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='2 m/s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The proposed approach can accommodate different prediction model for improved prediction accuracy and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' DISTRIBUTED COORDINATION A distributed coordination of CAVs to implement a proac- tive and cooperative lane change strategy is designed as follows: MPC Path-planners with three modes in which different terminal constraints and safety constraints as well as cost functions are used to plan trajectories of each CAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Each CAVs have corresponding finite state machines that manage the mode transitions of the planners and coordinate multi-CAV platoon maneuvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In this sec- tion, the FSM and MPC path-planner are formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Fur- thermore, we discuss methods to avoid trajectory conflicts between CAVs in distributed path planning of a multi- vehicle system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 Finite State Machine The CAVs in the platoon are indexed by i starting from the front-most CAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Note that the facilitator is always assigned i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The FSM makes use of parameters and binary inputs associated with every CAVs in the platoon and denoted by the superscript i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', n} for the i- th CAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The parameters and binary inputs used in our proposed FSM are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The desired minimum longitudinal inter-vehicular dis- tances are defined as the following: d1 des = n · dsafe (4) di des = (n − i) · dsafe, for i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', n (5) where dsafe is the user-defined safe vehicular distance and impacts the conservativeness of the controlled vehicle’s driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In the context of multi-CAV lane change, d1 des represents the desired free space that the facilitator must create.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' di des corresponds to the desired distance (relative to the facilitator) for the i-th CAV to execute a lane change into the free space created by the facilitator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' These inter-vehicular distances fully describe the relative positions of the multi-CAV platoon and the facilitator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Typically, the simplest descriptions of highway driving of an autonomous vehicle is categorized into two modes: Lane Keeping (LK) and Lane Change (LC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In lane keeping mode, a CAV stays in its current lane and attempts to follow the reference velocity while maintaining a safe dis- tance from the preceding vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In lane change mode, the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' FSM parameters and inputs Parameter Definition TL Target lane for lane change OL Original lane of the platoon ∆sT L Distance from the facilitator to the preceding vehicle in the TL ∆sOL Distance from the facilitator to the preceding vehicle in the OL d1 min(∆sT L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' ∆sOL) di Signed distance from i-th CAV to the facilitator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' for i : [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' n] d1 des Desired minimum distance from the facilitator to the preceding vehicles in TL or OL di des Desired minimum signed distance from i-th CAV to the facilitator for i : [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' n] FSM binary inputs Definition LCi feasible Lane change feasibility of the i-th CAV in the platoon LCi complete Lane change completion of the i-th CAV in the platoon LCP complete �n i=2 LCi complete ri satisfied di ≥ d1 des satisfied CAV attempts to change lanes into TL if such maneuver is safe and feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' An FSM that shows the transition map between the two modes for a multi-CAV platoon is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Note that this two-mode FSM is suffi- cient for a passive and opportunistic multi-CAV platoon lane change strategy, in which switch to Lane Change mode occurs if and only if a platoon-wise lane change is simultaneously feasible (�n i=1 LCi feasible is True).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' After completing a multi-CAV platoon lane change, the planner switches back to Lane Keeping mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Our proposed FSM has an additional mode: Gap Regula- tion (GR) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In Gap Regulation mode, the CAV regulates the inter-vehicular distance of interest (gap) to the desired longitudinal distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' By assigning different desired inter-vehicular distance per CAV, the multi-CAV platoon is regulated into a desired formation in a distributed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Furthermore, the mode transition logic is different for the facilitator and the multi-CAV pla- toon but not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The mode transition depends on not just the controlled CAV’s states but also on the states of other CAVs as a mechanism for coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' For instance, the multi-CAV platoon (excluding the facilitator) switches to Gap Regulation mode only after the facilitator has completed a lane change into the TL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In this mode, the facilitator modifies the environment and creates free space in the TL for the rest of the multi-CAV platoon to change lanes while the rest of the multi-CAV platoon gets in desired formation to execute a simultaneous lane change as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 1(b)-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The mode transition logic for the facilitator and the non- facilitator CAV are listed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 3(c) in separate columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' We denote the FSM mode of the i-th CAV at time t as zi F SM(t) ∈ {LK, LC, GR} and the vector containing FSM binary inputs listed in Table 1 for all CAVs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' i = {1, 2, · · · , n}) as xF SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The i-th CAV’s FSM mode transition at time t is denoted as zi F SM(t + ∆t) = ˜f i(zi F SM(t), xF SM), (6) Note that the proposed FSM in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 3 and Table 1 also allows simultaneous platoon-wise lane change when feasible by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Further, in cases of infeasibilities while in LC mode due to environmental factors, the mode transitions back to Lane Keeping mode and returns to OL by design for safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='2 MPC Path-planner Path planning is formulated as a constrained finite time optimization problem (CFTOP) to plan dynamically feasi- ble, smooth, and collision-free trajectories for each vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The MPC path-planner solves the CFTOP repeatedly in receding horizon at a constant frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The path- planner is designed using the MPC framework with an augmented state vector ˜zi k|t := [zi k|t ⊤, di k|t] ⊤ where zi k|t is defined as z of the i-th vehicle at time k predicted at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' di k|t is di at time k predicted at time t with the following dynamics d1 k+1|t = sNP C k+1|t − s1 k+1|t, (7) di k+1|t = s1 k+1|t − si k+1|t, for i ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' (8) Therefore, the augmented state dynamics ˜zi k+1|t := g(˜zi k|t, ui k|t, wk) is represented by (2) and di k+1|t which depends on the NPC prediction with uncertainty in (3) when i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' For i ̸= 1, di k+1|t is deterministic because it is the inter-vehicular distance between two CAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In a distributed multi-CAV system, the exact motion forecasts of other CAVs are communicated via V2V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Using the augmented state dynamics, the path-planner is defined as follows min ˜zi(·|t), ui(·|t), ϵi(·|t) t+N � k=t E[∥Q(˜zi k|t − ˜zi,des k|t )∥2 2] + cs · ϵi k|t+ t+N−1 � k=t ∥Ruui k|t∥2 2 + ∥R∆u∆ui k|t∥2 2 (9a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' ˜zi k+1|t = g(˜zi k|t, ui k|t, wk), (9b) ∆umin k|t ≤ ∆ui k|t ≤ ∆umax k|t , (9c) ui k|t ∈ U, ∀k = t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', t + N − 1 (9d) ˜zi t|t = ˜zi(t) (9e) zi k|t ∈ Z, ∀k = t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', t + N (9f) if zi F SM(t) ∈ {LC} : zi t+N|t ∈ Zf (9g) E[si,min k|t ] + dsafe ≤ si k|t ≤ E[si,max k|t ] − dsafe (9h) if zi F SM(t) ∈ {LK, GR} : dsafe − ϵi k|t ≤ E[si front,k|t] − si k|t (9i) ϵi k|t ≥ 0, ∀k = t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', t + N (9j) where ui(·|t) = {ui t|t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', ui t+N−1|t} is the control input sequence over the horizon for the i-th vehicle, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Further- more, ∆ui k|t := ui k|t −ui k−1|t and ˜zi,des k|t is the desired state of the i-th vehicle which is determined by the lane center line and the speed limit of the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Q := diag(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' cy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' cψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' cv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' cd) where each scalar elements represent the tracking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Transition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Facilitator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Platoon (non-Facilitators) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='~𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='∧ 𝐿𝐶𝑓𝑒𝑎𝑠𝑖𝑏𝑙𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='~𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑃 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='∧ 𝐿𝐶𝑓𝑒𝑎𝑠𝑖𝑏𝑙𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑃 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='∧ 𝐿𝐶𝑓𝑒𝑎𝑠𝑖𝑏𝑙𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='∧ ~𝑟𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑑 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='∧ ~𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑃 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='∧ 𝑟𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑑 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='| 𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑃 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='∧ ~𝑟𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑑 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='∧ ~𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑃 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='∧ ~𝑟𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑑 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='∧ ~𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='N/A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝐿𝐶𝑓𝑒𝑎𝑠𝑖𝑏𝑙𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑃 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='∧ 𝑟𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑑 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='∧ ~𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='(𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='∧ 𝑟𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑑 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=')|~𝐿𝐶𝑓𝑒𝑎𝑠𝑖𝑏𝑙𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='| ~𝐿𝐶𝑓𝑒𝑎𝑠𝑖𝑏𝑙𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Lane ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Keeping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='(LK) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Lane ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Change ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='(LC) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Gap ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Regulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='(GR) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Lane ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Keeping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='(LK) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Lane ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Change ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='(LC) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='ෑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑖=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝐿𝐶𝑓𝑒𝑎𝑠𝑖𝑏𝑙𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝐿𝐶𝑐𝑜𝑚𝑝𝑙𝑒𝑡𝑒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Finite state machines of path planner modes in a highway driving scenario: (a) a FSM representing a passive and opportunistic lane change for a platoon P, (b) FSM with an additional mode—Gap Regulation—and its switching conditions listed in (c) for the proposed proactive and cooperative lane change strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Refer to Table 1 for definitions of the mode switching conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' costs for ei y, ei ψ, vi, and di, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Ru R∆u are the input cost matrix and input rate cost matrix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Note that these cost matrices are positive definite matrices and tuning parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' (9e) sets the initial condition at time t/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Z is a feasible state set representing the road boundaries and vehicle dynamics limitations, U is a feasi- ble input set, and ∆umin |· and ∆umax |· are input rate limits that arise from vehicle dynamic limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' zi F SM(t) is the FSM mode of the i-th CAV at time t at which the CFTOP (9) is solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The FSM mode is fixed during the horizon and its transition is described by (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' As previously stated, the planner has three modes with different constraints and tracking costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Firstly, in LC mode, a state terminal constraint (9g) is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The terminal state set, Zf, is determined by the TL’s lateral displacement from the OL and curvature at si N|t and requires the controlled vehicle to reach the target lane at the end of the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' (9h) is applied to constrain the vehicle to the free space in the target lane and avoid collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The si,min k|t and si,max k|t are the longitudinal displacement of the rear and front (with respect to that of the controlled vehicle) vehicles in the TL at time k predicted at time t, respectively, if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The expectation- type constraint is applied because si,min k|t and si,max k|t may be stochastic NPC predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' This constraint ensures that the controlled vehicle remains in free space in the TL while changing lanes within the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Secondly, in LK and GR mode, (9i) and (9j) are applied for safe adaptive cruise control if there exists a preceding vehicle (its longitudinal position denoted as si front,k|t) in the same lane as the controlled vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Otherwise, (9i) and (9j) are not applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' To prevent infeasibility occurring from surrounding vehicles’ sudden deceleration, the safety constraint (9i) is relaxed by the time-varying slack variable ϵi k|t with the cost cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In LC mode, the desired ey is the lateral-displacement of the TL whereas it is zero (OL) in other modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Lastly, the tracking cost for di k|t is only included in GR mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Accordingly, cd = qd if in GR mode and cd = 0 otherwise, where qd ∈ R+ is a tracking cost for gap regulation and a design parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='3 Multi-CAV Conflict Prevention In distributed path planning of a multi-vehicle system, resolving trajectory conflicts between CAVs in the multi- CAV platoon is central to designing a safe and robust system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Trajectory conflicts in our lane change scenario may arise when lane change intention and motion pre- dictions of other CAVs are not communicated and are wrong, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' For instance, during a simultane- ous lane change of multiple CAVs, the controlled CAV’s independently-computed optimal lane change trajectory may conflict with other CAVs’ trajectories if their lane change intentions are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Therefore, during a simultaneous lane change, a CAV must plan its trajectory while considering other lane changes of other CAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In our distributed coordination method, conflicts are prevented by using the following implementation steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Lane change intentions of other CAVs are considered when planning a simultaneous lane change of multiple CAVs by using the virtual vehicle-based method (Debada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In this method, virtual vehicles are forecasts of CAVs onto their target lane and are considered when planning a lane change trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The controlled CAV plans a lane change trajectory while considering the virtual CAVs in the target lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' This imposes stricter constraint (9h) and reserve space for the virtual CAV’s corresponding real CAV to change lanes into the free space, if feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Since all CAVs mutually consider other CAVs’ virtual vehicles during simultaneous lane change, conflicts are prevented in their optimal lane change trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Furthermore, in our planner formulation, the exact motion predictions of other CAVs at time t, which are the state sequence zi(·|t) for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', n} of CFTOP (9), are obtained via V2V technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Thus, the exact planned motion of other CAVs at time t are known which allows the controlled CAV to plan a LC trajectory that avoids collision with other CAVs as well as their virtual vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' During a simultaneous lane change of multi-CAV platoon, either the rear, front, or both vehicles are other CAVs’ virtual vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Therefore, at least one of the si,min k|t and si,max k|t terms in (9h), which is the longitudinal motion predictions of rear and front vehicles in the target lane, is no longer an uncertain prediction but instead known exactly at time t during simultaneous lane changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' NUMERICAL SIMULATION We validate the proposed distributed coordination method and demonstrate the enhanced feasibility of platoon lane change using the proactive and cooperative multi-CAV platoon lane change strategy in comparison to that of using passive and opportunistic strategy (the baseline) through numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In the baseline strategy, the multi-CAV will change lanes only when it is feasible for all three CAVs simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The nonlinear optimization problem (9) is modeled using CasADi and solved using IPOPT (Andersson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Byrd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In this section, the NPC controller and numerical simulation scenario setup are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The following assumptions have been made in our numeri- cal simulations: 1) the multi-CAV platoon has 3 CAVs with identical lengths, 2) the Facilitator role is preassigned to one CAV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1 NPC Controller To close the loop in our simulation, the NPC vehicles are controlled using the Enhanced Intelligent Driver Model (EIDM), which is based on the intelligent driver model (IDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The EIDM relaxes the sudden deceleration in not safety-critical situations while inheriting IDM’s crash-free characteristics (Kesting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' This model com- bines the IDM and constant-acceleration heuristics, which assumes that the leading vehicle will not change its ac- celeration for the next few seconds, to relax the sudden deceleration in not safety-critical situations (Kesting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Furthermore, a scaling constant called coolness fac- tor, c, is a model parameter that determines the sensitivity with respect to changes of the gap with the vehicle in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Numerical simulation parameters Vehicle Parameters NPC Model Parameters Parameter Value Parameter Value lf 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='235 m c 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='0 lr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='235 m s0 2 m w 2 m v0 vmax scene vmin 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='0 m/s T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='0 s vmax 32 m/s a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='73 m/s2 amin 3 m/s2 b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='67 m/s2 amax 2 m/s2 σ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='2 m/s2 δmin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='4 Planner Parameters δmax 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='4 Parameter Value ∆amin 2 m/s3 N 40 ∆amax 2 m/s3 ∆t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='05 s ∆δmin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='3 rad/s dsafe 2 + (lf + lr) m ∆δmax 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='3 rad/s n 3 Scene Parameter cs 20 Parameter Value cy, cψ, qd 3 Lane Width 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='8 m cv 2 front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Kesting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' (2010) recommend c ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='95, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='00] to simulate realistic driving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' This car-following model outputs vehicle acceleration to emulate an adaptive cruise control behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The EIDM parameters used in the simulation are reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' s0 represents the minimum desired spacing between the controlled vehicle and the preceding vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' v0 is the desired velocity, T is the desired time headway, a is the maximum vehicle acceleration, and b is the comfortable braking deceleration where b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='2 NGSIM Dataset Simulation Our proposed planner was tested in a dense traffic scenario based on reconstructed NGSIM dataset (Montanino and Punzo, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The NGSIM dataset contains the recorded trajectories (sampling frequency of 10 Hz) of all vehicles passing a segment of freeway I-80 for 15 minutes (Mon- tanino and Punzo, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The initial scene is generated by randomly selecting the initial frame and two adjacent lanes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' lane 1 and 2) out of 5 total lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Then, three vehicles with the shortest inter-vehicle distances in the initial scene are selected as CAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The inter-vehicle distances must be smaller than the user-defined threshold which is set to 9 meters in our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' If the threshold condition is not satisfied, the scene is discarded and generated again randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' After selecting the CAVs, the desired velocity of all CAVs and NPCs are set to the maximum velocity of all vehicles in the initial scene: vmax scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The NPC vehi- cles are simulated using the NPC controller as described previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' CAVs and NPCs are simulated in closed loop with kinematic bicycle model by applying the planner and NPC controller at 20 Hz update rate using the parameters in Table 2 for 25 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Note that the NPC vehicles in the scene have different initial speed and time-varying acceleration, which renders it a more challenging scenario for multi-CAV platoon to change lanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In 200 randomly generated dense traffic scenarios, the baseline and our proposed lane change strategies were ap- plied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The performance metrics are the number of success- ful lane change completion of all three CAVs and the aver- age lane change competition time in successful lane change simulation runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Table 3 reports the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Over two-fold increase in number of lane change completions (a) (b) (c) (d) (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Top: A multi-CAV platoon changes lanes using the proactive and cooperative lane change strategy in a dense traffic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Note that orange vehicle 1, vehicle 2, and vehicle 3 correspond to CAV 1, CAV 2, and CAV 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Scene (a) shows the initial scene of lane 2 and 3 at frame 561 from the NGSIM dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Scene (b)-(c) show the facilitator’s (CAV 1) changing lanes and regulating the distance while CAV 2 and 3 get in the desired formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Scene (d) shows the completion of platoon-wise lane change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Bottom: The multi-CAV platoon vehicle’s states and inputs are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The labeled sections on the time axis correspond to the scenes described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Performance comparison of baseline and proposed lane change strategies Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' of LCP complete Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' LCP complete time Baseline 27 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='20 s Proposed 57 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='64 s using our proposed strategy is mainly attributed to the single vehicle lane change characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' It only requires a single vehicle (facilitator) to change lanes to initiate the coordinated maneuvers to enable multi-CAV lane change whereas the baseline controller requires three-vehicles to change lanes simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' After changing lanes, the fa- cilitator regulates the free space by proactively modifying the environment to render the lane change feasible for the other CAVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Moreover, multi-CAV platoon completed lane change using the proposed strategy in every traffic scenario that the baseline strategy successfully completed lane change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' On average, the baseline strategy takes 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='20 seconds to complete a multi-CAV platoon lane change while the proposed strategy takes 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='64 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 4 shows snapshots of platoon-wise lane change using the proposed strategy in a dense traffic scenario generated from NGSIM dataset as well as time series of states and inputs of the CAVs during the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' In this scenario, platoon-wise lane change using the baseline strategy is not feasible during the duration of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' CONCLUSION In this paper, we presented the concept of a facilitator and a proactive and cooperative multi-CAV platoon lane change strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' A distributed coordination method with MPC path-planners with three modes and a higher-level FSM to manage mode transitions were used to implement the proposed lane change strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Leveraging vehicle con- nectivity, the facilitator assists the multi-CAV platoon’s objective by proactively modifying the environment and enhances, as a result, the feasibility of the lane change objective as shown in numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Tested in 200 dense traffic scenarios randomly generated based on the NGSIM dataset, the proposed lane change strategy demonstrated over a two-fold increase in numbers of lane change completion of a three-CAV platoon in comparison to the passive and opportunistic lane change strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The simulation results substantiate the utility and potential of proactive and cooperative approach to multi-CAV lane change in certain traffic conditions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' dense traffic and small speed difference between OL and TL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' The future work will consider vehicle interaction model and a fa- cilitator selection algorithm to improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Experimental results are also envisaged for the future.' metadata={'source': 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+page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' An efficient lane change maneuver for platoons of vehicles in an automated highway system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' Dynamic Systems and Control, Volumes 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} +page_content='1115/imece2003- 41845.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE3T4oBgHgl3EQfGgn0/content/2301.04316v1.pdf'} diff --git a/e9E5T4oBgHgl3EQfhA8F/content/tmp_files/2301.05637v1.pdf.txt b/e9E5T4oBgHgl3EQfhA8F/content/tmp_files/2301.05637v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f71510580522e308dd91cbd737feeb1790ffac0b --- /dev/null +++ b/e9E5T4oBgHgl3EQfhA8F/content/tmp_files/2301.05637v1.pdf.txt @@ -0,0 +1,3794 @@ +arXiv:2301.05637v1 [math.GN] 15 Dec 2022 +Skorohod’s topologies on path space +Nic Freeman∗1 and Jan M. Swart†2 +1School of Mathematics and Statistics, University of Sheffield +2The Czech Academy of Sciences, Institute of Information Theory and Automation. +January 16, 2023 +Abstract +We introduce the path space over a general metrisable space. Elements of this space are +paths, which are pairs consisting of a closed subset of the real line and a cadlag function that +is defined on that subset and takes values in the metrisable space. We equip the space of +all paths with topologies that generalise Skorohod’s J1 and M1 topologies, prove that these +topologies are Polish, and derive compactness criteria. +The central idea is that the closed graph (in case of the J1 topology) and the filled-in graph +(in case of the M1 topology) of a path can naturally be viewed as totally ordered compact +sets. We define a variant of the Hausdorff metric that measures the distance between two +compact sets, each of which is equipped with a total order. We show that the topology +generated by this metric is Polish and derive a compactness criterion. Specialising to closed +or filled-in graphs then yields Skorohod’s J1 and M1 topologies, generalised to functions that +need not all be defined on the same domain. +MSC 2020. To be filled in. +Primary: 26A15; Secondary: 06A05, 54E35, 60G07 +Keywords. Skorohod topology, J1 topology, M1 topology, path space, Hausdorff metric. +Acknowledgments. Work sponsored by GAˇCR grant 22-12790S. +∗n.p.freeman@sheffield.ac.uk +†swart@utia.cas.cz +1 + +Contents +1 +Introduction +3 +2 +Preliminaries +4 +2.1 +The split real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2 +Cadlag functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.3 +The Hausdorff metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +2.4 +The ordered Hausdorff metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.5 +Betweenness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +2.6 +Squeezed space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +3 +Topologies on path space +11 +3.1 +Path space +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +3.2 +Metrics on path space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +3.3 +Compactness criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +3.4 +Paths on fixed domains +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +3.5 +Interpolation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +3.6 +Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +3.7 +Outline of the proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +4 +Proofs of the preliminary results +18 +4.1 +The split real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +4.2 +The Hausdorff metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +4.3 +The ordered Hausdorff metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +4.4 +The mismatch modulus +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +4.5 +Polishness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +4.6 +Compactness criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +4.7 +Cadlag curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +4.8 +Betweenness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +4.9 +Squeezed space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +5 +Proofs of the main results +37 +5.1 +Closed and filled-in graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +37 +5.2 +Polishness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +39 +5.3 +Compactness criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +43 +5.4 +Paths on fixed domains +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +45 +2 + +1 +Introduction +A function is cadlag (from the French “continue `a droit, limite `a gauche”) if it is right-continuous +with left limits. In his classical paper [Sko56], Skorohod introduced four topologies on the space +of real cadlag functions on a compact interval, which he called J1, J2, M1, and M2. Of these, +the J1 topology has proved to be the most natural in many situations, in particular, when +discussing convergence of Markov processes [EK86]. For this reason, Skorohod’s J1 topology +is now normally known as the “Skorohod topology”. +Classical textbook discussions of the +Skorohod topology can be found in [EK86, Section 3.5] and [Bil99, Section 12]. All four topologies +introduced by Skorohod are discussed in [Whi02, Section 11.5]. +Motivated by the theory of the Brownian web and net [FINR04, SSS17], we study paths, +which, roughly speaking, are cadlag functions that are defined on an arbitrary closed subset of the +real line and take values in a metrisable topological space X. We generalise Skorohod’s topologies +to the space of all paths over a fixed space X. Roughly, a sequence of paths converges when the +domains on which they are defined converge and moreover the paths themselves converge in the +sense of Skorohod’s J1 or M1 topologies. We show that the path space, equipped with such a +topology, is a Polish space, and we derive compactness criteria in terms of a suitable modulus +of continuity. Polish spaces play a crucial role in probability theorem, as they are required for +Prohorov’s theorem [Bil99, Thm 5.2], and sufficient for many other theorems requiring some +regularity of a measurable space. +In Section 2, we first develop the necessary topological material, which in Section 3 we then +apply to path space. The remaining Sections 4 and 5 contain proofs. +In Subsections 2.1 and 2.2, we develop an observation, due to Kolmogorov [Kol56], that +a cadlag function defined on an interval, together with its left-continuous modification, can +be viewed as a continuous function on a somewhat peculiar topological space introduced by +Alexandroff and Urysohn [AU29]. This makes it possible to allow cadlag functions to jump at +their initial times, which later simplifies the compactness criteria, and also provides the right +set-up to define paths whose domains may not be intervals. +In Subsection 2.3 we recall the Hausdorff metric, which measures the distance between two +compact subsets of a metric space. In Subsection 2.3, we introduce a variant of this metric that +measures the distance between two compact subsets that are moreover each equipped with a +total order. We will later apply this to the closed graph (in case of the J1 topology) or the +filled-in graph (in case of the M1 topology), which can naturally be viewed as totally ordered +compact sets. +The classical M1 topology is defined only for paths taking values in the real line. +For +paths taking values in more general spaces, there are several possible ways to generalise the M1 +topology. In Subsection 2.5, we introduce the concept of a “betweenness”, which will allow us to +treat the J1 topology and various possible variants of the M1 topology in a unified framework. +Skorohod and Kolmogorov [Sko56, Kol56] only considered cadlag functions defined on com- +pact time intervals. The extension to unbounded time intervals is important in applications, +but not completely trivial. For continuous paths, one defines locally uniform convergence of a +sequence of functions by requiring that their restrictions to any compact time interval converge +uniformly. +For the J1 and M1 topologies, this approach is not feasible, since the map that +restricts a function to a smaller time interval is not continuous. To solve this, in Subsection 2.6, +we use an idea first developed in the theory of the Brownian web [FINR04], which is to intro- +duce a topology on space-time that cares less about the spatial distance between two space-time +points if their time coordinates are large. This will allow us to view graphs of cadlag functions +as compact sets, even when they are defined on unbounded time intervals. +With all the right ingredients in place, in Section 3 we introduce and study the J1 and M1 +3 + +topologies on path space. In Subsection 3.1, we show that the closed and filled-in graphs of a +path can be viewed as totally ordered compact sets. In Subsection 3.2, we then define metrics +for the J1 and M1 topologies by measuring the distance between closed and filled-in graphs using +the ordered Hausdorff metric from Subsection 2.3. In Subsection 3.3, we derive compactness +criteria in terms of a suitable modulus of continuity, generalising results from [Sko56, Kol56]. In +Section 3.4, we specialise to cadlag functions defined on a fixed time interval and show how our +results relate to the classical textbook definitions of the J1 and M1 topologies, and also briefly +discuss the less commonly used J2 and M2 topologies. +2 +Preliminaries +2.1 +The split real line +Let Rs be the space that consists of all words of the form t⋆ where t ∈ R is a real number and +⋆ ∈ {−, +} is a sign. We think of Rs as obtained by cutting each point of the real line into two. +Consequently, we call Rs the split real line and call elements of Rs split real numbers. We denote +split real numbers either by words t⋆ consisting of a Roman letter and a sign, or by a single +Greek letter. In this case, if τ = t⋆, then we call τ := t the real part of τ and we call s(τ) := ⋆ +its sign. +We equip Rs with the lexicographic order, i.e., we set σ ≤ τ if and only if either σ < τ or +σ = τ and s(σ) ≤ s(τ), where {−, +} is equipped with the natural total order in which − ≤ +. +We write σ < τ if σ ≤ τ and σ ̸= τ. We use notation for intervals in Rs similar to the usual +notation for the real line, i.e., +(σ, ρ) := {τ ∈ Rs : σ < τ < ρ}, +[σ, ρ) := {τ ∈ Rs : σ ≤ τ < ρ}, +(σ, ρ] := {τ ∈ Rs : σ < τ ≤ ρ}, +[σ, ρ] := {τ ∈ Rs : σ ≤ τ ≤ ρ}. +(2.1) +Note that there is some redundancy in this notation: for example, (s−, t+) = [s+, t−]. We +equip Rs with the order topology, which means that by definition, a basis for the topology on Rs +is formed by all intervals of the form (σ, ρ) with σ, ρ ∈ Rs. We defer the (simple) proof of the +following lemma, and all further results stated in this section, to Section 4. +Lemma 2.1 (Convergence criterion) For τn ∈ Rs and t ∈ R, one has +(i) τn → t+ if and only if τ n → t and τn ≥ t+ for all n sufficiently large, +(ii) τn → t− if and only if τ n → t and τn ≤ t− for all n sufficiently large. +Intervals of the form (σ, ρ) (resp. [σ, ρ]) are open (resp. closed) in the topology on Rs. In par- +ticular, (s−, t+) = [s+, t−] is both open and closed. The following lemma lists some elementary +properties of Rs. +Lemma 2.2 (The split real line) The space Rs is first countable, Hausdorff, and separable, +but not second countable and not metrisable. Moreover, Rs is totally disconnected, meaning that +its only connected subsets are singletons. +We equip the product space Rd +s with the product topology. By definition, a subset A ⊂ Rd +s +is bounded if A ⊂ [σ, τ]d for some σ, τ ∈ Rs. The following proposition gives a characterisation +of the compact subsets of Rd +s, similar to the well-known characterisation of compact subsets of +Rd. +4 + +Proposition 2.3 (Compact sets) For a subset C ⊂ Rd +s, the following three claims are equiv- +alent: (i) C is compact, (ii) C is sequentially compact, and (iii) C is closed and bounded. +We can compactify Rs by adding two points ±∞, in such a way that −∞ < τ < +∞ for all +τ ∈ Rs, and then equipping Rs := Rs ∪ {−∞, +∞} with the topology generated by intervals of +the form (σ, τ), [−∞, τ), or (σ, +∞]. Instead of +∞ we also write ∞. We call Rs the extended +split real line. Note that we notationally distinguish the points at infinity ±∞ of the extended +split real line from the points ±∞ of the extended real line. We extend the functions τ �→ τ and +τ �→ s(τ) that assign to each split real number τ its real part τ and sign s(τ) to the extended +split real line Rs by setting ±∞ := ±∞ and s(−∞) := +, s(∞) = −. Note that with these +definitions Rs is naturally isomorphic to [0+, 1−]. The extended split real line provides us with +a natural way to denote half infinite intervals in Rs; for example, [σ, ∞) = {τ ∈ Rs : σ ≤ τ}. +2.2 +Cadlag functions +Let I ⊂ R be an interval, let X be a Hausdorff topological space, and let f : I → X be a +function. By definition, we say that f is cadlag if it is right-continuous with left limits, i.e.: +(i) f(t) = lim +n→∞ f(tn) whenever tn, t ∈ I satisfy tn −→ +n→∞ t and tn > t for all n, +(ii) f(t−) := lim +n→∞ f(tn) exists whenever tn, t ∈ I satisfy tn −→ +n→∞ t and tn < t for all n. +Similarly, a caglad function (from the French “continue `a gauche, limite `a droit”) is left- +continuous with right limits. For any closed interval I ⊂ R of nonzero length, we let DI(X) +denote the space of all functions f : I → X such that: +(i) f is cadlag, +(ii) if t := sup I < ∞, then f(t−) = f(t). +(2.2) +We impose condition (ii) in order to have a more symmetric definition, since cadlag functions +can by construction not have a jump at the left boundary of their domain. We let D− +I (X) denote +the space of all functions f : I → X such that: +(i) f is caglad, +(ii) if t := inf I > −∞, then f(t) = f(t+), +(2.3) +where f(t+) denotes the right limit of f at t. The left-continuous modification of a function +f ∈ DI(X) is the function f − ∈ D− +I (X) uniquely defined by the requirement that f −(t) := f(t−) +for all t ∈ I where the left limit is defined. Right-continuous modifications of functions in D− +I (X) +are defined similarly. +A cadlag function f ∈ DI(X) and its left-continuous modification f − +uniquely determine each other. Indeed, f is the right-continuous modification of f −. +If I ⊂ Rs is a closed subinterval of the split real line and X is a Hausdorff topological space, +then by Lemma 2.1, a function f : I → X is continuous if and only if +(i) f(τn) → f(t+) for all t+ ∈ I and τn ∈ I such that τn ≥ t+ for all n and τ n → t, +(ii) f(τn) → f(t−) for all t− ∈ I and τn ∈ I such that τn ≤ t− for all n and τ n → t. +We let CI(X) denote the space of continuous functions f : I → X. Continuous functions on a +closed subinterval of the split real line correspond more or less to cadlag functions on a closed +subinterval of the real line. To make this connection precise, for any closed interval I ⊂ R of +nonzero length, we define Iin ⊂ Rs by +Iin := +� +t− : (t − ε, t] ⊂ I for some ε > 0 +� +∪ +� +t+ : [t, t + ε) ⊂ I for some ε > 0 +� +. +(2.4) +In particular, if I = [s, u], then Iin = [s+, u−]. Then we have the following lemma. +5 + +Lemma 2.4 (Cadlag functions as continuous functions) Let I be a closed real interval of +nonzero length and let X be a Hausdorff topological space. Let f + ∈ DI(X) and let f − ∈ D− +I (X) +be its left-continuous modification. Then setting +f(t±) := f ±(t) +(t± ∈ Iin) +(2.5) +defines a function f ∈ CIin(X), and each function f ∈ CIin(X) is of this form. +In particular, if [s, u] is a compact real interval, then Lemma 2.4 says that there is a nat- +ural isomorphism between the space of cadlag functions D[s,u](X) and the space of continuous +functions C[s+,u−](X). An advantage of working with the split real line is that we can also easily +allow for functions that jump at the endpoints of their domain. Indeed, if we replace C[s+,u−](X) +by the slightly larger space C[s−,u+](X), then we obtain a space of functions that can also jump +at the endpoints s and u of the real interval [s, u]. +Although we will not need this in the present paper, we note that the split real line also +leads to a natural definition of cadlag functions of several variables, since we can simply define +them as continuous functions defined on (a subset of) the product space Rn +s . This seems much +simpler than the approach used by other authors such as [Neu71]. +2.3 +The Hausdorff metric +For any metric space (X, d), we let K+(X) denote the space of all nonempty compact subsets of +X. The Hausdorff metric dH on K+(X) is defined as +dH(K1, K2) := sup +x1∈K1 +d(x1, K2) ∨ sup +x2∈K2 +d(x2, K1), +(2.6) +where d(x, A) := infy∈A d(x, y) denotes the distance between a point x ∈ X and a set A ⊂ X. +We can alternatively define dH in terms of correspondences. A correspondence between two sets +A1, A2 is a set R ⊂ A1 × A2 such that +∀x1 ∈ A1 ∃x2 ∈ A2 s.t. (x1, x2) ∈ R +and +∀x2 ∈ A2 ∃x1 ∈ A1 s.t. (x1, x2) ∈ R. +(2.7) +We let Cor(A1, A2) denote the set of all correspondences between A1 and A2. +Lemma 2.5 (Hausdorff metric and correspondences) Let (X, d) be a metric space. Then +dH(K1, K2) = +inf +R∈Cor(K1,K2) +sup +(x1,x2)∈R +d(x1, x2). +(2.8) +Moreover, there exists an R ∈ Cor(K1, K2) such that dH(K1, K2) = max(x1,x2)∈R d(x1, x2). +We cite the following lemma from [SSS14, Lemma B.1]. +Lemma 2.6 (Convergence criterion) Let Kn, K ∈ K+(X) (n ≥ 1). Then Kn → K in the +Hausdorff topology if and only if there exists a C ∈ K+(X) such that Kn ⊂ C for all n ≥ 1 and +K = {x ∈ X : ∃xn ∈ Kn s.t. xn → x} += {x ∈ X : ∃xn ∈ Kn s.t. x is a cluster point of (xn)n∈N}. +(2.9) +Lemma 2.6 shows that if d and d′ are two metrics generating the same topology on X, then +the corresponding Hausdorff metrics dH and d′ +H generate the same topology on K+(X). We +call this topology the Hausdorff topology. Note the subtle difference between “the Hausdorff +topology” (the topology generated by the Hausdorff metric) and “a Hausdorff topology” (any +topology satisfying Hausdorff’s separation axiom). +The following lemma is [SSS14, Lemma B.2]. In particular, it shows that K+(X) is Polish if +X is. +6 + +Lemma 2.7 (Properties of the Hausdorff metric) +(a) If (X, d) is separable, then so is (K+(X), dH). +(b) If (X, d) is complete, then so is (K+(X), dH). +Recall that a set is called precompact if its closure is compact. +The following lemma is +[SSS14, Lemma B.3]. In particular, it shows that K+(X) is compact if X is. +Lemma 2.8 (Compactness in the Hausdorff topology) A set A ⊂ K+(X) is precompact +if and only if there exists a C ∈ K+(X) such that K ⊂ C for each K ∈ A. +The following lemma says connectedness is a property of compact sets that is preserved +under limits. +Lemma 2.9 (Preservation of connectedness) The set Kc(X) of all connected nonempty +compact subsets of X is a closed subset of K+(X). +2.4 +The ordered Hausdorff metric +We will need a variant of the Hausdorff metric that measures the distance between two compact +sets, each of which is equipped with a total order. For any metric space (X, d), we let Kpart(X) +denote the space of all pairs (K, ⪯) where K is a nonempty compact subset of X and ⪯ is a +partial order on K that is compatible with the topology in the sense that the set +K⟨2⟩ := +� +(x, y) ∈ K2 : x ⪯ y +� +(2.10) +is a closed subset of K2, equipped with the product topology. Note that we do not assume that +X is equipped with a partial order; in particular, the partial order on K does not have to come +from an order on X, although we always assume that the topology on K is the induced topology +from X. We will sometimes be sloppy and denote elements of Kpart(X) simply as K, where it +is implicitly understood that K is equipped with a partial order that is compatible with the +topology. We equip the space X 2 with the metric +d2� +(x1, y1), (x2, y2) +� +:= d(x1, x2) ∨ d(y1, y2), +(2.11) +which generates the product topology, and we equip the space K+(X 2) of compact nonempty +subsets of X 2 with the associated Hausdorff metric +d2 +H(A1, A2) := +sup +(x1,y1)∈A1 +d2� +(x1, y1), A2 +� +∨ +sup +(x2,y2)∈A2 +d2� +(x2, y2), A1 +� +� +A1, A2 ∈ K+(X 2) +� +. +(2.12) +An element (K, ⪯) of Kpart(X) is clearly uniquely determined by the compact set K⟨2⟩ ⊂ X 2 +defined in (2.10), so setting +dpart(K1, K2) := d2 +H(K⟨2⟩ +1 , K⟨2⟩ +2 ) +� +K1, K2 ∈ Kpart(X) +� +(2.13) +defines a metric dpart on Kpart(X). +We let Ktot(X) denote the space of all pairs (K, ⪯) ∈ Kpart(X) such that ⪯ is a total order +on K. There is a natural way to define a metric on Ktot(X) that is at first sight very different +from the definition in (2.13). Recall the definition of a correspondence from Subsection 2.3. +By definition, a correspondence R between two totally ordered spaces (K1, ⪯1) and (K2, ⪯2) is +monotone if +there are no (x1, x2), (y1, y2) ∈ R such that x1 ≺1 y1 and y2 ≺2 x2, +(2.14) +7 + +where x ≺ y means that x ⪯ y and x ̸= y. +We let Cor+(K1, K2) denote the space of all +monotone correspondences between two totally ordered spaces K1 and K2, and define a metric +dtot on Ktot(X) by +dtot(K1, K2) := +inf +R∈Cor+(K1,K2) +sup +(x1,x2)∈R +d(x1, x2) +� +K1, K2 ∈ Ktot(X) +� +. +(2.15) +The following theorem says that dpart and dtot generate the same topology on Ktot(X) and satisfy +dpart ≤ dtot, but they do not satisfy an opposite inequality of the form dtot ≤ Cdpart for any +C < ∞. +Theorem 2.10 (The ordered Hausdorff topology) Let (X, d) be a metric space. Then the +metrics dpart and dtot defined in (2.13) and (2.15) generate the same topology on Ktot(X). Also, +if d and d′ generate the same topology on X and dpart and d′ +part are defined in terms of d and d′ +as in (2.13), then dpart and d′ +part generate the same topology on Ktot(X). One has +dH(K1, K2) ≤ dpart(K1, K2) ≤ dtot(K1, K2) +� +K1, K2 ∈ Ktot(X) +� +. +(2.16) +If X = [0, 1], then for each ε > 0, there exist K1, K2 ∈ Ktot(X) such that dpart(K1, K2) ≤ +εdtot(K1, K2). +We call the topology on Ktot(X) generated by dpart or dtot the ordered Hausdorff topology. +The second claim of Theorem 2.10 says that this topology depends only on the topology on X +and not on the choice of metric on X. We recall that a topological space X is Polish if X is +separable and there exists a complete metric generating the topology on X. Note that being +Polish is a property of the topology and not a property of the metric. In fact, on each non- +compact Polish space X, there also exist non-complete metrics that generate the topology on +X.1 If (X, d) is complete, then as we will show in Lemma 4.13 below, Ktot(X) is not in general +complete in the metrics dpart or dtot. Nevertheless, we have the following result. +Proposition 2.11 (Preservation of Polishness) If X is a Polish space, then so is Ktot(X), +equipped with the ordered Hausdorff topology. +Our next result characterises the compact subsets of Ktot(X). For K ∈ Ktot(X) and ε > 0, +we define the mismatch modulus mε(K) as +mε(K) := sup +� +d(x1, y1) ∨ d(x2, y2) : x1, y1, x2, y2 ∈ K +d(x1, x2) ∨ d(y1, y2) ≤ ε, x1 ⪯ y1, y2 ⪯ x2 +� +. +(2.17) +Theorem 2.12 (Compact subsets) Let (X, d) be a metric space and let Ktot(X) be equipped +with the ordered Hausdorff topology. Then a set A ⊂ Ktot(X) is precompact if and only if +(i) ∃ compact C ⊂ X s.t. K ⊂ C ∀K ∈ A +and +(ii) lim +ε→0 sup +K∈A +mε(K) = 0. +(2.18) +Recall the definition of the space D[0,1](X) of cadlag functions f : [0, 1] → X in (2.2). A +cadlag parametrisation of an element K ∈ Ktot(X) is a function γ ∈ D[0,1](X) such that +K = +� +γ(t), γ−(t) : t ∈ [0, 1] +� +and +γ(s) ≺ γ(t) ∀0 ≤ s < t ≤ 1, +(2.19) +1Indeed, it is well-known that each separable metric space X is homeomorphic to a subset of a compact metric +space Y. The completion of X in the metric from Y is equal to the closure of X in Y, so unless X is compact, it +is not complete in the metric from Y. +8 + +where γ− denotes the caglad modification of γ. +Clearly, not every element of Ktot(X) has +a cadlag representation. For those that do, the following proposition gives an expression for +the metrics dH and dtot that will later help us make the connection between our definitions +and the classical definitions of the J1 and M1 topologies. Let Λ be the space of all bijections +λ : [0, 1] → [0, 1] and let Λ+ be the subset consisting of all bijections λ that are monotone in the +sense that s ≤ t implies λ(s) ≤ λ(t). Note that each λ ∈ Λ+ is continuous and strictly increasing +with λ(0) = 0 and λ(1) = 1. +Proposition 2.13 (Distance between cadlag curves) Let (X, d) be a metric space, and as- +sume that K1, K2 ∈ Ktot(X) have cadlag parametrisations γ1, γ2, respectively. Then +dH(K1, K2) = inf +λ∈Λ sup +t∈[0,1] +d +� +γ1(t), γ2 +� +λ(t) +�� +, +dtot(K1, K2) = inf +λ∈Λ+ sup +t∈[0,1] +d +� +γ1(t), γ2 +� +λ(t) +�� +. +(2.20) +2.5 +Betweenness +There are great similarities between Skorohod’s J1 and M1 topologies. In fact, it turns out to be +possible to treat them in a unified framework. To this aim, we introduce a natural concept that +we will call “betweenness” and that seems to be new in this context. It seems quite conceivable +it may have been invented in other contexts before, but we have been unable to find a reference. +If X is a set, then we define a betweenness on X to be a function that assigns to each pair x, z +of elements of X a subset ⟨x, z⟩ of X, such that the following axioms hold for all x, y, z ∈ X: +(i) ⟨x, z⟩ = ⟨z, x⟩, +(ii) x ∈ ⟨x, z⟩, +(iii) y ∈ ⟨x, z⟩ ⇒ ⟨x, y⟩ ∩ ⟨y, z⟩ = {y}, +(iv) y ∈ ⟨x, z⟩ ⇒ ⟨x, y⟩ ∪ ⟨y, z⟩ = ⟨x, z⟩. +If y ∈ ⟨x, z⟩, then we say that y lies between x and z. We call ⟨x, z⟩ the segment with endpoints +x and z. The following lemma lists some elementary consequences of the axioms (i)–(iv). +Lemma 2.14 (Elementary properties) Each betweenness satisfies, for each x, y, y′, z ∈ X: +(v) ⟨x, x⟩ = {x}, +(vi) y ∈ ⟨x, z⟩ ⇒ ⟨x, y⟩ ⊂ ⟨x, z⟩, +(vii) x ∈ ⟨y, z⟩ and y ∈ ⟨x, z⟩ ⇒ x = y. +(viii) y, y′ ∈ ⟨x, z⟩ and y′ ∈ ⟨x, y⟩ ⇒ y ∈ ⟨y′, z⟩. +For x, z ∈ X and y, y′ ∈ ⟨x, z⟩, one has +⟨x, y⟩ ⊂ ⟨x, y′⟩ ⇔ y ∈ ⟨x, y′⟩ ⇔ y′ ∈ ⟨y, z⟩ ⇔ ⟨y, z⟩ ⊃ ⟨y′, z⟩. +(2.21) +Setting y ≤x,z y′ if any of these equivalent conditions holds defines a total order on ⟨x, z⟩. +9 + +We next give some examples of betweennesses. For any set X, it is straightforward to check +that setting ⟨x, z⟩ := {x, z} defines a betweenness. We call this the trivial betweenness. If X is +a linear space, then it is easy to see that +⟨x, z⟩ := +� +(1 − p)x + pz : p ∈ [0, 1] +� +(x, z ∈ X) +(2.22) +defines a betweenness on X. We call this the linear betweenness. If (X, ≤) is a totally ordered +space, then one can check that setting +⟨x, z⟩ := +� +y ∈ X : x ≤ y ≤ z or z ≤ y ≤ x +� +(x, z ∈ X) +(2.23) +defines a betweenness on X. We call this the order betweenness. If (X, d) is a metric space, +then we recall that a geodesic in (X, d) is a subset Γ of X that is isometric to a compact real +interval, i.e., there exists an isometry γ : [s, u] → X (with s, u ∈ R, s ≤ u) such that Γ is the +image of [s, u] under γ. Clearly, γ is uniquely determined by Γ up to translations and mirror +images of the interval [s, u]. The points γ(s), γ(u) are called the endpoints of the geodesic Γ. We +say that a metric space has unique geodesics if for each x, z ∈ X, there exists a unique geodesic +Γ with endpoints x, z. We call the betweenness defined in the following lemma the geodesic +betweennesss. +Lemma 2.15 (Geodesic betweenness) Let (X, d) be a metric space with unique geodesics. +Then letting ⟨x, z⟩ denote the unique geodesic with endpoints x, z defines a betweenness on X. +As an example of spaces without a linear structure where Lemma 2.15 is applicable we +mention real-trees [DT96]. We say that a betweenness on a metrisable space X is generated by +an interpolation function if there exists a continuous function ϕ : X 2 × [0, 1] → X that satisfies +ϕ(x, z, 0) = x, ϕ(x, z, 1) = z, and +⟨x, z⟩ = +� +ϕ(x, z, p) : p ∈ [0, 1] +� +(x, z ∈ X). +(2.24) +A metric space is called proper if for each x ∈ X and r ≥ 0, the closed ball {y ∈ X : d(x, y) ≤ r} +is a compact subset of X. We do not know if the properness assumption in the following lemma +is needed, but it is certainly sufficient. +Lemma 2.16 (Interpolation functions) If X is a normed linear space, then the linear be- +tweenness is generated by an interpolation function. If X is a proper metric space with unique +geodesics, then the same is true for the geodesic betweenness. +If X is a metrisable space, then we say that a betweenness on X is compatible with the +topology if ⟨x, z⟩ is compact for each x, z ∈ X, and the map X 2 ∋ (x, z) �→ ⟨x, z⟩ ∈ K+(X) is +continuous with respect to the product topology on X 2 and the Hausdorff topology on K+(X). +Lemma 2.17 (Compatible betweennesses) If X is a metrisable space, then the trivial be- +tweenness is compatible with the topology. The same is true for any betweenness that is generated +by an interpolation function. If X is a closed subset of R, then the order betweenness on X is +compatible with the topology. +2.6 +Squeezed space +We will need to view graphs of functions as compact sets. This will require us to compactify +the real time axis by adding points at ±∞. At the same time, we want to equip space-time +X × R with a metric that cares less about the spatial distance between two points if their time +10 + +coordinates are very large. In the special case when X = R, such a topology has been introduced +in [FINR04, formula (3.4)]. Here, we generalise this to X being any metrisable space. +Let (X, d) be a metric space and let ∗ be a point not contained in X. Then we call +R(X) := (X × R) ∪ +� +(∗, −∞), (∗, ∞) +� +(2.25) +the squeezed space associated with X. Let dR be a metric that generates the topology on the +extended real line R and let φ : R → [0, ∞) be a continuous function such that φ(±∞) = 0 and +φ(t) > 0 for all t ∈ R. We define dsqz : R(X)2 → [0, ∞) by +dsqz +� +(x, s), (y, t) +� +:= +� +φ(s) ∧ φ(t) +�� +d(x, y) ∧ 1 +� ++ +��φ(s) − φ(t) +�� + dR(s, t), +(2.26) +where naturally the first term is zero if (x, s) or (y, t) are elements of +� +(∗, −∞), (∗, ∞) +� +(even +though d(x, y) is not defined in this case). +Lemma 2.18 (Squeezed space) Let (X, d) be a metric space. Then dsqz is a metric on R(X). +One has dsqz +� +(xn, tn), (x, t) +� +→ 0 if and only if: +(i) tn → t, +(ii) if t ∈ R, then xn → x. +Usually, we will only be interested in R(X) as a topological space. The conditions (i) and +(ii) show that the topology on R(X) depends only on the topology on X and not on the choice +of the metric d on X, the metric dR on R, and the function φ. Condition (ii) is trivially satisfied +if tn → −∞ or → +∞, i.e., we have that (xn, tn) → (∗, ±∞) if and only if tn → ±∞, with no +conditions on the sequence xn. The squeezed space R(R) plays an important role in the theory +of the Brownian web, see [SSS17, Figure 6.2]. +We need some elementary properties of squeezed space. The following lemma shows that +R(X) is Polish if X is. +Lemma 2.19 (Preservation of Polishness) +(a) If (X, d) is separable, then so is (R(X), dsqz). +(b) If (X, d) is complete, then so is (R(X), dsqz). +The following lemma identifies the compact subsets of R(X). In particular, the lemma shows +that R(X) is compact if X is compact. +Lemma 2.20 (Compactness criterion) A set A ⊂ R(X) is precompact if and only if for each +T < ∞, there exists a compact set K ⊂ X such that {x ∈ X : ∃t ∈ [−T, T] s.t. (x, t) ∈ A} ⊂ K. +3 +Topologies on path space +3.1 +Path space +For any set I ⊂ R, we let Is denote the subset of the split real line defined as Is := +� +t−, t+ : +t ∈ I}. Let X be a metrisable space. By definition, a path in X is an object that consists of +two parts: a closed subset I(π) ⊂ R (possibly empty) and a continuous function π : Is(π) → X. +The path with I(π) = ∅ is called the trivial path. We usually denote a path simply by π, which +includes both the function and its domain. We let Π(X) denote the space of all paths in X and +let +Πc(X) := +� +π ∈ Π(X) : π(t−) = π(t+) ∀t ∈ I(π) +� +(3.1) +11 + +denote the subspace consisting of paths without jumps. For π ∈ Πc(X) and t ∈ I(π) we simply +write π(t) := π(t−) = π(t+). Then π : I(π) → X is a continuous function. For this reason, we +call Πc(X) the space of continuous paths, even though using the split real line, we can also view +paths with jumps as continuous functions. We call +σπ := inf I(π) +and +τπ := sup I(π) +(3.2) +the starting time and final time of a path π, respectively. By convention, σπ = ∞ and τπ = −∞ +for the trivial path π. We let +Π|(X) := +� +π ∈ Π(X) : t ∈ I(π) ∀s, u ∈ I(π) and t ∈ R s.t. s < t < u +� +, +Π↑(X) := +� +π ∈ Π(X) : t ∈ I(π) ∀s ∈ I(π) and t ∈ R s.t. s < t +� +, +Π↓(X) := +� +π ∈ Π(X) : t ∈ I(π) ∀u ∈ I(π) and t ∈ R s.t. t < u +� +(3.3) +denote the sets of paths π for which I(π) is an interval, an interval that is unbounded from +above, and an interval that is unbounded from below, respectively. Note that all these sets +contain the trivial path. We also set +Π↕(X) := +� +π ∈ Π(X) : I(π) = R +� +, +(3.4) +which is Π↑(X) ∩ Π↓(X) minus the trivial path. We write Π| +c(X) := Π|(X) ∩ Πc etc. +By definition, the closed graph of a path π ∈ Π(X) is the set G(π) ⊂ R(X) defined as +G(π) := +� +(x, t) : t ∈ I(π), x ∈ {π(t−), π(t+)} +� +∪ +� +(∗, −∞), (∗, +∞) +� +. +(3.5) +Note that G(π) is nonempty, since we always add the points (∗, ±∞), even for the trivial path. +If X is equipped with a betweenness (see Subsection 2.5), then we define the filled-in graph2 of +a path π ∈ Π(X) as +Gf(π) := +� +(x, t) : t ∈ I(π), x ∈ ⟨π(t−), π(t+)⟩ +� +∪ +� +(∗, −∞), (∗, +∞) +� +. +(3.6) +Note that for the trivial betweenness, the filled-in and closed graphs coincide. This will allow +us to treat Skorohod’s J1 and M1 topologies in a unified framework. The filled-in graph Gf(π) +is naturally equipped with a total order, which is defined by setting (x, s) ⪯ (y, t) if either s < t +and x, y are arbitrary, or s = t and x ≤π(t−),π(t+) y, where ≤π(t−),π(t+) is the total order on the +segment ⟨π(t−), π(t+)⟩ defined in Lemma 2.14. Informally, the total order ⪯ corresponds to the +direction of time. Recall the definitions of Ktot and R(X) from Subsections 2.4 and 2.6. The +following lemma says that we can view Gf(π) as an element of the space Ktot(R(X)). +Lemma 3.1 (Filled-in graphs) Assume that X is a metrisable space that is equipped with a +betweenness that is compatible with the topology. Then for any path π, the filled-in graph Gf(π) +is a compact subset of the squeezed space R(X), and the total order ⪯ is compatible with the +induced topology on Gf(π). +A path π is uniquely determined by the totally ordered compact set +� +Gf(π), ⪯ +� +. If π ∈ Πc(X) +or if I(π) does not contain any isolated points, then π is uniquely determined by Gf(π) as a set, +but if t is an isolated point of I(π) and π(t−) ̸= π(t+), then one needs the order ⪯ to find out +which of the two endpoints of the segment ⟨π(t−), π(t+)⟩ is π(t−) and which is π(t+). The +following lemma gives another indication why it may be useful to view filled-in graphs as totally +ordered sets. +2Except for the points at infinity, this is what Whitt [Whi02] calls the completed graph. +12 + +Lemma 3.2 (Characterisation of graphs) Let X be a metrisable space that is equipped with +a betweennesss that is compatible with the topology. +Assume that (G, ⪯) ∈ Ktot(R(X)) and +(∗, ±∞) ∈ G. Then (G, ⪯) is the filled-in graph of a path π ∈ Π(X) if and only if the following +conditions are satisfied. +(i) For each t ∈ R and (x1, t), (x2, t), (x3, t) ∈ G with (x1, t) ⪯ (x2, t) ⪯ (x3, t), one has +x2 ∈ ⟨x1, x2⟩. +(ii) (x, s) ⪯ (y, t) for all (x, s), (y, t) ∈ G such that s < t. +Applying Lemma 3.2 to the trivial betweenness, it is easy to see that a nonempty compact +set G ⊂ R(X) with (∗, ±∞) ∈ G is the closed graph of a path if and only if it is possible to +equip G with a total order that is compatible with the topology, such that it satisfies (ii) above +and +(i)’ For each t ∈ R, the set {x ∈ X : (x, t) ∈ G} has at most two elements. +The total order ⪯ is essential here. To see this, assume that x, y ∈ X satisfy x ̸= y, and let +G := {x} × [−1, 1] ∪ {(y, 0)}. Then G is not the closed graph of a path π ∈ Π(X), while it +satisfies condition (i)’. However, it is not possible to equip G with a total order ⪯ as in (ii) so +that ⪯ is compatible with the topology. +3.2 +Metrics on path space +Let (X, d) be a metric space that is equipped with a betweennesss that is compatible with the +topology and let Π(X) be the path space defined in Section 3.1. We view the filled-in graph +Gf(π) of a path π as an element of the space Ktot(R(X)) of totally ordered compact subsets +of the squeezed space R(X) defined in Section 2.6. Let dsqz be any metric that generates the +topology on R(X), and let dpart and dtot be the metrics on Ktot(R(X)) defined in terms of dsqz +as in (2.13) and (2.15). Since a path is uniquely characterised by its filled-in graph (viewed as +an element of Ktot(R(X))), setting +dS +part(π1, π2) := dpart +� +Gf(π1), Gf(π2) +� +and +dS +tot(π1, π2) := dtot +� +Gf(π1), Gf(π2) +� +(3.7) +� +π1, π2 ∈ Π(X) +� +defines two metrics on the path space Π(X), that in view of Theorem 2.10 +generate the same topology. Letting dH denote the Hausdorff metric on K+(R(X)) associated +with the metric dsqz on R(X), we moreover define a pseudometric on Π(X) by setting +dH(π1, π2) := dH +� +Gf(π1), Gf(π2) +� +. +(3.8) +Restricted to the space Πc(X) of continuous paths, this is a metric. +We call the topology on the path space Π(X) generated by the metrics dS +part and dS +tot the +Skorohod topology associated with the given betweenness. In particular, we define the J1 topol- +ogy to be the Skorohod topology associated with the trivial betweenness. In the special case +when X = R, we define the M1 topology to be the Skorohod topology associated with the linear +betweenness. More generally, Skorohod topologies associated with a betweenness that is gener- +ated by an interpolation function may naturally be viewed as generalisations of the classical M1 +topology. In view of Theorem 2.10 and Lemma 2.18, the definition of a Skorohod topology only +depends on the topology on X and on the choice of the betweenness, and not on the precise +choice of the metrics on X and R(X). The following proposition says that Skorohod topologies +are Polish. +13 + +Proposition 3.3 (Skorohod topologies are Polish) If X is a Polish space, then so is Π(X), +equipped with the Skorohod topology, for any choice of the betweennesss that is compatible with +the topology. +Recall from (3.1) that Πc(X) denotes the space of continuous paths and that dH is a metric +on Πc(X). Since paths in Πc(X) make no jumps, the definitions of dS +part, dS +tot, and dH restricted to +Πc(X) do not depend on the choice of the betweenness. As a result of the following proposition, +all these metrics generate the same topology on Πc(X), so we simply call the resulting topology +the topology on Πc(X). The space Π↑ +c(R) of half-infinite paths with values in R, equipped with +the topology we have just defined, plays an important role in the theory of the Brownian web +and net [SSS14, Subsection 6.2.1]. +Proposition 3.4 (Space of continuous paths) For paths πn ∈ Π(X) and π ∈ Πc(X), the +following statements are equivalent: +(i) dS +part(πn, π) −→ +n→∞ 0, +(ii) dS +tot(πn, π) −→ +n→∞ 0, +(iii) dH(πn, π) −→ +n→∞ 0. +In particular, these metrics all generate the same topology on Πc(X). If X is a Polish space, +then so is Πc(X), equipped with this topology. +The final statement of the following lemma reveals a special property of the J1 topology that +does not hold for general Skorohod topologies. We note that it is not hard to check that Πc(X), +contrary to Π| +c(X), is in general not closed in the J1 topology. +Lemma 3.5 (Closed subspaces) Let X be a metrisable space that is equipped with a between- +ness that is compatible with the topology. Then Π|(X), Π↑(X), and Π↓(X) are closed subsets of +Π(X), equipped with the Skorohod topology. If the betweenness is the trivial betweenness, then +also Π| +c(X) is a closed subset of Π(X). +3.3 +Compactness criteria +In this subsection, we give criteria for compactness in the spaces Π(X) and Πc(X). These criteria +are similar to well-known results for spaces of functions defined on a fixed domain. Let (X, d) +be a metric space. We say that a set A ⊂ Π(X) satisfies the compact containment condition if +∀T < ∞ ∃ compact C ⊂ X s.t. π(t±) ∈ C ∀π ∈ A and t ∈ I(π) ∩ [−T, T]. +(3.9) +For each 0 < T < ∞ and δ > 0, we define the (traditional) modulus of continuity of a path +π ∈ Πc(X) as +mT,δ(π) := sup +� +d +� +π(t1), π(t2) +� +: t1, t2 ∈ I(π), −T ≤ t1 < t2 ≤ T, t2 − t1 ≤ δ +� +. +(3.10) +We say that a set A ⊂ Πc(X) is equicontinuous if +lim +δ→0 sup +π∈A +mT,δ(π) = 0 +∀T < ∞. +(3.11) +The following theorem generalises the classical Arzela-Ascoli theorem to sets of functions that +are not necessarily all defined on the same domain, which moreover does not need to be an +interval. +Theorem 3.6 (Arzela-Ascoli) Let X be a metric space. Then a set A ⊂ Πc(X) is precompact +if and only if it is equicontinuous and satisfies the compact containment condition. +14 + +For paths with jumps, it is possible to give a very similar compactness criterion. Assume that +X is equipped with a betweenness that is compatible with the topology. For each 0 < T < ∞ +and δ > 0, we define the Skorohod modulus of continuity as +mS +T,δ(π) := sup +� +d +� +π(τ2), ⟨π(τ1), π(τ3)⟩ +� +: τ1, τ2, τ3 ∈ Is(π), τ1 ≤ τ2 ≤ τ2, +−T ≤ τ 1, τ 3 ≤ T, τ 3 − τ 1 ≤ δ +� +, +(3.12) +where as before d(x, A) denotes the distance of a point x to a set A and ⟨x, y⟩ is the segment +with endpoints x and y. We say that a set A ⊂ Π(X) is Skorohod-equicontinuous if +lim +δ→0 sup +π∈A +mS +T,δ(π) = 0 +∀T < ∞. +(3.13) +Specialising these definitions to the trivial betweenness, for which ⟨π(τ1), π(τ3)⟩ = {π(τ1), π(τ3)}, +yields the definitions of the J1-modulus of continuity and J1-equicontinuity. If X = R, equip- +ped with the linear betweenness, then we speak of the M1-modulus of continuity and M1- +equicontinuity. +Theorem 3.7 (Compactness criterion) Let (X, d) be a metric space that is equipped with +a betweenness that is compatible with the topology. +Then a set A ⊂ Π(X) is precompact in +the Skorohod topology if and only if it is Skorohod-equicontinuous and satisfies the compact +containment condition. +3.4 +Paths on fixed domains +Let X be a metrisable space that is equipped with a betweenness that is compatible with the +topology. Let I be a closed real interval of positive length, let int(I) denote its interior and let +∂I := I\int(I) denote its boundary. Let DI(X) be the set of cadlag functions f : I → X defined +in (2.2). We have seen in Lemma 2.4 that we may identify DI(X) with the set of paths +� +π ∈ Π(X) : I(π) = I, π(t−) = π(t+) if t ∈ ∂I +� +. +(3.14) +In this identification, dS +part, dS +tot, and dH are metrics3 on DI(X). +We have already seen that +dS +part and dS +tot generate the same topology on the larger space Π(X) and hence the same is true +on DI(X). If X is equipped with the trivial betweenness, then we call the topology on DI(X) +generated by the metrics dS +part and dS +tot the J1 topology, and we call the topology generated by +dH the J2 topology. If X = R, equipped with the linear betweenness, then we call these the M1 +topology and M2 topology, respectively. +Skorohod [Sko56] only considered compact time intervals. It is easy to see that his definition +of the M2 topology [Sko56, Def 2.2.6] coincides with our definition. For the J1, J2, and M1 +topologies, the equivalence of [Sko56, Defs 2.2.2, 2.2.3, and 2.2.4] with our definitions follows +from Proposition 2.13. It is interesting to note that all previous treatments of the Skorohod +topology seem to have been based on variants of the metric dS +tot, while te fact that the metric +dS +part generates the same topology seems to have been overlooked. +Skorohod [Sko56] did not consider unbounded time intervals but other authors such as [EK86, +Whi02] have done so. To see that our definitions agree with their definitions, one can use the +following simple lemma. We note that for the Skorohod topologies, the restriction map that +restricts a function to a smaller time interval is in general not a continuous map, which is why +in (3.15) we have to restrict ourselves to continuity points of the limit function. +3For the metric dH, the assumption that I has positive length is essential, since otherwise this would only be +a pseudometric. +15 + +Lemma 3.8 (Convergence of restricted functions) Let (X, d) be a metric space that is +equipped with a betweenness that is compatible with the topology. Let g +�� +[0,t] denote the restriction +of a function g to the interval [0, t]. Then for all fn, f ∈ D[0,∞)(X), one has +dH(fn, f) −→ +n→∞ 0 +⇔ +dH� +fn +�� +[0,t], f +�� +[0,t] +� +−→ +n→∞ 0 +∀t > 0 s.t. f(t−) = f(t), +dS +tot(fn, f) −→ +n→∞ 0 +⇔ +dS +tot +� +fn +�� +[0,t], f +�� +[0,t] +� +−→ +n→∞ 0 +∀t > 0 s.t. f(t−) = f(t). +(3.15) +It is not hard to see that for f ∈ DI(X), the Skorohod modulus of continuity defined in +(3.12) can alternatively be written as +mS +T,δ(f) = sup +� +d +� +f(t2), ⟨f(t1), f(t3)⟩ +� +: t1, t2, t3 ∈ I +−T ≤ t1 < t2 < t3 ≤ T, t3 − t1 ≤ δ +� +. +(3.16) +Moreover, a set F ⊂ DI(X) satisfies the compact containment condition if and only if +∀T < ∞ ∃ compact C ⊂ X s.t. f(t) ∈ C ∀f ∈ F and t ∈ I. +(3.17) +In other words, these last two formulas say that in (3.9) and (3.12), it suffices to consider π(t+) +only. As a straightforward application of Theorem 3.7, we obtain the following. +Theorem 3.9 (Compactness criterion) Let (X, d) be a metric space that is equipped with a +betweenness that is compatible with the topology and let I be a closed real interval of positive +length. Then a set F ⊂ DI(X) is precompact in the Skorohod topology if and only if: +(i) the compact containment condition holds, +(ii) lim +δ→0 sup +f∈F +mS +T,δ(f) = 0 for all T < ∞, +(iii) lim +δ→0 sup +f∈F +sup +� +d +� +f(s), f(t) +� +: s ∈ I, |s − t| ≤ δ +� += 0 for all t ∈ ∂I. +Note that compared to Theorem 3.7, we need the extra condition (iii) to guarantee that a +sequence of functions in F cannot converge to a function with a discontinuity at a time t ∈ ∂I. If +in (3.14) we drop the condition that π(t−) = π(t+) for t ∈ ∂I, then condition (iii) of Theorem 3.9 +can be dropped. For the J1 topology on D[0,1], Theorem 3.9 was first proved by Kolmogorov +in [Kol56, Thm IV]. The analogue statement for the M1 topology was proved by Skorohod in +[Sko56, 2.7.3]. +3.5 +Interpolation +It often happens that a sequence of functions that are defined on a countable subset of R converge +to a limit that is defined on a subinterval of R. In such situations, to formulate what convergence +means, it is common practise to interpolate the approximating functions, so that all functions +are defined on the same domain. With the use of path space, one can compare functions that +are defined on different domains. In the present subsection, we show that in such situations, for +the J1 topology, there is no need to interpolate. For the M1 topology, on the other hand, it still +makes sense to interpolate. +Let ˆI denote the convex hull of a closed set I ⊂ R. Fix π ∈ Πc(X) and for each t ∈ ˆI(π)\I(π), +let +tl := sup{s ∈ I : s < t} +and +tr := inf{s ∈ I : s > t}, +(3.18) +16 + +where the subscripts l and r stand for “left” and “right”. Then we can uniquely define interpolated +paths πl ∈ DˆI(π)(X) and πr ∈ D− +ˆI(π)(X) by +πl(t+) := +� π(t) +if t ∈ I(π), +π(tl) +if t ∈ ˆI(π)\I(π), +πr(t−) := +� π(t) +if t ∈ I(π), +π(tr) +if t ∈ ˆI(π)\I(π). +(3.19) +As in (3.14), we can identify DˆI(π)(X) and D− +ˆI(π)(X) with subsets of Π(X) and hence view πl +and πr as paths. Let the metric on the squeezed space R(X) be defined as in (2.26) in terms +of a metric dR generating the topology on R and a function φ. +By setting up a monotone +correspondence, it is easy to see that for the J1 topology +dS +tot(π, πl) ≤ εl(π) := +sup +t∈ˆI(π)\I(π) +� +dR(t, tl) + |φ(t) − φ(tl)| +� +, +(3.20) +and similarly for πr (with a similar εr(π)). In particular, when πn ∈ Πc(X) and π ∈ Π|(X) +satisfy εl(πn) → 0, then with respect to the J1 topology one has πn → π if and only if πl +n → π. +In other words, no interpolation is needed. Indeed, convergence of the uninterpolated paths +πn → π gives more information since this also implies εl(πn) → 0. +We next consider Skorohod topologies associated with a betweenness that is generated by +an interpolation function ϕ. In this case, for any π ∈ Πc(X), we can define a continuously +interpolated path πϕ by I(πφ) := ˆI(π) and +πϕ(t+) := +� π(t) +if t ∈ I(π), +ϕ +� +π(tl), π(tr), p(t) +� +if t ∈ ˆI(π)\I(π), +where +p(t) := t − tl +tr − tl +. +(3.21) +By setting up a monotone correspondence, it is easy to see that +dS +tot(πϕ, πl), dS +tot(πϕ, πr) ≤ ε(π) := εl(π) ∨ εr(π). +(3.22) +Thus, when πn ∈ Πc(X) and π ∈ Π|(X) satisfy ε(πn) → 0, then with respect to the M1 topology, +the conditions πϕ +n → π, πl +n → π, and πr +n → π are all equivalent. In other words, for the M1 +topology, it makes sense to interpolate, but it does not matter if we interpolate in a continuous +way or in a piecewise constant manner. +3.6 +Open problems +By definition, a laglad function (from the French “limite `a gauche, limite `a droit”) is a function +that has both left- and right- limits in each point, but whose value in a point does not need to +be equal to either the left or right limit at that point. Whitt [Whi02, Chapter 15] introduces +a topology on spaces of laglad functions. It seems it should be possible to develop the theory +of laglad functions very much in parallel to the theory of cadlag functions, except that instead +of splitting each point of the real line into two points, as we did in the split real line, one now +needs a topological space where each point of the real line is replaced by three points. A slight +complication is that the closed graph of a laglad function cannot always be equipped with a +total order that is compatible with the topology, as pointed out below Lemma 3.2. However, +it seems likely this can be overcome and it should be possible to prove a compactnes criterion +similar to Theorem 3.7, this time involving a modulus of continuity that compares the function +values at four consecutive times, rather than three as for the Skorohod modulus of continuity. +Let X be a metrisable space. Recall from Subsection 2.4 that Kpart(X) denotes the space of +all compact subsets K ⊂ X that are equipped with a partial order ⪯ that is compatible with +17 + +the (induced) topology on K. For each finite partially ordered set (S, ≤), let KS denote the +space of all monotone functions f : S → K, i.e., functions such that i ≤ j implies f(i) ⪯ f(j). +Then K⟨2⟩, defined in (2.10), is the same as KS where S is the totally ordered space {1, 2}. +In Subsection 4.3 below, we more generally define K⟨m⟩ := KS with S the totally ordeed set +{1, . . . , m}. For each partially ordered set (S, ≤), similar to (2.13), we can define a pseudometric +dS by +dS(K1, K2) := dH(KS +1 , KS +2 ), +(3.23) +where dH is the Hausdorff metric on the product space X S, equipped with a product metric +as in (2.11). In Subsection 2.4, we show that the metrics dS with S a finite totally ordered +set with at least two elements all generate the same topology on the space Ktot(X) of totally +ordered compact sets. It seems this result does not generalise to partially ordered sets. A natural +idea is therefore to equip the larger space Kpart(X) with a topology such that Kn → K in the +topology on Kpart(X) if and only if dS(Kn, K) → 0 for every finite partially ordered set. Such a +topology can be generated by a metric, for example by setting d(K1, K2) := � +S rSdS(K1, K2) +where rS are positive weights such that the sum over all partially ordered finite sets � +S rS is +finite. It then seems interesting to study the associated “ordered” Gromov-Hausdorff distance +between two partially ordered sets K1, K2, which is the infimum of d(K1, K2) over all isometric +embeddings of K1 and K2 into a common metric space X. +As a final, minor open problem, we ask whether the properness assumption in Lemma 2.16 +can be removed. This may be known; we have just not managed to find this in the literature. +3.7 +Outline of the proofs +The results from Section 2 are proved in Section 4 and the from Section 3 are proved in Sec- +tion 5. More precisely, Lemmas 2.1 and 2.2, Proposition 2.3, and Lemma 2.4 are proved in +Subsection 4.1. Lemmas 2.5 and 2.9 are proved in Subsection 4.2. We cited Lemmas 2.6–2.8 +from [SSS14, Appendix B], so these don’t need proofs. +Theorem 2.10 is proved in Subsec- +tion 4.4. Proposition 2.11 is proved in Subsection 4.5. Theorem 2.12 is proved in Subsection 4.6. +Proposition 2.13 is proved in Subsection 4.7. Lemmas 2.14, 2.15, 2.16, and 2.17 are proved in +Subsection 4.8.Lemmas 2.18, 2.19, and 2.20 are proved in Subsection 4.9. +Lemmas 3.1 and 3.2 are proved in Subsection 5.1. Propositions 3.3 and 3.4 are proved in +Subsection 5.2, which also contains the proof of Lemma 3.5. Theorems 3.6 and 3.7 are proved +in Subsection 5.3. Lemma 3.8 and Theorem 3.9 are proved in Subsection 5.4. +4 +Proofs of the preliminary results +4.1 +The split real line +In this subsection, we prove Lemmas 2.1 and 2.2, Proposition 2.3, and Lemma 2.4, as well as one +more lemma that will be needed in what follows. We recall some basic definitions from topology. +A topology on a set X is a collection O of subsets of X that are called open and that have the +properties that ∅, X ∈ O and O is closed under finite intersections and arbitrary unions. If Y is +a subset of X, then the induced topology is defined as {O ∩Y : O ∈ O}. A basis for the topology +on X is a subset O′ ⊂ O such that each element of O can be written as the union of elements +of O′. The set Vx of neighbourhoods of a point x ∈ X is Vx := {V ⊂ X : x ∈ O ⊂ V for some +O ∈ O}. A fundamental system of neighbourhoods is a set V′ +x ⊂ Vx such that ∀V ∈ Vx ∃V ′ ∈ V′ +x +s.t. V ′ ⊂ V . +A Hausdorff topology is a topology that has the Hausdorff property, i.e., for all x1, x2 ∈ X +with x1 ̸= x2 there exist disjoint O1, O2 ∈ O such that x1 ∈ O1, x2 ∈ O2. +A topology is +18 + +first countable if each point has a countable fundamental system of neighbourhoods and second +countable if there exists a countable basis for the topology. A sequence converges to a limit, +denoted xn → x, if for each V ∈ Vx, there exists an m such that xn ∈ V for all n ≥ m. It suffices +to check this for a fundamental system of neighbourhoods. In a Hausdorff space, a sequence can +have at most one limit. +A set is closed if its complement is open and sequentially closed if it contains the limits of +all convergent sequences that lie inside it; in first countable spaces, the concepts are equivalent. +The closure of a set is the smallest closed set that contains it and a dense set is a set whose +closure is the whole space. A topological space X is separable if it contains a countable dense set +and connected if ∅, X are the only sets that are both open and closed. A set C ⊂ X is compact +if each covering with open sets has a finite subcover and sequentially compact if each sequence +in C has a subsequence that converges to a limit in C; in second countable spaces, the concepts +are equivalent. A metric defines a topology in the usual way; a topology that is generated by a +metric is called metrisable. +Proof of Lemma 2.1 By symmetry, it suffices to prove (i). +By definition, a basis for the +topology is formed by all intervals of the form (σ, ρ) with σ, ρ ∈ Rs. +If t+ ∈ (σ, ρ), then +σ < t+ < ρ and hence (t−, u+) ⊂ (σ, ρ) for some u > t. It follows that the sets of the form +(t−, u+) = [t+, u−] with u ∈ {t + n−1 : n ≥ 1} form a fundamental system of neighbourhoods +of t, which is easily seen to imply the claim. +Proof of Lemma 2.2 It is easy to see that Rs has the Hausdorff property. +In the proof +of Lemma 2.1, we have already seen that each point has a countable fundamental system of +neighbourhoods, so Rs is first countable. On the other hand, each basis of the topology must +for each t ∈ R contain an open set O such that t ∈ O ⊂ (t−, (t + 1)+) = [t+, (t + 1)−]. These +open sets are all distinct, so Rs is not second countable. By Lemma 2.1, the set {t+ : t ∈ Q} +is dense so Rs is separable. Since in metric spaces, separability implies second countability, we +conclude that Rs is not metrisable. Since for each t ∈ R, we can write Rs as the union of two +disjoint closed sets as Rs = (−∞, t−] ∪ [t+, ∞), we see that Rs is totally disconnected. +The next lemma prepares for the proof of Proposition 2.3. Even though Rs is not second +countable, it has a property that is almost as good. +Lemma 4.1 (Strong Lindel¨of property) Every open cover of a subset of Rs has a countable +subcover. +Proof This is proved in [AU29], but since the latter reference is not readily available to everyone, +including the present authors, we provide our own proof. The Sorgenfrey line is the set of real +numbers equipped with the lower limit topology that is generated by intervals of the form [a, b) +[SS95]. Similarly, the upper limit topology on R is generated by intervals of the form (a, b]. One +can check that the topology on Rs induces the lower limit topology on its subspace {t+ : t ∈ R} +and the upper limit topology on its subspace {t− : t ∈ R}. In view of this, it suffices to prove +that the Sorgenfrey line has the strong Lindel¨of property. This is well-known [Sor47] but for +completeness we provide a proof. Since sets of the form [a, b) form a basis for the topology, it +suffices to prove that if A ⊂ R and I is a collection of intervals of the form [a, b) that covers A, +then a countable subset of I covers A. +Fix n ≥ 1, let I′ denote the subset of I consisting of all intervals [a, b) ∈ I with b − a ≥ 2/n, +and let A′ be the union of all elements of I′. It suffices to show that for each n ≥ 1, a countable +subset of I′ already covers A′. Fix k ∈ Z, let A′′ := A′ ∩ [k/n, (k + 1)/n], and let I′′ denote +the subset of I′ consisting of all intervals [a, b) ∈ I′ with [a, b) ∩ [k/n, (k + 1)/n] ̸= ∅. It suffices +to show that for each k ∈ Z, a countable subset of I′′ already covers A′′. Since b − a ≥ 2/n +19 + +for all [b, a) ∈ I′′, each element of I′′ must contain either k/n or (k + 1)/n, or both. If I′′ +contains an element [a, b) that contains both k/n and (k + 1)/n we are done. Otherwise, we can +select countable subsets of {[a, b) ∈ I′′ : k/n ∈ [a, b)} and {[a, b) ∈ I′′ : (k + 1)/n ∈ [a, b)} that +cover A′′. +We note that the product space Rs × Rs, equipped with the product topology, does not have +the strong Lindel¨of property. Indeed, the collection of open sets +� +[t+, ∞−) × [−t+, ∞−) : t ∈ R +� +(4.1) +covers the set {(t+, −t+) : t ∈ R}, but no countable subset of (4.1) has this property. The +set {(s+, t+) : s, t ∈ R} with the induced topology from R2 +s is a well-known counterexample in +topology, known as the Sorgenfrey plane. +Proof of Proposition 2.3 We first prove the statement for Rs. In any first countable space, +being sequentially compact is equivalent to being countably compact, which means that every +countable open covering has a finite subcovering. By the strong Lindel¨of property, a subset of +Rs is compact if and only if it is countably compact, proving that (i) and (ii) are equivalent. +Since the map τ �→ τ, that assigns to a split real number τ its real part, is continuous, and +since the continuous image of a compact set is compact, (i) implies that C := {τ : τ ∈ C} is +closed and bounded. Since moreover a compact subset of a Hausdorff space is closed, (i) implies +(iii). +Property (iii) implies that each sequence τn ∈ C has a subsequence τ ′ +n such that τ ′ +n converges +to a limit t ∈ C. The τ ′ +n must then contain a further subsequence τ ′′ +n such that one of the following +three cases occurs: 1. τ ′′ +n < t for all n, 2. τ ′′ +n > t for all n, or 3. τ ′′ +n is constant. In either case, the +fact that C is closed implies that τ ′′ +n converges to a limit in C, proving the implication (iii)⇒(ii). +This completes the proof for Rs. +We saw before that the strong Lindel¨of property does not hold for Rd +s in dimensions d ≥ +2, so to prove the statement for these spaces we have to proceed differently. +Property (i) +implies countable compactness which by the fact that Rs and hence also Rd +s are first countable +is equivalent to (ii). The continuous image of a countably compact set is countably compact. +Applying this to the coordinate projections and using what we already know for Rs, we see that +(ii) implies that C is bounded. Since, moreover, in any first countable Hausdorff space, being +sequentially compact implies being closed, we see that (ii) implies (iii). By Tychonoff’s theorem +and what we already know for Rs, the set [s−, t+]d is compact for each −∞ < s < t < ∞. Since +a closed subset of a compact set is compact, (iii) implies (i). +Proof of Lemma 2.4 A function f : Iin → X is continuous if and only if +(i) f(τn) → f(t+) for all t+ ∈ Iin and τn ∈ Iin such that τ n → t and τn ≥ t+, +(ii) f(τn) → f(t−) for all t− ∈ Iin and τn ∈ Iin such that τ n → t and τn ≤ t−. +We see from this that for a given f ∈ CIin(X), setting +f ±(t) := f(t±) +if t± ∈ Iin, +f +(t) := f(t−) +if t = sup I < ∞, +f −(t) := f(t+) +if t = inf I > −∞ +(4.2) +defines f + ∈ DI(X) and f − ∈ D− +I (X) such that f − is the left-continuous modification of f + +and f is given by (2.5). +20 + +Conversely, if such f ± are given, then to see that f defined in (2.5) satisfies f ∈ CIin(X), by +symmetry, it suffices to check only condition (i) above. To show that f(τn) → f(t+), it suffices +to prove that each subsequence τ ′ +n contains a further subsequence τ ′′ +n such that f(τ ′′ +n) → f(t+). +In view of this, without loss of generality, we may assume that either s(τn) = + for all n or +s(τn) = − for all n. In the first case, we have f(τn) → f(t+) by the right-continuity of f +, +while in the second case we have f(τn) → f(t+) by the fact that f + is the right-continuous +modification of f −. +The following fact is well-known, but since we need this in what follows, for completeness +we include the proof. +Lemma 4.2 (Countably many discontinuities) Let (X, d) be a metric space, let I be a +closed real interval, and let f ∈ DI(X). Then the set {t ∈ I : f(t−) ̸= f(t)} is countable. +Proof For each ε > 0 and T < ∞, the set J := {t ∈ I ∩ [−T, T] : d +� +f(t−), f(t) +� +≥ ε} must be +finite, since otherwise there exist a strictly increasing or decreasing sequence tn ∈ J, which is +easily seen to contradict the cadlag property. +4.2 +The Hausdorff metric +In this subsection, we prove Lemmas 2.5 and 2.9, which are the only results from Subsection 2.3 +for which we did not give a reference, as well as Lemma 4.3 below that will be needed in what +follows. +Proof of Lemma 2.5 Let R ∈ Cor(K1, K2) and let D := sup(x1,x2)∈R d(x1, x2). +Then +d(x1, K2) ≤ D and d(x2, K1) ≤ D for each x1 ∈ K1, x2 ∈ K2, and hence dH(K1, K2) ≤ D. +On the other hand, by the compactness of K2 and the continuity of the function d(x1, · ), for +each x1 ∈ K1, there exists an x2 ∈ K2 such that d(x1, K2) = d(x1, x2). The same statement +holds with the roles of K1 and K2 interchanged, so setting +R := +� +(x1, x2) ∈ K1 × K2 : d(x1, x2) ∈ {d(x1, K2), d(x2, K1)} +� +(4.3) +defines a correspondence between K1 and K2. By the compactness of K1 and the continuity of the +map d( · , K2), there exists an x′ +1 ∈ K1 such that d(x′ +1, K2) = maxx1∈K1 d(x1, K2), and similarly +there exists an x′′ +2 ∈ K2 such that d(x′′ +2, K1) = maxx2∈K2 d(x2, K1). By our earlier arguments, +there exist x′ +2 ∈ K2 and x′′ +1 ∈ K1 such that d(x′ +1, K2) = d(x′ +1, x′ +2) and d(x′′ +2, K1) = d(x′′ +1, x′′ +2). +Then +dH(K1, K2) = d(x′ +1, x′ +2) ∨ d(x′′ +1, x′′ +2) = +max +(x1,x2)∈R d(x1, x2). +(4.4) +Proof of Lemma 2.9 Imagine that Kn, K ∈ K+(X) satisfy Kn → K. If K is not connected, +then there exist disjoint nonempty compact sets C1, C2 such that K = C1 ∪ C2. +Let ε := +d(C1, C2) = inf{d(x1, x2) : x1 ∈ C1, x2 ∈ C2}. By the compactness of C1 and C2, the infimum +is attained and ε > 0. Let Ui := {x ∈ X : d(x, Ci) ≤ ε/3} (i = 1, 2). Then U1, U2 are disjoint +closed sets. For all n large enough such that dH(Kn, K) ≤ ε/3, one has Kn ⊂ U1 ∪ U2 while +Kn ∩ U1 and Kn ∩ U2 are both nonempty, which proves that Kn is not connected. +We recall that the image of a compact set under a continuous map is compact. In what +follows, we will need the following simple observation. +Lemma 4.3 (Continuous image) Let X, Y be metrisable spaces and let ψ : X → Y be con- +tinuous. If Kn, K ∈ K+(X) satisfy Kn → K, then their images under ψ satisfy ψ(Kn) → ψ(K) +in K+(X). +21 + +Proof This follows easily from Lemma 2.6. Since Kn → K, there exists a compact set C ⊂ X +such that Kn ⊂ C for all n, and now ψ(C) ⊂ Y is a compact set such that ψ(Kn) ⊂ ψ(C) for +all n. By (2.9), it now suffices to check that: +(i) +ψ(K) ⊂ {y ∈ Y : ∃xn ∈ Kn s.t. ψ(xn) → y}, +(ii) +{y ∈ Y : ∃xn ∈ Kn s.t. y is a subsequential limit of (ψ(xn))n∈N} ⊂ ψ(K). +(4.5) +Here (i) follows from (2.9) and the continuity of ψ. +To prove (ii), if ψ(x′ +n) → y for some +subsequence x′ +n, then since Kn ⊂ C for all n there exists a further subsequence x′′ +n such that +x′′ +n → x for some x ∈ X. Then x ∈ K by (2.9) and hence ψ(x) = y by the continuity of ψ. +4.3 +The ordered Hausdorff metric +In this subsection, we study the metrics dpart and dtot defined in (2.13) and (2.15), preparing +for the proofs of Theorem 2.10, Proposition 2.11, and Theorem 2.12, which will be given in the +next subsection. Let (X, d) be a metric space. Generalising the definition in (2.10), for each +m ≥ 1 and K ∈ Kpart(X), we set +K⟨m⟩ := +� +(x1, . . . , xm) ∈ Km : x1 ⪯ · · · ⪯ xm +� +. +(4.6) +It is straightforward to check that K⟨m⟩ is a closed subset of Km and hence a compact subset +of X m. Generalising the definitions in (2.11) and (2.12), we equip X m with the metric +dm� +(x1, . . . , xm), (y1, . . . , ym) +� +:= +m +� +k=1 +d(xk, yk), +(4.7) +and we equip K+(X m) with the associated Hausdorff metric dm +H . Generalising the definition in +(2.13), for each m ≥ 1, we define a function d⟨m⟩ on Kpart(X)2 by +d⟨m⟩(K1, K2) := dm +H (K⟨m⟩ +1 +, K⟨m⟩ +2 +) +� +K1, K2 ∈ Kpart(X) +� +. +(4.8) +In particular, when m ≥ 2, this is a metric on Kpart(X) since (K, ⪯) is uniquely characterised +by K⟨m⟩ for m ≥ 2. On the other hand, d⟨1⟩(K1, K2) is simply the Hausdorff distance between +K1 and K2 as sets, which gives no information about the partial order. The following lemma +describes a simple property of the metric d⟨2⟩. +Lemma 4.4 (Ordered limit) Let X be a metrisable space. Assume that Kn, K ∈ Kpart(X) +satisfy d⟨2⟩(Kn, K) → 0 and that xn, yn ∈ Kn satisfy xn → x, yn → y, and xn ⪯ yn for all n. +Then x, y ∈ K satisfy x ⪯ y. +Proof Since d⟨2⟩(Kn, K) → 0, we have K⟨2⟩ +n +→ K⟨2⟩ and hence by Lemma 2.6 x, y ∈ K⟨2⟩, +which proves that x ⪯ y. +The folling lemma gives a one-sided bound between metrics of the form d⟨m⟩ for different +values of m. +Lemma 4.5 (One-sided bound) One has +d⟨m⟩(K1, K2) ≤ d⟨m+1⟩(K1, K2) +� +m ≥ 1, K1, K2 ∈ Kpart(X) +� +. +(4.9) +22 + +Proof By Lemma 2.5, there exists a correspondence R between K⟨m+1⟩ +1 +and K⟨m+1⟩ +2 +such that +dm+1(x, y) ≤ dm+1 +H +(K⟨m+1⟩ +1 +, K⟨m+1⟩ +2 +) for all (x, y) ∈ R. Let ψ : X m+1 → X denote the projection +ψ(x1, . . . , xm+1) := (x1, . . . , xm). Then (4.7) implies that +dm� +ψ(x), ψ(y) +� +≤ dm+1(x, y) +(x, y ∈ X m+1). +(4.10) +Since ψ(K⟨m+1⟩ +i +) = K⟨m⟩ +i +(i = 1, 2), it follows that R′ := +�� +ψ(x), ψ(y) +� +: (x, y) ∈ R +� +is a +correspondence between K⟨m⟩ +1 +and K⟨m⟩ +2 +such that dm(x′, y′) ≤ dm+1 +H +(K⟨m+1⟩ +1 +, K⟨m+1⟩ +2 +) for all +(x′, y′) ∈ R′. By Lemma 2.5, this proves that +dm +H (K⟨m⟩ +1 +, K⟨m⟩ +2 +) ≤ dm+1 +H +(K⟨m+1⟩ +1 +, K⟨m+1⟩ +2 +), +(4.11) +which in view of (4.8) implies the claim. +The following lemmas show that in general, the metrics d⟨m⟩ for different values of m are +not comparable. +More precisely, the one-sided bound in Lemma 4.5 is not matched by an +opposite inequality of the form d⟨m+1⟩(K1, K2) ≤ Cd⟨m⟩(K1, K2) for any finite constant C, and +convergence in d⟨m⟩ does not imply convergence in d⟨m+1⟩. +Lemma 4.6 (No opposite inequality) +Let X = [0, 1], equipped with the usual distance. +Then for each m ≥ 1 and 0 < ε ≤ 1/4, there exist K1, K2 ∈ Ktot(X) such that d⟨m⟩(K1, K2) ≤ ε +and d⟨m+1⟩(K1, K2) ≥ 1/2. +Proof We choose K1 = {x1, . . . , xm+1} with xk ∈ [0, ε] when k is even and xk ∈ [1 − ε, 1] if k is +odd, and we choose K2 = {y1, . . . , ym+1} with yk ∈ [0, ε] when k is odd and yk ∈ [1 − ε, 1] if k is +even. We equip K1 and K2 with total orders such that x1 ≺ · · · ≺ xm+1 and y1 ≺ · · · ≺ ym+1. +It is easy to see that +d +� +(x1, . . . , xm+1), K⟨m+1⟩ +2 +� +≥ 1/2, +(4.12) +and hence d⟨m+1⟩(K1, K2) ≥ 1/2. On the other hand, it is easy to see that for each (z1, . . . , zm) ∈ +K⟨m⟩ +1 +, there exists a (z′ +1, . . . , z′ +m) ∈ K⟨m⟩ +2 +such that |zk − z′ +k| ≤ ε for all k, and vice versa, so +d⟨m⟩(K1, K2) ≤ ε. +Lemma 4.7 (Different topologies) Let X = [0, 1], equipped with the usual distance. Then +for each m ≥ 1, there exist Kn ∈ Kpart(X) and K ∈ Ktot(X) such that d⟨m⟩(Kn, K) → 0 as +n → ∞ but d⟨m+1⟩(Kn, K) ≥ 1/2 for all n. +Proof It will be convenient to use the notation [m] := {1, . . . , m} (m ≥ 1). +We choose +x1, . . . , xm+1, all different, with xk ∈ [0, 1/4] when k is even and xk ∈ [3/4, 1] if k is odd. +We set K = {x1, . . . , xm+1} and we equip K with a total order by setting x1 ≺ · · · ≺ xm+1. We +choose points +� +xl +k(n) +�l∈[m+1] +k∈[m+1] +(4.13) +in [0, 1], all different, such that xl +k(n) → xk as n → ∞ for all k, l ∈ [m + 1], and we set +Kn := +� +xl +k : k, l ∈ [m + 1], k ̸= l +� +. +(4.14) +We equip Kn with a partial order such that +xl +k ⪯ xl′ +k′ +⇔ +k ⪯ k′ and l = l′. +(4.15) +23 + +Then it is easy to check that d⟨m⟩(Kn, K) → 0 as n → ∞ but d⟨m+1⟩(Kn, K) ≥ 1/2 for all n. +We note that d⟨m⟩(K1, K2) ≤ sup(x1,x2)∈K1×K2 d(x1, x2), which is finite by the continuity of +d and the compactness of K1 × K2. We use this and Lemma 4.5 to define d⟨∞⟩ as the increasing +limit +d⟨∞⟩(K1, K2) := lim +m→∞ d⟨m⟩(K1, K2) +� +K1, K2 ∈ Kpart(X) +� +. +(4.16) +It is straightforward to check that d⟨∞⟩ is a metric on Kpart(X): symmetry and the triangle +inequality follow by taking the limit in the corresponding properties of the metrics d⟨m⟩, and +d⟨∞⟩(K1, K2) = 0 clearly implies d⟨m⟩(K1, K2) = 0 for all m ≥ 1 and hence equality of K1 and +K2 as partially ordered spaces. In the special case that K1 and K2 are totally ordered, the +following proposition identifies d⟨∞⟩(K1, K2) as the metric dtot defined in (2.15). +Proposition 4.8 (Monotone correspondences) Let (X, d) be a metric space. Then one has +d⟨∞⟩(K1, K2) = dtot(K1, K2) for all K1, K2 ∈ Ktot(X). +The proof of Proposition 4.8 uses the following simple lemma. +Lemma 4.9 (Eventually ordered sequences) Let K be a compact metrisable set that is +equipped with a total order that is compatible with the topology. Assume that x ≺ y and that +xn, yn ∈ K satisfy xn → x and yn → y. Then xn ≺ yn for all n sufficiently large. +Proof Since ⪯ is a total order, if the statement is not true, then yn ⪯ xn for infinitely many +n, so we can select a subsequence such that yn ⪯ xn for all n. Taking the limit, using the fact +that the total order that is compatible with the topology, we find that y ⪯ x, which contradicts +x ≺ y. +Proof of Proposition 4.8 We first prove the inequality d⟨∞⟩(K1, K2) ≤ dtot(K1, K2). Let R +be a monotone correspondence between K1 and K2. Let x1, . . . , xm ∈ K1 satisfy x1 ⪯ · · · ⪯ xm. +Then we can choose x′ +1, . . . , x′ +m ∈ K2 such that (xk, x′ +k) ∈ R for all 1 ≤ k ≤ m, and moreover +x′ +k = x′ +k+1 whenever xk = xk+1 (1 ≤ k < m). Since R is monotone and K2 is totally ordered, +we must have x′ +1 ⪯ · · · ⪯ x′ +m. This shows that +dm� +(x1, . . . , xm), K⟨m⟩ +2 +� +≤ +sup +(x,x′)∈R +d(x, x′) +� +(x1, . . . , xm) ∈ K⟨m⟩ +1 +� +. +(4.17) +The same is true with the roles of K1 and K2 interchanged, so we conclude that +d⟨m⟩(K1, K2) = dm +H (K⟨m⟩ +1 +, K⟨m⟩ +2 +) ≤ +sup +(x,x′)∈R +d(x, x′). +(4.18) +Taking the infimum over all monotone correspondences between K1 and K2 and letting m → ∞ +we see that d⟨∞⟩(K1, K2) ≤ dtot(K1, K2). +To prove the opposite inequality, let εn be positive constants, tending to zero. Since K1 is +totally bounded, for each n, we can find an m(n) ≥ 1 and xn +1, . . . , xn +m(n) ∈ K1 such that +d +� +x, {xn +1, . . . , xn +m(n)} +� +≤ εn +∀x ∈ K1. +(4.19) +Since K1 is totally ordered, we can assume without loss of generality that xn +1 ⪯ · · · ⪯ xn +m(n). +Since +dm +H (K⟨m⟩ +1 +, K⟨m⟩ +2 +) = d⟨m⟩(K1, K2) ≤ d⟨∞⟩(K1, K2), +(4.20) +we can find yn +1 , . . . , yn +m(n) ∈ K2 with yn +1 ⪯ · · · ⪯ yn +m(n) such that +d(xn +k, yn +k) ≤ d⟨∞⟩(K1, K2) +(1 ≤ k ≤ m(n)). +(4.21) +24 + +Using the fact that K2 is totally bounded, adding points to {yn +1 , . . . , yn +m(n)} and making m(n) +larger if necessary, we can arrange things so that also +d +� +y, {yn +1 , . . . , yn +m(n)} +� +≤ ε +∀y ∈ K2. +(4.22) +Now using again (4.20) we can add corresponding points in K1 for the new points we have added +to K2 so that (4.21) remains true. Adding points will not spoil (4.19) so we can arrange things +such that (4.19), (4.20), and (4.22) are satisfied simultaneously. +Let Rn ⊂ K1 × K2 be the set +Rn := +� +(xn +k, yn +k) : 1 ≤ k ≤ m(n) +� +. +(4.23) +We claim that Rn is monotone in the sense that +there are no (xn +k, yn +k), (xn +l , xn +l ) ∈ Rn such that xn +k ≺ xn +l and yn +l ≺ yn +k. +(4.24) +Indeed, xn +k ≺ xn +l implies k < l and yn +l ≺ yn +k implies l < k, which is a contradiction. +Since K1 × K2 is compact, by Lemma 2.8, we can select a subsequence such that Rn → R +in the Hausdorff topology on K+(K1 × K2), for some compact set R ⊂ K1 × K2. We claim that +R is a correspondence between K1 and K2. Indeed, by (4.19), for each x ∈ K1, we can choose +k(n) such that xn +k(n) → x. By the compactness of K1 × K2, the sequence (xn +k(n), yn +k(n)) has at +least one cluster point (x, y), and by Lemma 2.6 (x, y) ∈ R. Similarly, for each y ∈ K2 there +exists an x ∈ K1 such that (x, y) ∈ R. +We next claim that R is monotone. Assume that conversely, there exist (x, y), (x′, y′) ∈ +R such that x ≺ x′ and y′ ≺ y. +Then by Lemma 2.6, there exist k(n), k′(n) such that +(xn +k(n), yn +k(n)) → (x, y) and (xn +k′(n), yn +k′(n)) → (x′, y′). By Lemma 4.9, xn +k(n) ≺ yn +k(n) and yn +k′(n) ≺ +xn +k′(n) for all n large enough, which contradicts (4.24). +Taking the limit in (4.21), using Lemma 2.6, we see that +d(x, y) ≤ d⟨∞⟩(K1, K2) +∀(x, y) ∈ R, +(4.25) +and hence by (2.15) dtot(K1, K2) ≤ d⟨∞⟩(K1, K2). +4.4 +The mismatch modulus +In this subsection, we prove Theorem 2.10. Generalising the definition in (2.17) for any K1, K2 ∈ +Ktot(X) and ε > 0, we define the mismatch modulus mε(K1, K2) by +mε(K1, K2) := sup +� +d(x1, y1) ∨ d(x2, y2) : x1, y1 ∈ K1, x2, y2 ∈ K2, +d(x1, x2) ∨ d(y1, y2) ≤ ε, x1 ⪯ y1, y2 ⪯ x2 +� +. +(4.26) +Lemma 4.10 (Convergence of the mismatch modulus) Let X be a metrisable space. As- +sume that Kn, K ∈ Kpart(X) satisfy d⟨2⟩(Kn, K) → 0. Then +mεn(Kn, K) −→ +n→∞ 0 +with +εn := d⟨1⟩(Kn, K). +(4.27) +Proof If (4.27) does not hold, then, by going to a subsequence, we can assume that there +exists a δ > 0 such that mεn(Kn, K) ≥ δ for all n. +It follows that for each n, we can +find x1(n), y1(n) ∈ Kn and x2(n), y2(n) ∈ K with d(x1(n), y1(n)) ∨ d(x2(n), y2(n)) ≥ δ and +d(x1(n), x2(n))∨d(y1(n), y2(n)) ≤ εn such that x1(n) ⪯ y1(n) and y2(n) ⪯ x2(n). By Lemma 4.5, +our assumption d⟨2⟩(Kn, K) → 0 implies εn = d⟨1⟩(Kn, K) → 0 and hence Kn → K. Therefore, +25 + +by Lemma 2.6, there exists a compact set C ⊂ X such that Kn ⊂ C for all n, so by going to a +subsequence, we can assume that x1(n), x2(n), y1(n), y2(n) converge to limits x1, x2, y1, y2 in X. +Since d(x1(n), x2(n))∨d(y1(n), y2(n)) ≤ εn → 0, we have x := x1 = x2 and y := y1 = y2. On the +other hand, our assumption that d(x1(n), y1(n)) ∨ d(y2(n), x2(n)) ≥ δ implies that d(x, y) ≥ δ. +This leads to a contradiction, since by the assumption that d⟨2⟩(Kn, K) → 0 and Lemma 4.4, +x1(n) ⪯ y1(n) and y2(n) ⪯ x2(n) imply x ⪯ y and y ⪯ x and hence x = y. +The following estimate essentially uses that the spaces are totally ordered. +Lemma 4.11 (Estimate in terms of mismatch modulus) Let X be a metrisable space and +let K1, K2 ∈ Ktot(X) satisfy d⟨1⟩(K1, K2) ≤ ε. Then +d⟨m⟩(K1, K2) ≤ mε(K1, K2) + ε +(m ≥ 1). +(4.28) +Proof By symmetry, it suffices to show that for each x1 = (x1 +1, . . . , xm +1 ) ∈ K⟨m⟩ +1 +, there exists +an x2 = (x1 +2, . . . , xm +2 ) ∈ K⟨m⟩ +2 +such that dm(x1, x2) ≤ mε(K1, K2) + ε. In view of (4.7), the +latter means that d(xk +1, xk +2) ≤ mε(K1, K2) + ε for all 1 ≤ k ≤ m. Since d⟨1⟩(K1, K2) ≤ ε, there +exists a z(1) ∈ K2 such that d(x1 +1, z(1)) ≤ ε. We set xi +2 = z(1) for all 1 ≤ i < I(1), where +I(1) := inf{i > 1 : d(xi +1, z(1)) > mε(K1, K2) + ε}. Using again that d⟨1⟩(K1, K2) ≤ ε, there +exists a z(2) ∈ K2 such that d(xI(1) +1 +, z(2)) ≤ ε. Then +d +� +z(1), z(2) +� +> d +� +xI(1) +1 +, z(2) +� +− ε ≥ mε(K1, K2) +(4.29) +and hence by the definition of mε(K1, K2) and the fact that x1 +1 ⪯ xI(1) +1 +we cannot have z(2) ⪯ +z(1). Since K2 is totally ordered, we conclude that z(1) ≺ z(2). This allows us to set xi +2 = z(2) +for all I(1) ≤ i < I(2), where I(2) := inf{i > I(1) : d(xi +1, z(2)) > mε(K1, K2) + ε}. Continuing +inductively, we obtain (x1 +2, . . . , xm +2 ) ∈ K⟨m⟩ +2 +such that d(xk +1, xk +2) ≤ mε(K1, K2)+ε for all 1 ≤ k ≤ +m. +As a consequence of Lemmas 4.10 and 4.11, we can prove that the metrics d⟨m⟩ with 2 ≤ +m ≤ ∞ all generate the same topology on Ktot(X). This may be a bit surprising in view of +Lemmas 4.6 and 4.7. As the latter shows, the restriction to totally ordered sets is essential in +the following lemma. +Lemma 4.12 (Convergence of totally ordered sets) Let X be a metrisable space. Then +Kn, K ∈ Ktot(X) satisfy d⟨2⟩(Kn, K) → 0 if and only if d⟨∞⟩(Kn, K) → 0. +Proof Since d⟨m⟩(Kn, K) ≤ d⟨m+1⟩(Kn, K) by Lemma 4.5 and since d⟨∞⟩(Kn, K) is defined +as the increasing limit of d⟨m⟩(Kn, K) as m → ∞, it is clear that d⟨∞⟩(Kn, K) → 0 implies +d⟨2⟩(Kn, K) → 0. To prove the opposite implication, it suffices to show that d⟨2⟩(Kn, K) → 0 +implies +sup +m≥1 +d⟨m⟩(Kn, K) −→ +n→∞ 0. +(4.30) +By Lemma 4.5, d⟨2⟩(Kn, K) → 0 implies εn := d⟨1⟩(Kn, K) → 0. Lemmas 4.10 and 4.11 now +imply that +sup +m≥1 +d⟨m⟩(Kn, K) ≤ mεn(Kn, K) + εn −→ +n→∞ 0. +(4.31) +Proof of Theorem 2.10 By the definition of dpart and Proposition 4.8 we have dpart = d⟨2⟩ +and dtot = d⟨∞⟩. By Lemma 4.12 both metrics generate the same topology on Ktot(X). If d +26 + +and d′ generate the same topology on X and dpart and d′ +part are defined in terms of d and d′ as +in (2.13), then by Lemmas 2.7 and 2.8, dpart and d′ +part generate the same topology on Ktot(X). +The inequalities (2.16) follow from the fact that d⟨1⟩(K1, K2) ≤ d⟨2⟩(K1, K2) ≤ d⟨∞⟩(K1, K2) by +Lemma 4.5. On the other hand, Lemma 4.6 shows that if X = [0, 1], then for each ε > 0 we can +find K1, K2 ∈ Ktot(X) such that +d⟨2⟩(K1, K2) ≤ ε +while +1/2 ≤ d⟨3⟩(K1, K2) ≤ d⟨∞⟩(K1, K2), +(4.32) +proving the final claim of the theorem. +4.5 +Polishness +In this subsection, we prove Proposition 2.11. We start with the following lemma, announced +in the introduction, that shows that even when (X, d) is complete, it is in general not true that +the metrics dpart and dtot are complete on Ktot(X). +Lemma 4.13 (Metric not complete) Let X = [0, 1], equipped with the usual distance. Then +the metrics d⟨m⟩ with 2 ≤ m ≤ ∞ are not complete on Ktot(X). +Proof It suffices to construct a Cauchy sequence that does not converge. In view of Lemma 4.5, +it suffices to construct a Cauchy sequence in the metric d⟨∞⟩, which by Proposition 4.8 equals +dtot. Let εn be positive constants converging to zero, and let Kn := {0, 1, εn} equipped with a +total order such that 0 ≺ 1 ≺ εn. For each n, m, we define a monotone correspondence Rn,m +between Kn and Km by Rn,m := {(0, 0), (1, 1), (εn, εm)}. Then +dtot(Kn, Km) ≤ +sup +(x1,x2)∈Rn,m +|x1 − x2| = |εn − εm|, +(4.33) +so Kn is clearly a Cauchy sequence in dtot. However, the sequence Kn does not converge in the +ordered Hausdorff topology. If it had a limit K, then (in view of Lemma 4.4) this would have to +be the set K = {0, 1} equipped with a total order such that 0 ⪯ 1 and 1 ⪯ 0, but such a totally +ordered set does not exist. +The proof of Proposition 2.11 needs some preparations. For each L ∈ K+(X 2) and ε ≥ 0, we +set +m⟨2⟩ +ε (L) := sup +� +d(x1, y1) ∨ d(x2, y2) : (x1, y1), (y2, x2) ∈ L, d(x1, x2) ∨ d(y1, y2) ≤ ε +� +. +(4.34) +In particular, this implies m⟨2⟩ +ε (K⟨2⟩) = mε(K) (K ∈ Ktot(X)). +Lemma 4.14 (Right continuity) For any metric space X and L ∈ K+(X 2), the function +[0, ∞) ∋ ε → m⟨2⟩ +ε (L) is right-continuous. +Proof The function ε �→ m⟨2⟩ +ε (L) is clearly nondecreasing, so it suffices to prove that +m⟨2⟩ +ε (L) ≥ lim +η↓ε m⟨2⟩ +η (L) +(ε ≥ 0) +(4.35) +where the limit exist by monotonicity. Fix εn > ε such that εn → ε. Then for each δ > 0 +and for each n, we can choose (xn +1, yn +1 ), (yn +2 , xn +2) ∈ L such that d(xn +1, xn +2) ∨ d(yn +1 , yn +2 ) ≤ εn and +d(xn +1, yn +1 ) ∨ d(xn +2, yn +2 ) ≥ m⟨2⟩ +ηn (L) − δ. Since L is compact, by going to a subsequence, we can +assume that (xn +1, yn +1 ) → (x1, y1) and (yn +2 , xn +2) → (y2, x2) for some (x1, y1), (y2, x2) ∈ L. Then +d(x1, x2)∨d(y1, y2) ≤ ε and d(x1, y1)∨d(x2, y2) ≥ limη↓ε m⟨2⟩ +η (L)−δ, which proves that m⟨2⟩ +ε (L) ≥ +limη↓ε m⟨2⟩ +η (L) − δ. Since δ > 0 is arbitrary, this implies (4.35). +27 + +Lemma 4.15 (Upper semi-continuity) Let X be a metric space and let Ln, L ∈ K+(X 2) +satisfy Ln → L. Then +m⟨2⟩ +ε (L) ≥ lim sup +n→∞ +m⟨2⟩ +ε (Ln) +(ε ≥ 0). +(4.36) +Proof By the compactness of [0, ∞] we can select a subsequence for which limn→∞ m⟨2⟩ +ε (Ln) +exists and is equal to the limit superior of the original sequence. Let δn > 0 converge to zero +and pick (xn +1, yn +1 ), (yn +2 , xn +2) ∈ Ln such that d(xn +1, xn +2) ∨ d(yn +1 , yn +2 ) ≤ ε and d(xn +1, yn +1 ) ∨ d(xn +2, yn +2 ) ≥ +m⟨2⟩ +ε (Ln) − δn. By Lemma 2.6, there exists a compact C ⊂ X 2 such that Ln ⊂ C for all n, so by +going to a further subsequence we can assume that (xn +1, yn +1 ) → (x1, y1) and (yn +2 , xn +2) → (y2, x2) for +some (x1, y1), (y2, x2) ∈ X 2. Then (x1, y1), (y2, x2) ∈ L by Lemma 2.6, d(x1, x2) ∨ d(y1, y2) ≤ ε, +and hence +m⟨2⟩ +ε (L) ≥ d(x1, y1) ∨ d(x2, y2) ≥ lim +n→∞ +� +m⟨2⟩ +ε (Ln) − δn). +(4.37) +Since δn → 0, this proves (4.36). +Before we can prove Proposition 2.11 we need one more lemma. For any metric space (X, d), +we define L(X) ⊂ K+(X 2) by +L(X) := +� +K⟨2⟩ : K ∈ Ktot(X) +� +, +(4.38) +and we let L(X) denote the closure of L(X) in the metric space +� +K+(X 2), d2 +H +� +. +Lemma 4.16 (Totally ordered sets) For any metric space X, one has +L(X) = +� +L ∈ L(X) : m⟨2⟩ +0 (L) = 0 +� +. +(4.39) +Proof To prove the inclusion ⊂ in (4.39), it suffices to observe that +m⟨2⟩ +0 (K⟨2⟩) = sup +� +d(x, y) : (x, y), (y, x) ∈ K⟨2⟩� += 0 +(4.40) +for all K ∈ Ktot(X), since x ⪯ y and y ⪯ x imply x = y. +We next prove the inclusion ⊃ in (4.39). Assume that L ∈ L(X) satisfies m⟨2⟩ +0 (L) = 0. Since +L ∈ L(X), there exist Kn ∈ Ktot(X) such that K⟨2⟩ +n +→ L in the topology on K+(X 2). Let +πi(x1, x2) := xi (i = 1, 2) denote the coordinate projections πi : X 2 → X. Since π1(K⟨2⟩ +n ) = +Kn = π2(K⟨2⟩ +n ) for each n, using Lemma 4.3, we see that Kn → K in the Hausdorff topology on +K+(X), where K := π1(L) = π2(L). We define a relation ⪯ on K by setting x ⪯ y if and only +if (x, y) ∈ L. To complete the proof, it suffices to show that ⪯ is a total order on K, i.e., +(i) for each x, y ∈ K, either x ⪯ y or y ⪯ x, or both, +(ii) x ⪯ y and y ⪯ x imply x = y, +(iii) x ⪯ y ⪯ z imply x ⪯ z. +To prove (i), let x, y ∈ K. Since Kn → K in the Hausdorff topology on K+(X), by Lemma 2.6, +there exist xn, yn ∈ Kn such that xn → x and yn → y. Since Kn is totally ordered, either +xn ⪯ yn happens for infinitely many n, or yn ⪯ xn happens for infinitely many n, or both. Since +K⟨2⟩ +n +→ L in the Hausdorff topology on K+(X 2), by Lemma 2.6, it folows that either x ⪯ y or +y ⪯ x, or both. Property (ii) follows immediately from the fact that sup{d(x, y) : (x, y), (y, x) ∈ +L} = m⟨2⟩ +0 (L) = 0. To prove (iii), assume that x, y, z ∈ K satisfy x ⪯ y ⪯ z. If x = y or +y = z then trivially x ⪯ z, so without loss of generality we may assume that x ̸= y ̸= z. Since +28 + +Kn → K in the Hausdorff topology on K+(X), by Lemma 2.6, there exist xn, yn, zn ∈ Kn such +that xn → x, yn → y, and zn → z. Since Kn is totally ordered, for each n either xn ⪯ yn, or +yn ⪯ xn, or both. But yn ⪯ xn can happen only for finitely many n since otherwise the fact that +K⟨2⟩ +n +→ L in the Hausdorff topology on K+(X 2) and Lemma 2.6 would imply that y ⪯ x, which +together with our assumptions x ⪯ y and x ̸= y contradicts (ii). We conclude that xn ⪯ yn +for all n sufficiently large and by the same argument also yn ⪯ zn for all n sufficiently large. It +follows that xn ⪯ zn for all n sufficiently large. Since K⟨2⟩ +n +→ L in the Hausdorff topology on +K+(X 2), Lemma 2.6 shows that (x, z) ∈ L and hence x ⪯ z. +We are now ready to prove Proposition 2.11. We need to recall one well-known fact. A +subset A ⊂ X of a topological space X is called a Gδ-set if A is a countable intersection of open +sets. Our proof of Proposition 2.11 makes use of the following fact, that we cite from [Bou58, +§6 No. 1, Theorem. 1] (see also [Oxt80, Thms 12.1 and 12.3]). +Lemma 4.17 (Subsets of Polish spaces) A subset Y ⊂ X of a Polish space X is Polish in +the induced topology if and only if Y is a Gδ-subset of X. +Proof of Proposition 2.11 We first observe that if X is Polish, then so is X 2, equipped with +the product topology, and hence, by Lemma 2.7, also K+(X 2). Therefore, since +� +Ktot(X), dpart +� +is isometric to +� +L(X), d2 +H +� +defined in (4.38), in view of Lemma 4.17, it suffices to show that L(X) +is a Gδ-subset of K+(X 2). +It follows from Lemma 4.15 that for each ε, δ > 0, the set +Aδ,ε := +� +L ∈ K+(X 2) : m⟨2⟩ +ε (L) ≥ δ +� +(4.41) +is a closed subset of K+(X 2) and hence its complement Ac +ε,δ is open. As a consequence, +G := +∞ +� +n=1 +∞ +� +m=1 +Ac +1/n,1/m +(4.42) +is a Gδ-set. Since each closed set is a Gδ-set, and the intersection of two Gδ-sets is a Gδ-set, +using Lemmas 4.14 and 4.16, we conclude that +L(X) ∩ G = +� +L ∈ L(X) : ∀δ > 0 ∃ε > 0 s.t. m⟨2⟩ +ε (L) < δ +� += +� +L ∈ L(X) : lim +ε→0 m⟨2⟩ +ε (L) = 0 +� += +� +L ∈ L(X) : m⟨2⟩ +0 (L) = 0 +� += L(X) +(4.43) +is a Gδ-set. +4.6 +Compactness criterion +In this subsection, we prove Theorem 2.12. +Proof of Theorem 2.12 Since the map K �→ K⟨2⟩ is a homeomorphism from Ktot(X) to L(X), +equipped with the induced topology from K+(X 2), a set A ⊂ K⟨2⟩ is precompact if and only if +B := {K⟨2⟩ : K ∈ A} is a precompact subset of K+(X 2) and its closure B is contained in L(X). +We will show that B is a precompact subset of K+(X 2) if and only if (2.18 (i) holds. Moreover, +if (2.18 (i) holds, then B is contained in L(X) if and only if (2.18 (ii) holds. +If (2.18 (i) holds, then C2 is a compact subset of X 2 and K⟨2⟩ ⊂ C2 for all K ∈ A, so +B is a precompact subset of K+(X 2) by Lemma 2.8. Conversely, if B is a precompact subset +of K+(X 2), then by Lemma 2.8 there exists a compact subset D ⊂ X 2 such that K⟨2⟩ ⊂ D +29 + +for all K ∈ A. Without loss of generality, we may assume that D is of the form D = C2 for +some compact subset C of X; for example, we may take for C the union of the two coordinate +projections of D. Then K ⊂ C for all K ∈ A, proving that (2.18 (i) holds. +To complete the proof, assume that (2.18 (i) holds. We must show that B is contained in +L(X) if and only if (2.18 (ii) holds. Assume, first, that (2.18 (ii) does not hold. Then we can +find a δ > 0 and εn > 0 tending to zero, as well as Kn ∈ A, such that m⟨2⟩ +εn (Kn) ≥ δ for each +n. By (2.18 (i), going to a subsequence if necessary, we can assume that K⟨2⟩ +n +→ L for some +L ∈ K+(X 2). Now Lemma 4.15 implies that +m⟨2⟩ +ε (L) ≥ lim sup +n→∞ m⟨2⟩ +ε (K⟨2⟩ +n ) ≥ lim sup +n→∞ m⟨2⟩ +εn (K⟨2⟩ +n ) ≥ δ +(4.44) +for each ε > 0, so letting ε ↓ 0, using Lemma 4.14, we conclude that m⟨2⟩ +0 (L) ≥ δ. +By +Lemma 4.16, we conclude that L ̸∈ L(X) and hence B is not contained in L(X). +Assume now that (2.18) (ii) holds. We must show that B is contained in L(X). Assume +that K⟨2⟩ +n +→ L for some Kn ∈ A. Then clearly L ∈ L(X) so by Lemma 4.16 it suffices to +prove that m⟨2⟩ +0 (L) = 0. Assume that, conversely, there exist x, y ∈ X with x ̸= y such that +(x, y), (y, x) ∈ L. Then by Lemma 2.6, there exist xn +1, xn +2, yn +1 , yn +2 ∈ Kn with xn +1 ⪯ yn +1 and yn +2 ⪯ xn +2 +such that xn +i → x and yn +i → y as n → ∞ (i = 1, 2). Then for each ε > 0, we can choose n large +enough such that d(xn +i , x) ≤ ε/2 (i = 1, 2). It follows that d(xn +1, yn +1 ) ∨ d(xn +2, yn +2 ) ≥ d(x, y) − ε +and d(xn +1, xn +2) ∨ d(yn +1 , yn +2 ) ≥ ε so that mε(Kn) ≥ d(x, y) − ε. This clearly contradicts (2.18) (ii), +so we conclude that m⟨2⟩ +0 (L) = 0 as required. +4.7 +Cadlag curves +In this subsection, we prove Proposition 2.13. Let (X, d) be a metric space. If R is any subset +of X 2, then let us call +dist(R) := +sup +(x1,x2)∈R +dsqz(x1, x2) +(4.45) +the distortion of R. Then (2.8) and (2.15) say that +dH(K1, K2) = +inf +R∈Corr(K1,K2) dist(R) +and +dtot(K1, K2) = +inf +R∈Corr+(K1,K2) dist(R), +(4.46) +where Corr(K1, K2) and Corr+(K1, K2) denote the sets of all (monotone) correspondences be- +tween K1 and K2. Let R denote the closure of a set R ⊂ X 2. Then +dist(R) = dist(R) +(R ⊂ X 2). +(4.47) +Indeed, the inequality ≤ is trivial, while the opposite inequality follows from the fact that for +each (x1, x2) ∈ R, there exist (xn +1, xn +2) ∈ R such that (xn +1, xn +2) → (x1, x2) and hence d(xn +1, xn +2) → +d(x1, x2). +We need one preparatory lemma. +Lemma 4.18 (Fine partition) Let (X, d) be a metric space. Then for each γ ∈ D[0,1](X) and +ε > 0, there exist t0 < 0 < t1 < · · · < tn−1 < 1 < tn such that +sup +� +d +� +γ(s), γ(s′) +� +: 1 ≤ k ≤ n, s, t ∈ [0, 1] ∩ [tk−1, tk) +� +< ε. +(4.48) +30 + +Proof By Lemma 2.4, writing γ(t+) := γ(t) and γ(t−) := γ−(t), where γ− is the caglad +modification of γ, we can view γ as a continuous function on the split real interval [0−, 1+], +that moreover satisfies γ(0−) = γ(0+) and γ(1−) = γ(1+). Fix ε > 0 and let +R := +� +(s, t) ∈ R2 : s < t, s ̸= 0, t ̸= 1, d +� +γ(σ), γ(τ) +� +< ε ∀σ, τ ∈ [s+, t−] ∩ [0−, 1+] +� +. +(4.49) +Using the properties of γ, it is easy to see that +� +(s,t)∈R +[s+, t−] ⊃ [0−, 1+]. +(4.50) +Since the intervals [s+, t−] are open in the topology of the split real line and since [0−, 1+] is +compact by Proposition 2.3, there exists a finite subcover, i.e., there exists a finite set S ⊂ R +such that +� +(s,t)∈S +[s+, t−] ⊃ [0−, 1+]. +(4.51) +Let T := {s : (s, t) ∈ S} ∪ {t : (s, t) ∈ S}. +Then, letting t0 denote the largest element of +T ∩ (−∞, 0), ordering the elements of T ∩ (0, 1) as t1 < · · · < tn−1, and letting tn denote the +smallest element of T ∩ (1, ∞), we obtain times t0 < · · · < tn as in (4.48). +Proof of Proposition 2.13 If γ1, γ2 are cadlag parametrisations of K1, K2, and λ ∈ Λ, then +let us set +Rλ := +�� +γ1(t), γ2 +� +λ(t) +�� +: t ∈ [0, 1] +� += +�� +γ1 +� +λ−1(t) +� +, γ2(t) +� +: t ∈ [0, 1] +� +, +(4.52) +and let Rλ denote its closure. We claim that Rλ is a correspondence between K1 and K2. To +see this, let γ− +i +denote the caglad modification of γi (i = 1, 2). By the definition of a cadlag +parametrisation, each element x1 ∈ K1 is of the form x1 = γ1(t) or = γ− +1 (t) for some t ∈ [0, 1]. If +x1 = γ1(t), then clearly there exists an x2 ∈ K2 such that (x1, x2) ∈ Rλ, namely x2 := γ2(λ(t)). +If x1 = γ− +1 (t), then we can choose tn ↑ t and set xn +1 := γ1(tn). Then xn +1 → x1 by the left +continuity of γ− +1 . We have already seen that there exist xn +2 ∈ K2 such that (xn +1, xn +2) ∈ Rλ. Since +K2 is compact, by going to a subsequence, we can assume that xn +2 → x2 for some x2 ∈ K2. +Then (x1, x2) ∈ Rλ. In the same way, we see that for each x2 ∈ K2, there exists an x1 ∈ K1 +such that (x1, x2) ∈ Rλ. This completes the proof that Rλ is a correspondence. Using the fact +that the total orders on K1 and K2 are compatible with the topology, it is easy to see that Rλ +is monotone if λ ∈ Λ+. +Using these facts as well as (4.46) and (4.47), we see that +dH(K1, K2) ≤ inf +λ∈Λ dist(Rλ) = inf +λ∈Λ dist(Rλ) = inf +λ∈Λ sup +t∈[0,1] +d +� +γ1(t), γ2 +� +λ(t) +�� +, +dtot(K1, K2) ≤ inf +λ∈Λ+dist(Rλ) = inf +λ∈Λ+dist(Rλ) = inf +λ∈Λ+ sup +t∈[0,1] +d +� +γ1(t), γ2 +� +λ(t) +�� +. +(4.53) +To complete the proof, we must show that: +(i) For each R ∈ Corr(K1, K2) and ε > 0, there exists a λ ∈ Λ such that dist(Rλ) ≤ dist(R)+ε. +(ii) For each R ∈ Corr+(K1, K2) and ε > 0, there exists a λ ∈ Λ+ such that dist(Rλ) ≤ +dist(R) + ε. +We first prove (i). Fix R ∈ Corr(K1, K2) and ε > 0. By Lemma 4.18, for i = 1, 2, there exist +ti +0 < 0 < ti +1 < · · · < ti +ni−1 < 1 < ti +ni such that +sup +� +d +� +γi(s), γi(s′) +� +: 1 ≤ k ≤ ni, s, t ∈ [0, 1] ∩ [ti +k−1, ti +k) +� +< ε/2 +(i = 1, 2). +(4.54) +31 + +For 1 ≤ k ≤ ni, let us write Ki +k := {γi(t) : t ∈ [0, 1] ∩ [ti +k−1, ti +k)} (i = 1, 2). We can choose +a correspondence S between {1, . . . , n1} and {1, . . . , n2} such that for each (k1, k2) ∈ S, there +exists an (x1, x2) ∈ R with xi ∈ Ki +ki (i = 1, 2). Then +sup +(k1,k2)∈S +sup +x1∈K1 +x2∈K2 +d(x1, x2) ≤ dist(R) + ε. +(4.55) +By refining the partitions ti +0, . . . , ti +ni, we can make sure that for each k1 ∈ {1, . . . , n1}, there is a +unique k2 ∈ {1, . . . , n2} such that (k1, k2) ∈ S, and vice versa. We can then construct a bijection +λ : [0, 1] → [0, 1] such that for each (k1, k2) ∈ S, the restriction of λ to [0, 1] ∩ [t1 +k1−1, t1 +k1) is a +bijection to [0, 1] ∩ [t2 +k2−1, t2 +k2). Then (4.55) implies that dist(Rλ) ≤ dist(R) + ε, completing the +proof of (i). +To also prove (ii), we observe that if R is a monotone correspondence, then S as we initially +constructed it will be a monotone correspondence between {1, . . . , n1} and {1, . . . , n2}, and +monotonicity will be preserved after we refine the partitions so that they have the same size and +S is a bijection. Now λ can be chosen monotone too, completing the proof of (ii). +4.8 +Betweenness +In this subsection, we prove Lemmas 2.14, 2.15, 2.16, and 2.17, as well as Lemma 4.19 below +that will be used in the proof of Lemma 3.1. +Proof of Lemma 2.14 Clearly, (ii) and (iii) imply (v) and (iv) implies (vi). To prove (vii), +we first observe that x ∈ ⟨y, z⟩ and y ∈ ⟨x, z⟩ imply by (vi) ⟨x, z⟩ ⊂ ⟨y, z⟩ ⊂ ⟨x, z⟩ and hence +⟨x, z⟩ = ⟨y, z⟩. Using this as well as the assumptions y ∈ ⟨x, z⟩ and x ∈ ⟨y, z⟩ we can conclude +by (i) and (iii) that {y} = ⟨x, y⟩ ∩ ⟨y, z⟩ = ⟨y, x⟩ ∩ ⟨x, z⟩ = {x} and hence x = y. +To prove (viii), assume that y, y′ ∈ ⟨x, z⟩ and y′ ∈ ⟨x, y⟩. The statement is trivial if y = y′ +so without loss of generality we assume that y ̸= y′. Since y′ ∈ ⟨x, z⟩ we have by (iv) that +⟨x, z⟩ = ⟨x, y′⟩ ∪ ⟨y′, z⟩. Since also y ∈ ⟨x, z⟩ we must have either y ∈ ⟨x, y′⟩, or y ∈ ⟨y′, z⟩, or +both. The first possibility would by (i) and (vii) and the fact that y′ ∈ ⟨x, y⟩ imply that y = y′, +which contradicts our assumptions, so we conclude that y ∈ ⟨y′, z⟩. +The first implication ⇒ in (2.21) follow from the fact that y ∈ ⟨x, y⟩ by (i) and (ii), while +the reverse implication follows from (vi). The second equivalence in (2.21) follows from (viii) +and the third equivalence follows from the first one, by the symmetry (i). It is clear that (2.21) +defines a partial order ≤x,z on ⟨x, z⟩. By (iv), if y, y′ ∈ ⟨x, z⟩, then at least one of the conditions +y′ ∈ ⟨x, y⟩ and y′ ∈ ⟨y, z⟩ must hold, which shows that ≤x,z is a total order. +Proof of Lemma 2.15 We need to check that our definition satisfies axioms (i)–(iv) of a +betweenness. Axioms (i) and (ii) are trivial. To prove (iii) and (iv), set r := d(x, z) and let +γ : [0, r] → X be the unique isometry such that γ(0) = x and γ(r) = z. Since an isometry is +one-to-one, there exists a unique p ∈ [0, r] such that γ(p) = y. Clearly, the restrictions of γ to +[0, p] and [p, r] are isometries, so ⟨x, y⟩ = {γ(t) : 0 ≤ t ≤ p} and ⟨y, z⟩ = {γ(t) : p ≤ t ≤ r}. +From these observations, axioms (iii) and (iv) follow immediately. +We skip the proof of Lemma 2.16 for the moment and first prove Lemma 2.17 and the already +announced Lemma 4.19. +Proof of Lemma 2.17 For the trivial betweenness, ⟨x, z⟩ = {x, z} is clearly compact for each +x, z ∈ X, and the continuity of the map (x, z) �→ {x, z} with respect to the Hausdorff topology +follows immediately from Lemma 2.6. +32 + +If a betweenness is generated by an interpolation function, then ⟨x, z⟩, being the image +of [0, 1] under the continuous map p �→ ϕ(x, z, p), is clearly compact for all x, z ∈ Z. +Let +xn → x and zn → z. To show that ⟨xn, zn⟩ → ⟨x, z⟩ in the Hausdorff topology, we check the +conditions of Lemma 2.6. Since xn → x and zn → z, the sets A := {x} ∪ {xn : n ∈ N} and +B := {z} ∪ {zn : n ∈ N} are compact. Let ϕ(A × B × [0, 1]) denote the image of A × B × [0, 1] +under ϕ, which is compact. Clearly ⟨xn, zn⟩ ⊂ ϕ(A × B × [0, 1]) for all n. To complete the +argument, it suffices to show that +� +y ∈ X : ∃yn ∈ ⟨xn, zn⟩ s.t. y is a cluster point of (yn)n∈N +� +⊂ ⟨x, z⟩ ⊂ +� +y ∈ X : ∃yn ∈ ⟨xn, zn⟩ s.t. yn → y +� +. +(4.56) +For the first inclusion, assume that y is a cluster point of yn = ϕ(xn, zn, pn). By going to a +subsequence, we can assume that pn → p for some p ∈ [0, 1]. Then y = ϕ(x, z, p) ∈ ⟨x, z⟩. For +the second inclusion, assume that y = ϕ(x, z, p) for some p ∈ [0, 1]. Then yn := ϕ(xn, zn, p) ∈ +⟨xn, zn⟩ converge to y, completing the proof that each betweenness that is generated by an +interpolation function is compatible with the topology. +For the final statement of the lemma, assume that X is a closed subset of R. Then clearly +⟨x, z⟩ := [x, z] ∩ X is compact for each x, z ∈ R. Let xn → x and zn → z. To show that +⟨xn, zn⟩ → ⟨x, z⟩ in the Hausdorff topology, we again check the conditions of Lemma 2.6. Clearly, +⟨xn, zn⟩ ⊂ [S, T] ∩ X for each n, where S := infn xn and T := supn zn, so to complete the +argument, it again suffices to check (4.56). For the first inclusion, assume that y ∈ X is a cluster +point of yn ∈ ⟨xn, zn⟩. Since xn ≤ yn ≤ zn for each n, taking the limit, we see that x ≤ y ≤ z +and hence y ∈ ⟨x, z⟩. For the second inclusion, assume that y ∈ ⟨x, z⟩. If x < y < z, then +xn < y < zn for all n large enough, so setting yn := y for n large enough and yn := xn otherwise +proves that y is al element of the set on the right-hand side of (4.56). If y ∈ {x, z}, then setting +yn := xn or := zn proves the same claim, so the proof is complete. +The following lemma will be used in the proof of Lemma 3.1, and is also of independent +interest. +Lemma 4.19 (Segments as ordered sets) Let X be a metrisable space that is equipped with +a betweenness that is compatible with the topology. Then for each x, z ∈ X, the segment ⟨x, z⟩ +equipped with the total order ≤x,z is an element of Ktot(X), and the map (x, z) �→ ⟨x, z⟩ is +continuous with respect to the product topology on X 2 and the topology on Ktot(X). +Proof To show that ⟨x, z⟩ is an element of Ktot(X), we must show that the total order ≤x,z +is compatible with the induced topology on ⟨x, z⟩. +Assume that yn, y′ +n, y, y′ ∈ ⟨x, z⟩ satisfy +yn → y, y′ +n → y′, and yn ≤x,z y′ +n for all n. Then yn ∈ ⟨x, y′ +n⟩ for all n. Since the betweenness is +compatible with the topology, ⟨x, y′ +n⟩ → ⟨x, y′⟩ in the Hausdorff topology, which by Lemma 2.6 +implies that y ∈ ⟨x, y′⟩ and hence y ≤x,z y′. This shows that the total order ≤x,z is compatible +with the induced topology on ⟨x, z⟩. +To show that the map (x, z) �→ ⟨x, z⟩ is continuous with respect to the topology on Ktot(X), +assume that xn → x, zn → z. We will show that +dpart +� +⟨xn, zn⟩, ⟨x, z⟩ +� +−→ +n→∞ 0, +(4.57) +which is equivalent to the statement that ⟨xn, zn⟩⟨2⟩ converges to ⟨x, z⟩⟨2⟩ in the Hausdorff +topology on K+(X 2). We apply Lemma 2.6. Since ⟨xn, zn⟩ → ⟨x, z⟩, there exists a compact +C ⊂ X such that ⟨xn, zn⟩ ⊂ C for all n and hence ⟨xn, zn⟩⟨2⟩ ⊂ C2 for all n. Thus, it suffices to +33 + +check that (compare (4.56)) +� +(y, y′)∈X 2 : ∃yn, y′ +n ∈ ⟨xn, zn⟩ with yn ≤xn,zn y′ +n s.t. (y, y′) is a cluster point of (yn, y′ +n)n∈N +� +⊂ ⟨x, z⟩⟨2⟩ ⊂ +� +(y, y′) ∈ X : ∃yn, y′ +n ∈ ⟨xn, zn⟩ with yn ≤xn,zn y′ +n s.t. (yn, y′ +n) → (y, y′) +� +. +(4.58) +If (y, y′) is an element of the set on the left-hand side of (4.58) and (yn, y′ +n) fulfill the conditions of +the definition of this set, then by going to a subsequence we may assume that (yn, y′ +n) → (y, y′). +Then y, y′ ∈ ⟨x, z⟩ since ⟨xn, zn⟩ → ⟨x, z⟩. Moreover yn ≤xn,zn y′ +n means yn ∈ ⟨xn, y′ +n⟩. Since +yn → y and ⟨xn, y′ +n⟩ → ⟨x, y′⟩, this implies y ∈ ⟨x, y′⟩ and hence y ≤x,z y′, proving that +(y, y′) ∈ ⟨x, z⟩⟨2⟩. +To prove the second inclusion in (4.58), assume that (y, y′) ∈ ⟨x, z⟩⟨2⟩. Since ⟨xn, zn⟩ → ⟨x, z⟩, +there exist yn, y′ +n ∈ ⟨xn, zn⟩ such that yn → y and y′ +n → y′. We now distinguish two cases: y ̸= y′ +and y = y′. If y ̸= y′, then we claim that yn ≤xn,zn y′ +n for all n large enough. Indeed, in the +opposite case, since ≤xn,zn is a total order, by going to a subsequence, we can assume that +y′ +n ≤xn,zn yn for all n, which by the arguments we have already seen implies y′ ≤x,z y, so that +by the fact that (y, y′) ∈ ⟨x, z⟩⟨2⟩ we must have y = y′, contradicting our assumption. Since +yn ≤xn,zn y′ +n for all n large enough, changing the definitions of yn, y′ +n for finitely many n, we see +that there exist yn, y′ +n ∈ ⟨xn, zn⟩ with yn ≤xn,zn y′ +n such that s.t. (yn, y′ +n) → (y, y′). In the case +y = y′ the argument is even simpler, since now (yn, yn) → (y, y′) while obviously yn ≤xn,zn yn +for all n. +In this subsection, it only remains to prove Lemma 2.16. The statement about the linear +betweenness is trivial, but before we can prove the statement about the geodesic betweenness, +we first need a better understanding of metric spaces with unique geodesics, which is provided +by Proposition 4.20 below. In any metric space (X, d), for all x, z ∈ X and ε ≥ 0, we define +ηx,z(ε) := sup +� +d(y1, y2) : +� +d(x, y1) ∧ d(x, y2) +� ++ +� +d(y1, z) ∧ d(y2, z) +� +≤ d(x, z) + ε +� +. +(4.59) +In other words, this is the largest distance between two points y1, y2 ∈ X for which there exist +constants r, r′ ≥ 0 with r + r′ ≤ d(x, z) + ε such that d(x, yi) ≤ r and d(yi, z) ≤ r′ (i = 1, 2). +Proposition 4.20 (Unique geodesics) Let (X, d) be a metric space. Consider the following +conditions. +(i) For all x, z ∈ X and r, r′ ≥ 0 with r + r′ = d(x, z), there exists an y ∈ X such that +d(x, y) = r and d(y, z) = r′. +(ii) ηx,z(0) = 0 for all x, z ∈ X. +(ii)’ lim +ε→0 ηx,z(ε) = 0 for all x, z ∈ X. +Then (X, d) has unique geodesics if and only if (i) and (ii) hold. Moreover, (ii)’ implies (ii), +and if (X, d) is a proper metric space, then (ii)’ implies (ii). +Proof In any metric space (X, d), let us introduce the notation +⟨x, z⟩ := +� +y ∈ X : d(x, y) + d(y, z) = d(x, z) +� +. +(4.60) +We claim that +y ∈ ⟨x, z⟩, y′ ∈ ⟨x, y⟩, y′′ ∈ ⟨y, z⟩ +⇒ +y ∈ ⟨y′, y′′⟩. +(4.61) +34 + +To see this, we note that if the assumptions in (4.61) hold but the conclusion does not, then by +the triangle inequality +d(x, z) = d(x, y) + d(y, z) = d(x, y′) + d(y′, y) + d(y, y′′) + d(y′′, z) +> d(x, y′) + d(y′, y′′) + d(y′′, z), +(4.62) +which contradicts the triangle inequality. Now let (X, d) be a metric space with unique geodesics +and let ⟨⟨x, z⟩⟩ denote the unique geodesic with endpoints x, z. Clearly ⟨⟨x, z⟩⟩ ⊂ ⟨x, z⟩. We claim +that +y ∈ ⟨x, z⟩ +⇒ +⟨⟨x, y⟩⟩ ∪ ⟨⟨y, z⟩⟩ = ⟨⟨x, z⟩⟩. +(4.63) +To see this, let r := d(x, y), r′ := d(y, z), and let γ : [0, r] → X and γ′′ : [r, r + r′] → X be +the unique isometries with γ(0) = x, γ(r) = γ′(r) = y, and γ′(r + r′) = z. We claim that +γ′′ : [0, r + r′] → X defined as γ′′(t) = γ(t) for t ∈ [0, r] and := γ′(t) for t ∈ [r, r + r′] is an +isometry. So see this, let 0 ≤ t′ < t′′ ≤ r + r′. We need to show that d +� +γ′′(t′), γ′′(t′′) +� += t′′ − t′. +This is clear when t′′ ≤ r or r ≤ t′′, while in the remaining case t′ < r < t′′ the claim follows +from (4.61). +We now prove that if (X, d) is a metric space with unique geodesics, then conditions (i) +and (ii) are satisfied. Condition (i) is trivial. To prove (ii), let x, z ∈ X, let r, r′ ≥ 0 satisfy +r + r′ := d(x, z), and assume that y1, y2 ∈ X satisfy d(x, yi) = r, d(yi, z) = r′ (i = 1, 2). By +(4.63), there exist isometries γi : [0, r + r′] → X with γi(0) = x, γi(r) = yi, and γi(r + r′) = z, +so by the assumption that (X, d) has unique geodesics we conclude that y1 = y2, proving (ii). +Conversely, if (X, d) is a metric space for which (i) and (ii) hold, then for each x, z ∈ X with +r := d(x, z), we can uniquely define γx,z : [0, r] → X by +γx,z(t) := y +with +d(x, y) = t, d(y, z) = r − t. +(4.64) +Clearly, if γ : [0, r] → X is an isometry with γ(0) = x and γ(r) = z, then we must have γ = γx,z, +so to prove that (X, d) has unique geodesics, it suffices to show that γx,z is an isometry. Let +0 ≤ t1 ≤ t2 ≤ r and let yi := γx,z(ti) (i = 1, 2). Set y′ +1 := γx,y2(t1). Then d(x, y′ +1) = t1 and +d(y′ +1, z) ≤ d(y′ +1, y2) + d(y2, z) = r − t1, which by the assumption (ii) implies y′ +1 = y1. Since +d(y′ +1, y2) = t2 − t1, this proves that γx,z is an isometry. This completes the proof that a metric +space (X, d) has unique geodesics if and only if (i) and (ii) hold. +Trivially, (ii)’ implies (ii), so to complete the proof of the proposition, it suffices to prove that +for proper metric spaces, (ii) implies (ii)’. Assume that (ii)’ does not hold for some x, z ∈ X. Let +0 < εn ≤ 1 satisfy εn → 0. Then for some δ > 0, we can find yn +1 , yn +2 ∈ X with d(yn +1 , yn +2 ) ≥ δ, as +well as rn, r′ +n ≥ 0 with rn+r′ +n ≤ d(x, z)+εn, such that d(x, yn +i ) ≤ rn and d(yn +i , z) ≤ r′ +n (i = 1, 2). +Since d(x, yn +i ) ≤ d(x, z) + 1 for all n, by the properness assumption, we can select a subsequence +such that yn +i → yi for some y1, y2 ∈ X. Since rn + r′ +n ≤ d(x, z) + 1, by going to a further +subsequence, we can assume that rn → r and r′ +n → r′ for some r, r′ ≥ 0 with r + r′ = d(x, z). +Then d(x, yi) ≤ r and d(yi, z) ≤ r′ while d(y1, y2) ≥ δ which shows that ηx,z(0) ≥ δ, violating +(ii). +If a metric space (X, d) has unique geodesics, then by conditions (i) and (ii) of Proposi- +tion 4.20, we can uniquely define a function ϕ : X 2 × [0, 1] → X by +ϕ(x, z, p) := y +with +d(x, y) = pd(x, z) and d(y, z) = (1 − p)d(x, z). +(4.65) +Lemma 2.16 is now implied by Proposition 4.20 and the following lemma (the statement in +Lemma 2.16 about normed linear spaces being trivial). +35 + +Lemma 4.21 (Geodesic interpolation function) Let (X, d) be a metric space with unique +geodesics and let ϕ be defined as in (4.65). Then for each x, z ∈ X, the unique geodesic with +endpoints x, z is given by {ϕ(x, z, p) : p ∈ [0, 1]}. If condition (ii)’ of Proposition 4.20 is satisfied, +then ϕ : X 2 × [0, 1] → X is continuous. +Proof Let x, z ∈ X and r := d(x, z). We observe that {ϕ(x, z, p) : p ∈ [0, 1]} = +� +γx,z(t) : t ∈ +[0, r]} where γx,z is defined as in (4.64). It has already been shown in the proof of Proposition 4.20 +that this is the unique geodesic with endpoints x, z. Therefore, to complete the proof, it suffices +to show that condition (ii)’ of Proposition 4.20 implies that ϕ is continuous. +Assume that xn, x, zn, z ∈ X and pn, p ∈ [0, 1] satisfy xn → x, zn → z, and pn → p. Set +yn := ϕ(xn, zn, pn) and y := ϕ(x, z, p). We have to show that yn → y. We observe that +d(x, yn) + d(yn, z) ≤ d(xn, yn) + d(yn, zn) + d(x, xn) + d(z, zn) += d(xn, zn) + d(x, xn) + d(z, zn) −→ +n→∞ d(x, z). +(4.66) +Thus, for each ε > 0, we can find an m such that d(x, yn) + d(yn, z) ≤ d(x, z) + ε for all n ≥ m. +Since moreover d(x, y) + d(y, z) = d(x, z), it follows that d(yn, y) ≤ ηx,z(ε) for all n ≥ m. Since +ε > 0 is arbitrary, by (ii)’, this implies d(yn, y) → 0. +4.9 +Squeezed space +In this subsection, we prove Lemmas 2.18, 2.19, and 2.20. +Proof of Lemma 2.18 We first prove that dsqz is a metric on R(X). For brevity, we write +d′(x, y) := d(x, y) ∧ 1. Then d′ is a metric on E. The only nontrivial statement that we have to +prove is the triangle inequality, and it suffices to prove this for the function +ρ +� +(x, s), (y, t) +� +:= +� +φ(s) ∧ φ(t) +� +d′(x, y) + +��φ(s) − φ(t) +��. +We estimate +ρ +� +(x, s), (z, u) +� +≤ +� +φ(s) ∧ φ(u) +�� +d′(x, y) + d′(y, z) +� ++ +��φ(s) − φ(u) +��. +(4.67) +If φ(t) ≥ φ(s) ∧ φ(u), then φ(s) ∧ φ(u) is less than φ(s) ∧ φ(t) and also less than φ(t) ∧ φ(u), so +we can simply estimate the expression in (4.67) from above by +� +φ(s) ∧ φ(t) +� +d′(x, y) + +� +φ(t) ∧ φ(u) +� +d′(y, z) +� ++ +��φ(s) − φ(t) +�� + +��φ(t) − φ(u) +�� +and we are done. On the other hand, if φ(t) < φ(s) ∧ φ(u), then +��φ(s) − φ(t) +�� + +��φ(t) − φ(u) +�� = +��φ(s) − φ(u) +�� + 2 +� +φ(s) ∧ φ(u) − φ(t) +� +. +Using the fact that d′ ≤ 1, we can now estimate the right-hand side of (4.67) from above by +φ(t) +� +d′(x, y) + d′(y, z) +� ++ 2 +� +φ(s) ∧ φ(u) − φ(t) +� ++ +��φ(s) − φ(u) +�� += +� +φ(s) ∧ φ(t) +� +d′(x, y) + +� +φ(t) ∧ φ(u) +� +d′(y, z) ++ +��φ(s) − φ(t) +�� + +��φ(t) − φ(u) +��, +and again we are done. This completes the proof that dsqz is a metric on R(X). +It remains to prove that +� +φ(tn) ∧ φ(t) +�� +d(xn, x) ∧ 1 +� ++ +��φ(tn) − φ(t) +�� + dR(tn, t) −→ +n→∞ 0 +(4.68) +36 + +if and only if conditions (i) and (ii) of the lemma are satisfied. Because of the third term on the +left-hand side, a necessary condition for (2.26) is that tn → t, and this condition also guarantees +that the second term tends to zero. If t ∈ {−∞, +∞}, then this is all one needs since the first +term now tends to zero regardless of the values of xn and x, but if t ∈ R, then one needs in +addition that d(xn, x) → 0. +Proof of Lemma 2.19 If D is a countable dense subset of (X, d), then D × Q is a countable +dense subset of (R(X), dsqz), proving (a). +To prove (b), let (xn, tn) be a Cauchy sequence in (R(X), dsqz). +Then by (2.26) tn is a +Cauchy sequence in R and hence tn → t for some t ∈ R. If t ∈ R, then by (2.26) xn is a Cauchy +sequence in (X, d) so by the completeness of the latter, xn → x for some x ∈ X. By Lemma 2.18, +it follows that (xn, tn) converges, proving the completeness of (R(X), dsqz). +Proof of Lemma 2.20 Assume that A ⊂ R(X) has the property that for each T < ∞, there +exists a compact set K ⊂ X such that {x ∈ X : (x, t) ∈ A, t ∈ [−T, T]} ⊂ K. To show that A +is precompact, we will show that each sequence (xn, tn) ∈ A has a convergent subsequence. By +the compactness of R, we can select a subsequence (x′ +n, t′ +n) such that t′ +n → t for some t ∈ R. If +t = ±∞, then by Lemma 2.18 (x′ +n, t′ +n) → (∗, ±∞) and we are done. Otherwise, there exists a +T < ∞ such that t′ +n ∈ [−T, T] for all n large enough. By assumption, there then exists a compact +set K ⊂ X such that x′ +n ∈ K for all n large enough, so we can select a further subsequence such +that (x′′ +n, t′′ +n) converges to a limit (x, t) ∈ X × R. +Assume, on the other hand, that A ⊂ R(X) has the property that for some T < ∞, there +does not exist a compact set K ⊂ X such that {x ∈ X : (x, t) ∈ A, t ∈ [−T, T]} ⊂ K. Set +B := +� +x ∈ X : (x, t) ∈ A for some t ∈ [−T, T] +� +The closure of B cannot be compact, since this would contradict our assumption. It follows that +there exists a sequence xn ∈ B that does not contain a convergent subsequence, and there exist +tn ∈ [−T, T] such that (xn, tn) ∈ A. But then, in view of Lemma 2.18, the sequence (xn, tn) +cannot contain a convergent subsequence either, proving that A is not precompact. +5 +Proofs of the main results +5.1 +Closed and filled-in graphs +In this subsection, we prove Lemmas 3.1 and 3.2, as well an analogue of Lemma 3.2 that will +later be used in the proof of Theorem 3.6. +Proof of Lemma 3.1 We will show that each sequence (xn, tn) ∈ Gf(π) has a subsequence that +converges to a limit in Gf(π). Since I(π) is closed, we can select a subsequence such that tn → t for +some t ∈ I(π)∪{−∞, ∞}. If t = ±∞, then Lemma 2.18 tells us that (xn, tn) → (∗, ±∞) ∈ Gf(π) +so we are done, so from now on we can assume that t ∈ R. By going to a further subsequence, +we can assume that we are in one of the following three cases: (i) tn < t for all n, (ii) tn > t for +all n, and (iii) tn = t for all n. In case (i), we use the cadlag property of π and the fact that the +betweenness is compatible with the topology to see, using Lemma 2.14 (v), that +xn ∈ ⟨π(tn−), π(tn+)⟩ −→ +n→∞ {π(t−)} +(5.1) +from which we conclude that (xn, tn) converges to +� +π(t−), t +� +∈ Gf(π). In case (ii), the same +argument shows that (xn, tn) converges to +� +π(t+), t +� +∈ Gf(π). In case (iii), finally, using the +compactness of ⟨π(t−), π(t+)⟩, we can select a further subsequence such that (xn, t) → (x, t) for +37 + +some x ∈ ⟨π(t−), π(t+)⟩. Since also in this case the limit (x, t) is an element of Gf(π), we are +done. +To see that the total order ⪯ on Gf(π) is compatible with the (induced) topology on Gf(π), +it suffices to show that +S := +�� +(x, s), (y, t) +� +∈ Gf(π) : (x, s) ≺ (y, t) +� +(5.2) +is an open subset of Gf(π)2. If +� +(x, s), (y, t) +� +is an element of S, then either: (i) s < t, or: (ii) +s = t ∈ R and x ̸= y. In case (i), we can choose s < S < T < t. Then +O := +�� +(x′, s′), (y′, t′) +� +∈ Gf(π) : s′ < S, T < t′� +(5.3) +is an open subset of Gf(π) such that +� +(x, s), (y, t) +� +∈ O ⊂ S. In case (ii), we recall that by +definition (x, t) ⪯ (y, t) if x ≤π(t−),π(t+) z, where by Lemma 4.19 the total order ≤π(t−),π(t+) +on ⟨π(t−), π(t+)⟩ is compatible with the topology. It follows that we can choose ε > 0 small +enough such that for z ∈ ⟨π(t−), π(t+)⟩, if d(z, x) < ε then (z, t) ≺ (y, t), while if d(z, y) < ε +then (x, t) ≺ (z, t). Next, we use the cadlag property of π to choose δ > 0 small enough such +that d +� +π(s±), x +� +> ε for all t < s < t + δ and d +� +π(s±), y +� +> ε for all t − δ < s < t. Then +O := +�� +(x′, s), (y′, u) +� +∈ Gf(π) : |s − t| ∨ |u − t| < δ, d(x′, x) ∨ d(y′, y) < ε +� +(5.4) +is an open subset of Gf(π) such that +� +(x, s), (y, t) +� +∈ O ⊂ S. Together, these observations prove +that S is an open subset of Gf(π)2. +The following lemma is similar to Lemma 3.2. +Lemma 5.1 (Characterisation of continuous graphs) Let X be a metrisable space. As- +sume that G ∈ K+(R(X)) and (∗, ±∞) ∈ G. Then G is the closed graph of a path π ∈ Πc(X) if +and only if for each t ∈ R, the set {x ∈ X : (x, t) ∈ G} has at most one element. +Proof Clearly, if G is the closed graph of a path π ∈ Πc(X), then for each t ∈ R, the set {x ∈ +X : (x, t) ∈ G} has at most one element. Conversely, if G has this property and (∗, ±∞) ∈ G, +then we define +I(π) := +� +t ∈ R : ∃x ∈ X s.t. (x, t) ∈ G +� +, +(5.5) +and we use the fact that G contains at each time at most one point to define π : I(π) → X by +� +π(t) +� +:= {x ∈ X : (x, t) ∈ G}. +(5.6) +We observe that I(π) is the intersection of R with the image of G under the continuous map +(x, t) �→ t. Since the continuous image of a compact set is compact, this proves that I(π) is +closed. To complete the proof, it suffices to show that π(tn) → π(t) for all tn, t ∈ I(π) such +that tn → t. It suffices to show that {π(tn) : n ∈ N} is precompact and its only cluster point is +π(t). Equivalently, we may show that each subsequence of π(tn) contains a further subsequence +that converges to π(t). By the compactness of G, for any subsequence, we can select a further +subsequence such that π(tn) → x for some x ∈ X with (x, t) ∈ G. But then x = π(t) by (5.6). +Proof of Lemma 3.2 By Lemma 3.1, the filled-in graph of a path π ∈ Π(X) corresponds to an +element of Ktot(R(X)). Properties (i) and (ii) now follow from the definition of the total order +⪯ on π and property (vi) of Lemma 2.14. +Assume that conversely, (G, ⪯) ∈ Ktot(X) contains (∗, ±∞) and satisfies (i) and (ii). We +claim that for each t ∈ R, there exist unique π(t−), π(t+) ∈ X such that π(t−) ⪯ π(t+) and +St := +� +x ∈ X : (x, t) ∈ G +� += ⟨π(t−), π(t+)⟩. +(5.7) +38 + +Indeed, St is a compact metrisable set, so we can choose a countable dense set {xn : n ∈ N} ⊂ St. +Set y0 := x0 and define yn as the maximum of xn and yn−1 in the total order ⪯ (n ≥ 1). By +the compactness of St, by going to a subsequence, we can assume that yn → π(t+) for some +π(t+) ∈ St. Then y′ ⪯ y for all y′ ∈ D and hence also for all y′ ∈ St since the order is compatible +with the topology. In the same way, we see that St has a (necessarily unique) minimal element +π(t−). By (ii), we conclude that St = ⟨π(t−), π(t+)⟩. We now define +I(π) := +� +t ∈ R : ∃x ∈ X s.t. (x, t) ∈ G +� +and +Is(π) := +� +t± : t ∈ I(π) +� +, +(5.8) +and we use the claim we have just proved to define π : Is(π) → X by +⟨π(t−), π(t+)⟩ := {x ∈ X : (x, t) ∈ G} +with +π(t−) ⪯ π(t+) +� +t ∈ I(π) +� +. +(5.9) +Since I(π) is the intersection of R with the image of G under the continuous map (x, t) �→ t, +which is compact, we see that I(π) is closed. +To complete the proof, it suffices to show that π : Is(π) → X is continuous. By symmetry, +it suffices to show that if τn ∈ Is(π) and t ∈ I(π) satisfy τ n > t for all n and τ n → t as n → ∞, +then π(τn) → π(t+). As in the proof of Lemma 5.1, it suffices to show that each subsequence +of π(τn) contains a further subsequence that converges to π(t+). By the compactness of G, for +any subsequence, we can select a further subsequence such that π(τn) → x for some x ∈ X such +that (x, t) ∈ G. By (iii), we have +� +π(t+), t +� +⪯ +� +π(τn), τ n +� +for all n, so using the fact that the +total order is compatible with the topology, we see that +� +π(t+), t +� +⪯ (x, t), which using the fact +that π(t+) is the maximal element of ⟨π(t−), π(t+)⟩ with respect to the order ⪯ identifies x as +π(t+). +5.2 +Polishness +In this subsection, we prove Propositions 3.3 and 3.4, and Lemma 3.5. Let X be a metrisable +space that is equipped with a betweenness that is compatible with the topology. By a slight +abuse of notation, for any G ∈ Ktot(R(X)), we set +mT,δ(G) := sup +� +d(x1, x2) : (xi, ti) ∈ G, −T ≤ ti ≤ T ∀i = 1, 2, +(x1, t1) ⪯ (x2, t2), t2 − t1 ≤ δ +� +mS +T,δ(G) := sup +� +d +� +x2, ⟨x1, x3⟩ +� +: (xi, ti) ∈ G and − T ≤ ti ≤ T ∀i = 1, 2, 3, +(x1, t1) ⪯ (x2, t2) ⪯ (x3, t3), t3 − t1 ≤ δ +� +. +(5.10) +Then mT,δ(Gf(π)) = mT,δ(π) for each π ∈ Πc(X) and mS +T,δ(Gf(π)) = mS +T,δ(π) for each π ∈ Π(X). +The following lemma is similar to Lemma 4.15. +Lemma 5.2 (Upper semi-continuity) Let X be a metrisable space and assume that Gn, G ∈ +Ktot(R(X)) satisfy Gn → G. Then, for each T < ∞ and δ > 0, +mT,δ(G) ≥ lim sup +n→∞ +mT,δ(Gn) +and +mS +T,δ(G) ≥ lim sup +n→∞ +mS +T,δ(Gn). +(5.11) +Proof We only prove the statement for the Skorohod modulus of continuity. The proof for +the traditional modulus of continuity is basically the same, but a bit simpler. The proof will +be very similar to the proof of Lemma 4.15. +By the compactness of [0, ∞] we can select a +subsequence for which limn→∞ mS +T,δ(Gn) exists and is equal to the limit superior of the original +sequence. Let εn > 0 converge to zero and pick (xn +i , tn +i ) ∈ Gn with −T ≤ tn +i ≤ T (i = 1, 2, 3), +39 + +(xn +1, tn +1) ⪯ (xn +2, tn +2) ⪯ (xn +3, tn +3), and tn +3 − tn +1 ≤ δ, such that d +� +xn +2, {xn +1, xn +3} +� +≥ mS +T,δ(Gn) − εn. Since +Gn → G in the topology on Ktot(R(X)), by the first inequality in (2.16) they also converge in the +topology on K+(R(X)), so by Lemma 2.6 there exists a compact C ⊂ R(X) such that Gn ⊂ C for +all n. It follows that we can select a subsequence such that (xn +i , tn +i ) → (xi, ti) for some (xi, ti) ∈ G +(i = 1, 2, 3). Recall from Proposition 4.8 that dtot = d⟨∞⟩ so by Lemma 4.5, convergence in +Ktot(R(X)) implies that G⟨m⟩ +n +→ G⟨m⟩ in the Hausdorff topology for any 1 ≤ m ≤ ∞. +In +particular, G⟨3⟩ +n +→ G⟨3⟩, which by Lemma 2.6 implies (x1, t1) ⪯ (x2, t2) ⪯ (x3, t3). Moreover +−T ≤ ti ≤ T (i = 1, 2, 3) and t3 − t1 ≤ δ, so +mS +T,δ(G) ≥ d +� +x2, {x1, x3} +� +≥ lim +n→∞ +� +mS +T,δ(Gn) − εn +� += lim +n→∞ mS +T,δ(Gn), +(5.12) +where we have used that the betweenness is compatible with the topology. Since we have chosen +our subsequence such that the right-hand side is equal to the limit superior of the original +sequence, this proves the claim. +Proof of Proposition 3.3 We observe that if X is Polish, then, by Lemma 2.19, so is R(X) +and hence, by Proposition 2.11, also Ktot(R(X)). By identifying a path with its filled-in graph, +we can identity Π(X) with a subset of Ktot(R(X)). The Skorohod topology on Π(X) is then the +induced topology from Ktot(R(X)). Therefore, in view of Lemma 4.17, it suffices to show that +Π(X), viewed as a subset of Ktot(R(X)), is a Gδ-subset of the latter. +We start by showing that condition (i) of Lemma 3.2 can be replaced by +(i)’ lim +δ→0 mS +T,δ(G) = 0 ∀T < ∞. +To see this, we argue as follows. If (i) does not hold, then for some t ∈ R, there exist (xi, ti) ∈ G +(i = 1, 2, 3) with (x1, t1) ⪯ (x2, t2) ⪯ (x3, t3) and x2 ̸∈ ⟨x1, x3⟩, which implies that mS +T,δ(G) ≥ +d +� +x2, ⟨x1, x3⟩ +� +> 0 for all δ > 0 and T < ∞ such that −T ≤ t ≤ T, so (i)’ clearly does not hold. +Conversely, if (i)’ does not hold, then for some T < ∞ and ε > 0 we can choose δn > 0 tending to +zero and (xn +i , tn +i ) ∈ G (i = 1, 2, 3) with (xn +1, tn +1) ⪯ (xn +2, tn +2) ⪯ (xn +3, tn +3) such that d +� +xn +2, ⟨xn +1, xn +3⟩ +� +≥ +ε. By the compactness of G, we can select a subsequence such that (xn +i , tn +i ) → (xi, ti) (i = 1, 2, 3). +Then clearly t1 = t2 = t3 =: t for some −T ≤ t ≤ T. Since the order is compatible with the +topology moreover (x1, t1) ⪯ (x2, t2) ⪯ (x3, t3). The fact that the betweenness is compatible +with the topology allows us to conclude that d +� +x2, ⟨x1, x3⟩ +� += limn→∞ d +� +xn +2, ⟨xn +1, xn +3⟩ +� +≥ ε. This +shows that x2 ̸∈ ⟨x1, x3⟩ and hence (i) does not hold. +Let H denote the set of all elements of Ktot(R(X)) that satisfy condition (ii) of Lemma 3.2. +By what we have just proved, +Π(X) = +� +G ∈ H : lim +δ→0 mS +T,δ(G) = 0 ∀T < ∞ +� +. +(5.13) +It follows from Lemma 5.2 that for each T < ∞ and ε, δ > 0, the set +GT,ε,δ := +� +G ∈ Ktot(R(X)) : mS +T,δ(L) ≥ ε +� +(5.14) +is a closed subset of K+(X 2) and hence its complement Ac +ε,δ is open. As a consequence, +G := +∞ +� +N=1 +∞ +� +n=1 +∞ +� +m=1 +Gc +N,1/n,1/m +(5.15) +is a Gδ-set. Formula (5.13) says that Π(X) = G ∩ H. It is easy to see that H is a closed subset +of Ktot(R(X)), and hence a Gδ-set. Since the intersection of two Gδ-sets is a Gδ-set, this yields +the statement we wanted to prove. +Proof of Proposition 3.4 By Theorem 2.10, dS +part and dS +tot generate the same topology on Π(X), +and by (2.16) convergence in any of these two metrics implies convergence in dH. Therefore, +40 + +to show that the conditions (i)–(iii) are equivalent, it suffices to show that if πn ∈ Π(X) and +π ∈ Πc(X), then (iii) implies (i). More precisely, we will show that for any betweenness on X +that is compatible with the topology, Gf(πn) → G(π) in the Hausdorff topology implies +Gf(πn)⟨2⟩ −→ +n→∞ G(π)⟨2⟩ +(5.16) +in the Hausdorff topology. +By Lemma 2.6, convergence of Gf(πn) implies the existence of a +compact set C ⊂ R(X) such that Gf(πn) ⊂ C for all n, which implies Gf(πn)⟨2⟩ ⊂ C2. To +complete the proof, by Lemma 2.6, we need to prove the following two statements. +(i) For every +� +(x, s), (y, t) +� +∈ G(π)⟨2⟩, there exist +� +(xn, sn), (yn, tn) +� +∈ Gf(πn)⟨2⟩ such that +� +(xn, sn), (yn, tn) +� +→ +� +(x, s), (y, t) +� +. +(ii) If a sequence +� +(xn, sn), (yn, tn) +� +∈ Gf(πn)⟨2⟩ has a cluster point +� +(x, s), (y, t) +� +∈ R(X)2, +then +� +(x, s), (y, t) +� +∈ G(π)⟨2⟩. +To prove (i), we use the fact that Gf(πn) → G(π) to find (xn, sn), (yn, tn) ∈ Gf(πn) such that +(xn, sn) → (x, s) and (yn, tn) → (y, t). If s < t, then +� +(xn, sn), (yn, tn) +� +∈ Gf(πn)⟨2⟩ for n large +enough, so (i) follows. On the other hand, if s = t, then +� +(xn, sn), (xn, sn) +� +∈ Gf(πn)⟨2⟩ so (i) +also holds in this case. +To prove (ii), we use the fact that Gf(πn) → G(π) to conclude that any cluster point +� +(x, s), (y, t) +� +satisfies (x, s), (y, t) ∈ G(π) with s ≤ t, and hence by the continuity of π either +s < t or (x, s) = (y, t), from which we conclude that +� +(x, s), (y, t) +� +∈ G(π)⟨2⟩. +It remains to prove that Πc(X) is Polish. This is very similar to the proof of Proposition 3.3, +but simpler, so we only sketch the argument. By Lemma 5.1, we may identify Πc(X) with the +subset of K+(R(X)) consisting of all G that contain (∗, ±∞) and have the property that for +each t ∈ R, the set {x ∈ X : (x, t) ∈ G} has at most one element. In this identification, the +Hausdorff metric on K+(R(X)) induces the metric dH which generates the topology on Πc(X). +We claim that the condition that for each t ∈ R, the set {x ∈ X : (x, t) ∈ G} has at most one +element is equivalent to +lim +δ→0 mT,δ(G) = 0 ∀T < ∞. +(5.17) +This follows by the same sort of argument as in the proof of Proposition 3.3, where it was shown +that condition (i)’ there is equivalent to condition (i) of Lemma 3.2. Thus, identifying a path +with its closed graph, we have +Πc(X) = +� +G ∈ K+(X) : lim +δ→0 mT,δ(G) = 0 ∀T < ∞ +� +. +(5.18) +By the same argument as in the proof of Proposition 3.3, it follows that Πc(X) is a Gδ-subset +of K+(X) and hence, by Lemmas 2.7 and 4.17, a Polish space if X is Polish. +Our next aim is the proof of Lemma 3.5. The proof of the final statement of that lemma +needs a bit of preparation. Assume that π ∈ Π(X) is not the trivial path. Then we define +G∗(π) := +� +(x, t) ∈ G(π) : t ∈ I(π) +� +, +(5.19) +where I(π) denotes the closure of I(π) in R. This is almost the same as the closed graph G(π), +except that we include the points at infinity (∗, ±∞) only if their time coordinate lies in I(π). +Since G∗(π) is a subset of G(π), it is naturally equipped with a total order that is compatible +with the topology, so we can view it as an element of the space Ktot(X). The reason why we +usually work with G(π) instead of G∗(π) is that if we would use the latter throughout, we would +41 + +end up with a space of paths that contains three trivial paths, whose graphs would be {(∗, −∞)}, +{(∗, −∞)}, and the union of these two. The following lemma says that as long as we restrict +ourselves to nontrivial paths whose domain is an interval, it does not matter which definition of +the closed graph we use. +Lemma 5.3 (Convergence of graphs) Assume that πn, π ∈ Π|(X) and that π is not the +trivial path. Then G(πn) → G(π) in the topology om K+(R(X)) if and only if G∗(πn) → G∗(π) +in the topology om K+(R(X)). +Proof Let π1, π2 ∈ Π(X). Each correspondence between G∗(π1) and G∗(π2) can be extended +to a correspondence between G(π1) and G(π2) by adding the points +� +(∗, −∞), (∗, −∞) +� +and +� +(∗, +∞), (∗, +∞) +� +. Adding these extra points does not change the supremum in (2.8), so we +see that +dH +� +G(π1), G(π2) +� +≤ dH +� +G∗(π1), G∗(π2) +� +� +π1, π2 ∈ Π(X) +� +. +(5.20) +From this we see immediately that G∗(πn) → G∗(π) in the Hausdorff topology implies G(πn) → +G(π) in the Hausdorff topology. This part of the argument holds for general πn, π ∈ Π(X). +To complete the proof, we must show that if πn, π ∈ Π|(X) and π is not the trivial path, then, +conversely, G(πn) → G(π) in the Hausdorff topology implies G∗(πn) → G∗(π) in the Hausdorff +topology. By Lemma 2.6, there exists a compact C ⊂ R(X) such that G(πn) ⊂ C for all n. +Since the G∗(πn) are subsets of G(πn), they are contained in C too, so by Lemma 2.6 it suffices +to show that: +(i) For each (x, t) ∈ G∗(π), there exist (xn, tn) ∈ G∗(πn) such that (xn, tn) → (x, t). +(ii) If (x, t) is a cluster point of (xn, tn) ∈ G∗(πn), then (x, t) ∈ G∗(π). +To prove (i), we observe that since G∗(π) ⊂ G(π), for each (x, t) ∈ G∗(π), there exist (xn, tn) ∈ +G(πn) such that (xn, tn) → (x, t). If t ∈ R, then tn ∈ R for n large enough and hence (xn, tn) ∈ +G∗(πn) and we are done. If t = ∞, then by the fact that π ∈ Π|(X) and π is not the trivial +path, we see that for each T < ∞, we must have I(πn) ∩ [T, ∞) ̸= ∅ for all n large enough. +Using this, we see that there exist (xn, tn) ∈ G(πn) with tn < ∞ such that (xn, tn) → (x, ∞). +But then (xn, tn) ∈ G∗(πn), as required. The proof when t = −∞ is the same, so the proof of +(i) is complete. +To prove (ii), we observe that since G∗(πn) ⊂ G(πn) → G(π), if (x, t) is a cluster point +of (xn, tn) ∈ G∗(πn), then (x, t) ∈ G(π). +If t ∈ R, then clearly (x, t) ∈ G∗(π) and we are +done. If t = ∞, then by the assumption that π is not trivial we can choose (y, s) ∈ G(π) with +s ∈ R, and by the assumption that G(πn) → G(π) we can choose (yn, sn) ∈ G(πn) such that +(yn, sn) → (y, s). Since πn ∈ Π|(X) for all n, it follows that I(πn) contains (sn, tn) for each +n and hence by the assumption that G(πn) → G(π), the domain I(π) contains (s, ∞), which +implies that (∗, ∞) ∈ G∗(π). The argument when t = −∞ is the same so we are done. +The following lemma reveals a pleasant property of G∗(π) that G(π) does not have. +Lemma 5.4 (Connected graphs) Assume that π ∈ Π(X) is not the trivial path. Then π ∈ +Π| +c(X) if and only if G∗(π) is connected. +Proof If π ∈ Π| +c(X), then G∗(π) is the image of the compact set I(π) under the continuous map +from R to R(X) given by t �→ +� +π(t), t) (with ±∞ �→ (∗, ±∞)). Since I(π) is connected and the +continuous image of a connected set is connected, we conclude that G∗(π) is connected. +Conversely, if G∗(π) is connected, then I(π) must be connected and hence π ∈ Π|(X). To see +that π is moreover continuous, assume that conversely, π(t−) ̸= π(t+) for some t ∈ I(π). Then +42 + +we can define new paths π′, π′′ with domains I(π′) := (−∞, t] ∩ I(π) and I(π′′) := [0, ∞) ∩ I(π), +by setting π′(s) := π(s) and π′′(s) := π(s) for s ̸= t, and π′(t±) := π(t−) and π′′(t±) := +π(t+). By Lemma 3.1, G∗(π′) and G∗(π′′) are compact sets. Since G∗(π′) ∩ G∗(π′′) = ∅ and +G∗(π′) ∪ G∗(π′′) = G∗(π), this proves that G∗(π) is not connected. +Proof of Lemma 3.5 By the first inequality in (2.16), convergence πn → π in Π(X) implies +convergence of Gf(πn) to Gf(π) in the Hausdorff topology, which by Lemma 4.3 implies conver- +gence of I(πn) ∪ {±∞} to I(π) ∪ {±∞} in K+(R). Using Lemma 2.6, it is easy to see that if In +are closed subintervals of R such that In ∪ {±∞} converges in K+(R) to a limit, then this limit +must be of the form I ∪{±∞} for some (possibly empty) closed interval I ⊂ R. This shows that +Π|(X) is closed and in the same way we also see that Π↑(X) and Π↓(X) are closed. +Now assume that the betweenness is the trivial betweenness. Assume that πn ∈ Π| +c converge +to π ∈ Π(X). We need to show that π ∈ Π| +c. This is certainly true if π is the trivial path, so +we assume from now on that π is nontrivial. By the first inequality in (2.16), convergence of +πn to π in the topology on Π(X) implies that G(πn) → G(π) in the Hausdorff topology, which +by Lemma 5.3 implies that also G∗(πn) → G∗(π) in the Hausdorff topology. By Lemma 5.4, the +graphs G∗(πn) are connected and hence Lemma 2.9 implies that G∗(π) is connected, which by +Lemma 5.4 implies that π ∈ Π| +c. +5.3 +Compactness criteria +In this subsection, we prove Theorems 3.6 and 3.7. Lemmas 3.2 and 5.1 will play an essential +role here. +Proof of Theorem 3.6 By Lemma 5.1, we may identify Πc(X) with the subset of K+(R(X)) +consisting of all G that contain (∗, ±∞) and have the property that for each t ∈ R, the set +{x ∈ X : (x, t) ∈ G} has at most one element. +In this identification, the Hausdorff metric +on K+(R(X)) induces the metric dH on Πc(X), which by the definition above Proposition 3.4 +generates the topology on Πc(X). For any G ∈ K+(R(X)), we define +mT,δ(G) := sup +� +d(x1, x2) : (x1, t1), (x2, t2) ∈ G, −T ≤ t1 < t2 ≤ T, t2 − t1 ≤ δ +� +. +(5.21) +In the special case that G is (the closed graph of) a path in Πc(X), this coincides with the +definition of the modulus of continuity in (3.10). +Let A ⊂ Πc(X). Then A is precompact if and only if its closure A in K+(R(X)) is compact +and A ⊂ Πc(X). By Lemmas 2.8 and 2.20, A is a compact subset of K+(R(X)) if and only if +A satisfies the compact containment condition. Therefore, to complete the proof, it suffices to +show that if A ⊂ Πc(X) satisfies the compact containment condition, then A ⊂ Πc(X) if and +only if A is equicontinuous. +Assume that A satisfies the compact containment condition and is equicontinuous, and that +Gn ∈ A satisfy Gn → G for some G ∈ K+(R(X)). We need to show that G ∈ Πc(X). Clearly +(∗, ±∞) ∈ G, so it suffices to show that for each t ∈ R, the set {x ∈ X : (x, t) ∈ G} has at most +one element. Assume that conversely, there exist (x1, t), (x2, t) ∈ G with t ∈ R and x1 ̸= x2. +Then by Lemma 2.6, there exist (xn +1, tn +1), (xn +2, tn +2) ∈ Gn such that (xn +i , tn +i ) → (xi, t) as n → ∞ +(i = 1, 2). Choose T < ∞ such that −T < t < T. Then for each δ > 0 we have for n large +enough that −T < tn +1, tn +2 < T, |tn +1 − tn +2| ≤ δ, and d(xn +1, xn +2) ≥ d(x1, x2)/2. This proves that +sup +π∈A +mT,δ(π) ≥ d(x1, x2)/2 +∀δ > 0, +(5.22) +contradicting the equicontinuity of A. +43 + +Assume, on the other hand, that A satisfies the compact containment condition and is not +equicontinuous. Let δn be positive constants tending to zero. Since A is not equicontinuous, +for some T < ∞ and ε > 0 we can find Gn ∈ A such that mT,δn(Gn) ≥ ε for all n. Since A +satisfies the compact containment condition, A is a compact subset of K+(R(X)), so by going to +a subsequence we may assume that Gn → G for some G ∈ K+(R(X)). Since mT,δn(Gn) ≥ ε we +can find (xn +1, tn +1), (xn +2, tn +2) ∈ Gn such that −T ≤ tn +1 < tn +2 ≤ T, tn +2 −tn +1 ≤ δn, and d(xn +1, xn +2) ≥ ε. By +Lemma 2.6, going to a further subsequence if necessary, we can assume that (xn +i , tn +i ) → (xi, t) as +n → ∞ (i = 1, 2) for some (x1, t), (x2, t) ∈ G. Then −T ≤ t ≤ T and d(x1, x2) ≥ ε, which shows +that {x ∈ X : (x, t) ∈ G} has more than one element and hence A is not contained in Πc(X). +Proof of Theorem 3.7 By Lemma 3.2, identifying a path with its filled-in graph, we may +identify Π(X) with the subset of Ktot(R(X)) consisting of all (G, ⪯) that contain (∗, ±∞) and +satisfy conditions (i) and (ii) of the lemma. In this identification, the metrics dpart and dtot on +Ktot(R(X)) induce metrics dS +part and dS +tot on Π(X) that both generate the Skorohod topology. +Let A ⊂ Π(X). Then A is precompact in the Skorohod topology if and only if its closure A +in Ktot(R(X)) is compact and A ⊂ Π(X). By Theorem 2.12 and Lemma 2.20, A is a compact +subset of Ktot(R(X)) if and only if A satisfies the compact containment condition and +lim +ε→0 sup +G∈A +mε(G) = 0, +(5.23) +where mε(G) denotes the mismatch modulus of G. To complete the proof we will prove the +following three statements. +I Assume that A is satisfies the compact containment condition and is Skorohod-equicon- +tinuous. Then A satisfies (5.23). +II Assume that A is Skorohod-equicontinuous. Then A ⊂ Π(X). +III Assume that A is a compact subset of Ktot(R(X)) and that A ⊂ Π(X). +Then A is +Skorohod-equicontinuous. +Now if A satisfies the compact containment condition and is Skorohod-equicontinuous, then by +our earlier remarks I implies that A is a compact subset of Ktot(R(X)) and II implies that +A ⊂ Π(X), so A is precompact in the Skorohod topology. Conversely, if A is precompact in the +Skorohod topology, then A is a compact subset of Ktot(R(X)) and hence by our earlier remarks +A satisfies the compact containment condition, and moreover A ⊂ Π(X) which by III implies +that A is Skorohod-equicontinuous. +We start by proving I. Since A satisfies the compact containment condition, by Lemma 2.20, +we see that there exists a compact C ⊂ R(X) such that G ⊂ C for all G ∈ A. +Since +supG∈A mε(G) is nondecreasing as a function of ε, the limit in (5.23) always exists. +Let εn +be positive constants, tending to zero. +If the limit in (5.23) is positive, then there exists a +δ > 0 such that for each n we can find a G ∈ A and (xn +i , sn +i ), (yn +i , tn +i ) ∈ G (i = 1, 2) with +(xn +1, sn +1) ⪯ (yn +1 , tn +1), (yn +2 , tn +2) ⪯ (xn +2, sn +2) such that +dsqz +� +(xn +1, sn +1), (xn +2, sn +2) +� +∨ dsqz +� +(yn +1 , tn +1), (yn +2 , tn +2) +� +≤ εn, +dsqz +� +(xn +i , sn +i ), (yn +i , tn +i ) +� +≥ δ +(i = 1, 2). +Since G ⊂ C for all G ∈ A, by going to a subsequence, we may assume that (xn +i , sn +i ) → (x, s) and +(yn +i , tn +i ) → (y, t) (i = 1, 2) for some (x, s), (y, t) ∈ R(X). Then dsqz +� +(x, s), (y, t) +� +≥ δ and hence +(x, s) ̸= (y, t). Since (xn +1, sn +1) ⪯ (yn +1 , tn +1) and (yn +2 , tn +2) ⪯ (xn +2, sn +2) we have sn +1 ≤ tn +1 and tn +2 ≤ sn +2 for +all n which implies s = t and hence x ̸= y, since (x, t) ̸= (y, t). By the structure of R(X), this +44 + +implies t ∈ R. Let (xn +−, sn +−) (resp. (xn ++, sn ++)) be the smallest (resp. largest) of the points (xn +i , sn +i ) +(i = 1, 2) with respect to the order ⪯, and define (yn +±, tn +±) similarly. Since G is totally ordered, +by going to a subsequence, we can assume that we are in one of the following two cases. 1. +(xn +−, sn +−) ⪯ (yn +−, tn +−) for all n, or 2. (yn +−, tn +−) ⪯ (xn +−, sn +−) for all n. Let us assume that we are in +case 1. Then (xn +−, sn +−) ⪯ (yn +−, tn +−) ⪯ (xn ++, sn ++) for all n. Since the betweenness is compatible with +the topology, +d +� +yn +−, ⟨xn +−, xn ++⟩ +� +−→ +n→∞ d(y, x) > 0, +(5.24) +which contradicts the Skorohod-equicontinuity. Case 2 is completely the same, exchanging the +roles of x and y. +We next prove II. Assume that πn ∈ A converge in Ktot(R(X)) to a limit G. Recall from +Subsection 4.3 that dpart = d⟨2⟩ ≤ d⟨m⟩ ≤ d⟨∞⟩ = dtot for all m ≥ 2. In particular, πn → G +in Ktot(X) implies that π⟨m⟩ +n +→ G⟨m⟩ in the Hausdorff topology for all m ≥ 2. +It suffices +to check that G satisfies conditions (i) and (ii) of Lemma 3.2. +Condition (ii) easily follows +from the fact that π⟨2⟩ +n +→ G⟨2⟩ in the Hausdorff topology, using Lemma 2.6. +It remains to +prove that G satisfies condition (i) of Lemma 3.2. Assume that conversely, for some t ∈ R, +there exist (x1, t), (x2, t), (x3, t) ∈ G with (x1, t) ⪯ (x2, t) ⪯ (x3, t) such that x2 ̸∈ ⟨x1, x3⟩. +Since π⟨3⟩ +n +→ G⟨3⟩ in the Hausdorff topology, by Lemma 2.6 there exist τ n +i ∈ Is(πn) ∪ {±∞} +(i = 1, 2, 3) with τ n +1 ≤ τ n +2 ≤ τ n +3 such that τ n +i → t and πn(τ n +i ) → xi (i = 1, 2, 3). Using the fact +that the betweenness is compatible with the topology, we see that +d +� +πn(τ n +2 ), ⟨πn(τ n +1 ), πn(τ n +3 )⟩ +� +−→ +n→∞ d +� +x2, ⟨x1, x3⟩ +� +, +(5.25) +which is easily seen to contradict Skorohod-equicontinuity, completing the proof of II. +To prove III, finallly, we will show that if A is a compact subset of Ktot(R(X)) and A is +not Skorohod-equicontinuous, then A is not contained in Π(X). Let δn be positive constants, +tending to zero. Since A is not Skorohod-equicontinuous, there exists a ε > 0, T < ∞, πn ∈ A, +and τ n +i +∈ Is(πn) (i = 1, 2, 3) such that τ1 ≤ τ2 ≤ τ3, −T ≤ τ n +1, τ n +3 ≤ T, τ n +3 − τ n +1 ≤ δn, +and d +� +π(τ n +2 ), ⟨π(τ n +1 ), π(τ n +3 )⟩ +� +≥ ε. Since A is a compact subset of Ktot(R(X)), we can select a +subsequence such that πn → G for some G ∈ Ktot(R(X)). Then π⟨3⟩ +n +→ G⟨3⟩ in the Hausdorff +topology, so by Lemma 2.6 there exist C ⊂ R(X)3 such that π⟨3⟩ +n +⊂ C for all n. It follows that +by going to a further subsequence we can assume that +� +πn(τ n +i ), τ n +i +� +→ (xi, ti) as n → ∞ for some +(xi, t) ∈ G (i = 1, 2, 3) with (x1, t1) ⪯ (x2, t2) ⪯ (x3, t3) and −T ≤ t ≤ T. Using the fact that +the betweenness is compatible with the topology, we see that d(x2, ⟨x1, x3⟩) ≥ ε, which shows +that G is not the filled-in graph of a path π ∈ Π(X) and hence A is not contained in Π(X). +5.4 +Paths on fixed domains +In this subsection, we prove Lemma 3.8 and Theorem 3.9. +Proof of Lemma 3.8 We start by proving the statement for dH. Let T := {t > 0 : f(t−) = +f(t)}, which is dense in [0, ∞) by Lemma 4.2. Let G := Gf(f), Gn := Gf(fn), Gt := Gf(f +�� +[0,t]), +and Gt +n := Gf(fn +�� +[0,t]). +We first prove the implication ⇒. Let t ∈ T. Since Gn → G, by Lemma 2.6, there exists a +compact set C such that Gn ⊂ C for all n, and hence also Gt +n ⊂ C for all n. By Lemma 2.8, it +follows that {Gt +n : n ∈ N} is compact, so to prove that Gt +n → Gt, it suffices to show that Gt is +the only subsequential limit of the Gt +n. Let G∗ be such a subsequential limit. Since Gt +n ⊂ Gn it +is clear from Lemma 2.6 that G∗ ⊂ G. Let ψ denote the projection ψ(x, s) := s and let ψ(G∗) +denote the image of G∗ under ψ. By Lemma 4.3, ψ(G∗) = [0, t]. It follows that G∗ ⊂ Gt. To +45 + +prove the opposite inclusion, assume that (y, s) ∈ Gt. If s < t, then we use that by Lemma 2.6, +there exist (xn, sn) ∈ Gn such that (xn, sn) → (x, s). Since s < t, we have (xn, sn) ∈ Gt +n for +n large enough and hence (x, s) ∈ G∗. If s = t, then we use that ψ(G∗) = [0, t] to conclude +that there must be at least one y′ ∈ X such that (y′, t) ∈ G∗. Since G∗ ⊂ Gt we must have +y, y′ ∈ ⟨f(t−), f(t)⟩ = {f(t)}, where we have used that t ∈ T, so we conclude that y′ = y, +concluding the proof that G∗ = Gt. +We next prove the implication ⇐. Since Gt +n → Gt for each t ∈ T, using Lemmas 2.6 and +2.20, we see that there exists a compact set C such that Gn ⊂ C for all n, so by Lemma 2.8 it +suffices to show that if G∗ is a subsequential limit of the Gn, then G∗ = G. Since Gt +n ⊂ Gn for +each n it is clear from Lemma 2.6 that Gt ⊂ G∗ for each t ∈ T. We claim that conversely, for +each (x, s) ∈ G∗ and s < t ∈ T, we have (x, s) ∈ Gt. Indeed, by Lemma 2.6 for some subsequence +there exist (xn, sn) ∈ Gn such that (xn, sn) → (x, s). Since sn < t for n large enough, it follows +that (x, s) ∈ Gt. These arguments show that {(x, t) ∈ G∗ : t < ∞} = {(x, t) ∈ G : t < ∞}, +which is enough to conclude G∗ = G. +We next prove the statement for dS +tot. Let π := f, πn := fn, πt := f +�� +[0,t], and πt +n := fn +�� +[0,t], +which we view as elements of the path space Π(X). We first prove the implication ⇒. By +Theorem 3.7, dS +tot(πn, π) → 0 implies that {πn : n ∈ N} is Skorohod-equicontinuous and satisfies +the compact containment condition, which implies the same is true for {πt +n : n ∈ N} for any +t ∈ T. Therefore, by Theorem 3.7, it suffices to show that all subsequential limits of the πt +n are +equal to πt. Since by Theorem 2.10, convergence in dS +tot implies convergence in dH +tot, we can use +what we have already proved for dH +tot to draw the desired conclusion. The implication ⇐ follows +in the same way, where now we use that if {πt +n : n ∈ N} is Skorohod-equicontinuous and satisfies +the compact containment condition for any t ∈ T, then the same is true for {πn : n ∈ N}. +Proof of Theorem 3.9 We view DI(X) as a subset of Π(X) as in (3.14). Then F is compact +as a subset of DI(X) if and only if its closure F in the larger space Π(X) is compact and satisfies +F ⊂ DI(X). By Theorem 3.7, F is a compact subset of Π(X) if an only if conditions (i) and +(ii) hold. To complete the proof, we will show that, assuming (i) and (ii), one has F ⊂ DI(X) +if and only if (iii) holds. +We first show that (iii) implies that F ⊂ DI(X). Assume that (iii) holds and let fn ∈ DI(X) +and π ∈ Π(X) satisfy fn → π in the Skorohod topology associated with the given betweenness. +Then clearly I(π) = I. To show that π ∈ DI(X) assume that conversely π(t−) ̸= π(t+) for +some t ∈ ∂I. Then, by the fact that dS +part(fn, π) → 0, there exist sn, tn ∈ I with sn < tn such +that fn(sn) → π(t−) and fn(tn) → π(t+), which is easily seen to contradict (iii). +Assume, on the other hand, that (iii) does not hold for some t ∈ ∂I. Let δn be positive +constants, tending to zero. Then there exists an ε > 0 such that for each n, we can find fn ∈ F +and sn ∈ I with |sn − t| ≤ δn and d +� +fn(sn), fn(t) +� +≥ ε. By (i) and (ii), F is compact in Π(X) +so by going to a subsequence we can assume that fn → π for some π ∈ Π(X). By (i), we can +moreover assume that fn(sn) → x and fn(t) → y for some x, y ∈ X. Then d(x, y) ≥ ε and +(x, t), (y, t) ∈ Gf(π), which by Lemma 2.14 (v) shows that π(t−) ̸= π(t+) and hence F is not +contained in DI(X). +Acknowledgments +We thank Jan Seidler for useful discussions and for his help in understanding [AU29]. +46 + +References +[AU29] +P. Alexandroff and P. Urysohn. M´emoire sur les espaces topologiques compacts, d´edi´e +`a Monsieur D. Egoroff. Verhandelingen Amsterdam 14(1) (1929). +[Bil99] +P. Billingsley. Convergence of Probability Measures. 2nd ed. John Wiley & Sons, 1999. +[Bou58] +N. Bourbaki. ´El´ements de Math´ematique. VIII. Part. 1: Les Structures Fondamen- +tales de l’Analyse. Livre III: Topologie G´en´erale. Chap. 9: Utilisation des Nombres +R´eels en Topologie G´en´erale. 2i´eme ´ed. Actualit´es Scientifiques et Industrielles 1045. +Hermann & Cie, Paris, 1958. +[DT96] +A.W.M. Dress and W.F. Terhalle. The real tree. Adv. Math. 120 (1996), 283–301. +[EK86] +S.N. 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Schertzer, R. Sun and J.M. Swart. Stochastic flows in the Brownian web and net. +Mem. Am. Math. Soc. Vol. 227 (2014), Nr. 1065. +[SSS17] +E. Schertzer, R. Sun and J.M. Swart. The Brownian web, the Brownian net, and +their universality. Pages 270–368 in: P. Contucci and C. Giardin`a (Eds.) Advances in +Disordered Systems, Random Processes and Some Applications. Cambridge University +Press, 2017. +[Whi02] +W. Whitt. Stochastic-Process Limits. An introduction to stochastic-process limits and +their application to queues. Springer, New York, 2002. +47 + diff --git a/e9E5T4oBgHgl3EQfhA8F/content/tmp_files/load_file.txt b/e9E5T4oBgHgl3EQfhA8F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..be9e0a538d3b1c7f4245c5051c9d90a46bed8ab0 --- /dev/null +++ b/e9E5T4oBgHgl3EQfhA8F/content/tmp_files/load_file.txt @@ -0,0 +1,3051 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf,len=3050 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='05637v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='GN] 15 Dec 2022 Skorohod’s topologies on path space Nic Freeman∗1 and Jan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Swart†2 1School of Mathematics and Statistics, University of Sheffield 2The Czech Academy of Sciences, Institute of Information Theory and Automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' January 16, 2023 Abstract We introduce the path space over a general metrisable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Elements of this space are paths, which are pairs consisting of a closed subset of the real line and a cadlag function that is defined on that subset and takes values in the metrisable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We equip the space of all paths with topologies that generalise Skorohod’s J1 and M1 topologies, prove that these topologies are Polish, and derive compactness criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The central idea is that the closed graph (in case of the J1 topology) and the filled-in graph (in case of the M1 topology) of a path can naturally be viewed as totally ordered compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We define a variant of the Hausdorff metric that measures the distance between two compact sets, each of which is equipped with a total order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We show that the topology generated by this metric is Polish and derive a compactness criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Specialising to closed or filled-in graphs then yields Skorohod’s J1 and M1 topologies, generalised to functions that need not all be defined on the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' MSC 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To be filled in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Primary: 26A15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Secondary: 06A05, 54E35, 60G07 Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Skorohod topology, J1 topology, M1 topology, path space, Hausdorff metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Work sponsored by GAˇCR grant 22-12790S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' ∗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='freeman@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='uk †swart@utia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='cas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='cz 1 Contents 1 Introduction 3 2 Preliminaries 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 The split real line .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 Cadlag functions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 43 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 Paths on fixed domains .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 45 2 1 Introduction A function is cadlag (from the French “continue `a droit, limite `a gauche”) if it is right-continuous with left limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In his classical paper [Sko56], Skorohod introduced four topologies on the space of real cadlag functions on a compact interval, which he called J1, J2, M1, and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Of these, the J1 topology has proved to be the most natural in many situations, in particular, when discussing convergence of Markov processes [EK86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For this reason, Skorohod’s J1 topology is now normally known as the “Skorohod topology”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Classical textbook discussions of the Skorohod topology can be found in [EK86, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5] and [Bil99, Section 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' All four topologies introduced by Skorohod are discussed in [Whi02, Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Motivated by the theory of the Brownian web and net [FINR04, SSS17], we study paths, which, roughly speaking, are cadlag functions that are defined on an arbitrary closed subset of the real line and take values in a metrisable topological space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We generalise Skorohod’s topologies to the space of all paths over a fixed space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Roughly, a sequence of paths converges when the domains on which they are defined converge and moreover the paths themselves converge in the sense of Skorohod’s J1 or M1 topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We show that the path space, equipped with such a topology, is a Polish space, and we derive compactness criteria in terms of a suitable modulus of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Polish spaces play a crucial role in probability theorem, as they are required for Prohorov’s theorem [Bil99, Thm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2], and sufficient for many other theorems requiring some regularity of a measurable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In Section 2, we first develop the necessary topological material, which in Section 3 we then apply to path space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The remaining Sections 4 and 5 contain proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In Subsections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2, we develop an observation, due to Kolmogorov [Kol56], that a cadlag function defined on an interval, together with its left-continuous modification, can be viewed as a continuous function on a somewhat peculiar topological space introduced by Alexandroff and Urysohn [AU29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This makes it possible to allow cadlag functions to jump at their initial times, which later simplifies the compactness criteria, and also provides the right set-up to define paths whose domains may not be intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 we recall the Hausdorff metric, which measures the distance between two compact subsets of a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3, we introduce a variant of this metric that measures the distance between two compact subsets that are moreover each equipped with a total order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We will later apply this to the closed graph (in case of the J1 topology) or the filled-in graph (in case of the M1 topology), which can naturally be viewed as totally ordered compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The classical M1 topology is defined only for paths taking values in the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For paths taking values in more general spaces, there are several possible ways to generalise the M1 topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5, we introduce the concept of a “betweenness”, which will allow us to treat the J1 topology and various possible variants of the M1 topology in a unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Skorohod and Kolmogorov [Sko56, Kol56] only considered cadlag functions defined on com- pact time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The extension to unbounded time intervals is important in applications, but not completely trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For continuous paths, one defines locally uniform convergence of a sequence of functions by requiring that their restrictions to any compact time interval converge uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For the J1 and M1 topologies, this approach is not feasible, since the map that restricts a function to a smaller time interval is not continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To solve this, in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, we use an idea first developed in the theory of the Brownian web [FINR04], which is to intro- duce a topology on space-time that cares less about the spatial distance between two space-time points if their time coordinates are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This will allow us to view graphs of cadlag functions as compact sets, even when they are defined on unbounded time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' With all the right ingredients in place, in Section 3 we introduce and study the J1 and M1 3 topologies on path space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1, we show that the closed and filled-in graphs of a path can be viewed as totally ordered compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2, we then define metrics for the J1 and M1 topologies by measuring the distance between closed and filled-in graphs using the ordered Hausdorff metric from Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3, we derive compactness criteria in terms of a suitable modulus of continuity, generalising results from [Sko56, Kol56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4, we specialise to cadlag functions defined on a fixed time interval and show how our results relate to the classical textbook definitions of the J1 and M1 topologies, and also briefly discuss the less commonly used J2 and M2 topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 The split real line Let Rs be the space that consists of all words of the form t⋆ where t ∈ R is a real number and ⋆ ∈ {−, +} is a sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We think of Rs as obtained by cutting each point of the real line into two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Consequently, we call Rs the split real line and call elements of Rs split real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We denote split real numbers either by words t⋆ consisting of a Roman letter and a sign, or by a single Greek letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In this case, if τ = t⋆, then we call τ := t the real part of τ and we call s(τ) := ⋆ its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We equip Rs with the lexicographic order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=', we set σ ≤ τ if and only if either σ < τ or σ = τ and s(σ) ≤ s(τ), where {−, +} is equipped with the natural total order in which − ≤ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We write σ < τ if σ ≤ τ and σ ̸= τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We use notation for intervals in Rs similar to the usual notation for the real line, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=', (σ, ρ) := {τ ∈ Rs : σ < τ < ρ}, [σ, ρ) := {τ ∈ Rs : σ ≤ τ < ρ}, (σ, ρ] := {τ ∈ Rs : σ < τ ≤ ρ}, [σ, ρ] := {τ ∈ Rs : σ ≤ τ ≤ ρ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1) Note that there is some redundancy in this notation: for example, (s−, t+) = [s+, t−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We equip Rs with the order topology, which means that by definition, a basis for the topology on Rs is formed by all intervals of the form (σ, ρ) with σ, ρ ∈ Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We defer the (simple) proof of the following lemma, and all further results stated in this section, to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 (Convergence criterion) For τn ∈ Rs and t ∈ R, one has (i) τn → t+ if and only if τ n → t and τn ≥ t+ for all n sufficiently large, (ii) τn → t− if and only if τ n → t and τn ≤ t− for all n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Intervals of the form (σ, ρ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' [σ, ρ]) are open (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' closed) in the topology on Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In par- ticular, (s−, t+) = [s+, t−] is both open and closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma lists some elementary properties of Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 (The split real line) The space Rs is first countable, Hausdorff, and separable, but not second countable and not metrisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Moreover, Rs is totally disconnected, meaning that its only connected subsets are singletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We equip the product space Rd s with the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By definition, a subset A ⊂ Rd s is bounded if A ⊂ [σ, τ]d for some σ, τ ∈ Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following proposition gives a characterisation of the compact subsets of Rd s, similar to the well-known characterisation of compact subsets of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 4 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 (Compact sets) For a subset C ⊂ Rd s, the following three claims are equiv- alent: (i) C is compact, (ii) C is sequentially compact, and (iii) C is closed and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We can compactify Rs by adding two points ±∞, in such a way that −∞ < τ < +∞ for all τ ∈ Rs, and then equipping Rs := Rs ∪ {−∞, +∞} with the topology generated by intervals of the form (σ, τ), [−∞, τ), or (σ, +∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Instead of +∞ we also write ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We call Rs the extended split real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Note that we notationally distinguish the points at infinity ±∞ of the extended split real line from the points ±∞ of the extended real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We extend the functions τ �→ τ and τ �→ s(τ) that assign to each split real number τ its real part τ and sign s(τ) to the extended split real line Rs by setting ±∞ := ±∞ and s(−∞) := +, s(∞) = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Note that with these definitions Rs is naturally isomorphic to [0+, 1−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The extended split real line provides us with a natural way to denote half infinite intervals in Rs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' for example, [σ, ∞) = {τ ∈ Rs : σ ≤ τ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 Cadlag functions Let I ⊂ R be an interval, let X be a Hausdorff topological space, and let f : I → X be a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By definition, we say that f is cadlag if it is right-continuous with left limits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' : (i) f(t) = lim n→∞ f(tn) whenever tn, t ∈ I satisfy tn −→ n→∞ t and tn > t for all n, (ii) f(t−) := lim n→∞ f(tn) exists whenever tn, t ∈ I satisfy tn −→ n→∞ t and tn < t for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Similarly, a caglad function (from the French “continue `a gauche, limite `a droit”) is left- continuous with right limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For any closed interval I ⊂ R of nonzero length, we let DI(X) denote the space of all functions f : I → X such that: (i) f is cadlag, (ii) if t := sup I < ∞, then f(t−) = f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2) We impose condition (ii) in order to have a more symmetric definition, since cadlag functions can by construction not have a jump at the left boundary of their domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We let D− I (X) denote the space of all functions f : I → X such that: (i) f is caglad, (ii) if t := inf I > −∞, then f(t) = f(t+), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3) where f(t+) denotes the right limit of f at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The left-continuous modification of a function f ∈ DI(X) is the function f − ∈ D− I (X) uniquely defined by the requirement that f −(t) := f(t−) for all t ∈ I where the left limit is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Right-continuous modifications of functions in D− I (X) are defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A cadlag function f ∈ DI(X) and its left-continuous modification f − uniquely determine each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Indeed, f is the right-continuous modification of f −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If I ⊂ Rs is a closed subinterval of the split real line and X is a Hausdorff topological space, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1, a function f : I → X is continuous if and only if (i) f(τn) → f(t+) for all t+ ∈ I and τn ∈ I such that τn ≥ t+ for all n and τ n → t, (ii) f(τn) → f(t−) for all t− ∈ I and τn ∈ I such that τn ≤ t− for all n and τ n → t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We let CI(X) denote the space of continuous functions f : I → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Continuous functions on a closed subinterval of the split real line correspond more or less to cadlag functions on a closed subinterval of the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To make this connection precise, for any closed interval I ⊂ R of nonzero length, we define Iin ⊂ Rs by Iin := � t− : (t − ε, t] ⊂ I for some ε > 0 � ∪ � t+ : [t, t + ε) ⊂ I for some ε > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4) In particular, if I = [s, u], then Iin = [s+, u−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 5 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 (Cadlag functions as continuous functions) Let I be a closed real interval of nonzero length and let X be a Hausdorff topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let f + ∈ DI(X) and let f − ∈ D− I (X) be its left-continuous modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then setting f(t±) := f ±(t) (t± ∈ Iin) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5) defines a function f ∈ CIin(X), and each function f ∈ CIin(X) is of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In particular, if [s, u] is a compact real interval, then Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 says that there is a nat- ural isomorphism between the space of cadlag functions D[s,u](X) and the space of continuous functions C[s+,u−](X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' An advantage of working with the split real line is that we can also easily allow for functions that jump at the endpoints of their domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Indeed, if we replace C[s+,u−](X) by the slightly larger space C[s−,u+](X), then we obtain a space of functions that can also jump at the endpoints s and u of the real interval [s, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Although we will not need this in the present paper, we note that the split real line also leads to a natural definition of cadlag functions of several variables, since we can simply define them as continuous functions defined on (a subset of) the product space Rn s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This seems much simpler than the approach used by other authors such as [Neu71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 The Hausdorff metric For any metric space (X, d), we let K+(X) denote the space of all nonempty compact subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The Hausdorff metric dH on K+(X) is defined as dH(K1, K2) := sup x1∈K1 d(x1, K2) ∨ sup x2∈K2 d(x2, K1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6) where d(x, A) := infy∈A d(x, y) denotes the distance between a point x ∈ X and a set A ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We can alternatively define dH in terms of correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A correspondence between two sets A1, A2 is a set R ⊂ A1 × A2 such that ∀x1 ∈ A1 ∃x2 ∈ A2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (x1, x2) ∈ R and ∀x2 ∈ A2 ∃x1 ∈ A1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (x1, x2) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7) We let Cor(A1, A2) denote the set of all correspondences between A1 and A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5 (Hausdorff metric and correspondences) Let (X, d) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then dH(K1, K2) = inf R∈Cor(K1,K2) sup (x1,x2)∈R d(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8) Moreover, there exists an R ∈ Cor(K1, K2) such that dH(K1, K2) = max(x1,x2)∈R d(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We cite the following lemma from [SSS14, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 (Convergence criterion) Let Kn, K ∈ K+(X) (n ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then Kn → K in the Hausdorff topology if and only if there exists a C ∈ K+(X) such that Kn ⊂ C for all n ≥ 1 and K = {x ∈ X : ∃xn ∈ Kn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' xn → x} = {x ∈ X : ∃xn ∈ Kn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' x is a cluster point of (xn)n∈N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 shows that if d and d′ are two metrics generating the same topology on X, then the corresponding Hausdorff metrics dH and d′ H generate the same topology on K+(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We call this topology the Hausdorff topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Note the subtle difference between “the Hausdorff topology” (the topology generated by the Hausdorff metric) and “a Hausdorff topology” (any topology satisfying Hausdorff’s separation axiom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma is [SSS14, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In particular, it shows that K+(X) is Polish if X is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 6 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7 (Properties of the Hausdorff metric) (a) If (X, d) is separable, then so is (K+(X), dH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (b) If (X, d) is complete, then so is (K+(X), dH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Recall that a set is called precompact if its closure is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma is [SSS14, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In particular, it shows that K+(X) is compact if X is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 (Compactness in the Hausdorff topology) A set A ⊂ K+(X) is precompact if and only if there exists a C ∈ K+(X) such that K ⊂ C for each K ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma says connectedness is a property of compact sets that is preserved under limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9 (Preservation of connectedness) The set Kc(X) of all connected nonempty compact subsets of X is a closed subset of K+(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 The ordered Hausdorff metric We will need a variant of the Hausdorff metric that measures the distance between two compact sets, each of which is equipped with a total order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For any metric space (X, d), we let Kpart(X) denote the space of all pairs (K, ⪯) where K is a nonempty compact subset of X and ⪯ is a partial order on K that is compatible with the topology in the sense that the set K⟨2⟩ := � (x, y) ∈ K2 : x ⪯ y � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10) is a closed subset of K2, equipped with the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Note that we do not assume that X is equipped with a partial order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' in particular, the partial order on K does not have to come from an order on X, although we always assume that the topology on K is the induced topology from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We will sometimes be sloppy and denote elements of Kpart(X) simply as K, where it is implicitly understood that K is equipped with a partial order that is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We equip the space X 2 with the metric d2� (x1, y1), (x2, y2) � := d(x1, x2) ∨ d(y1, y2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11) which generates the product topology, and we equip the space K+(X 2) of compact nonempty subsets of X 2 with the associated Hausdorff metric d2 H(A1, A2) := sup (x1,y1)∈A1 d2� (x1, y1), A2 � ∨ sup (x2,y2)∈A2 d2� (x2, y2), A1 � � A1, A2 ∈ K+(X 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12) An element (K, ⪯) of Kpart(X) is clearly uniquely determined by the compact set K⟨2⟩ ⊂ X 2 defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10), so setting dpart(K1, K2) := d2 H(K⟨2⟩ 1 , K⟨2⟩ 2 ) � K1, K2 ∈ Kpart(X) � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13) defines a metric dpart on Kpart(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We let Ktot(X) denote the space of all pairs (K, ⪯) ∈ Kpart(X) such that ⪯ is a total order on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' There is a natural way to define a metric on Ktot(X) that is at first sight very different from the definition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Recall the definition of a correspondence from Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By definition, a correspondence R between two totally ordered spaces (K1, ⪯1) and (K2, ⪯2) is monotone if there are no (x1, x2), (y1, y2) ∈ R such that x1 ≺1 y1 and y2 ≺2 x2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14) 7 where x ≺ y means that x ⪯ y and x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We let Cor+(K1, K2) denote the space of all monotone correspondences between two totally ordered spaces K1 and K2, and define a metric dtot on Ktot(X) by dtot(K1, K2) := inf R∈Cor+(K1,K2) sup (x1,x2)∈R d(x1, x2) � K1, K2 ∈ Ktot(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15) The following theorem says that dpart and dtot generate the same topology on Ktot(X) and satisfy dpart ≤ dtot, but they do not satisfy an opposite inequality of the form dtot ≤ Cdpart for any C < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10 (The ordered Hausdorff topology) Let (X, d) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then the metrics dpart and dtot defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15) generate the same topology on Ktot(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Also, if d and d′ generate the same topology on X and dpart and d′ part are defined in terms of d and d′ as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13), then dpart and d′ part generate the same topology on Ktot(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' One has dH(K1, K2) ≤ dpart(K1, K2) ≤ dtot(K1, K2) � K1, K2 ∈ Ktot(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16) If X = [0, 1], then for each ε > 0, there exist K1, K2 ∈ Ktot(X) such that dpart(K1, K2) ≤ εdtot(K1, K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We call the topology on Ktot(X) generated by dpart or dtot the ordered Hausdorff topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The second claim of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10 says that this topology depends only on the topology on X and not on the choice of metric on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We recall that a topological space X is Polish if X is separable and there exists a complete metric generating the topology on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Note that being Polish is a property of the topology and not a property of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In fact, on each non- compact Polish space X, there also exist non-complete metrics that generate the topology on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 If (X, d) is complete, then as we will show in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13 below, Ktot(X) is not in general complete in the metrics dpart or dtot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Nevertheless, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11 (Preservation of Polishness) If X is a Polish space, then so is Ktot(X), equipped with the ordered Hausdorff topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Our next result characterises the compact subsets of Ktot(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For K ∈ Ktot(X) and ε > 0, we define the mismatch modulus mε(K) as mε(K) := sup � d(x1, y1) ∨ d(x2, y2) : x1, y1, x2, y2 ∈ K d(x1, x2) ∨ d(y1, y2) ≤ ε, x1 ⪯ y1, y2 ⪯ x2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='17) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12 (Compact subsets) Let (X, d) be a metric space and let Ktot(X) be equipped with the ordered Hausdorff topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then a set A ⊂ Ktot(X) is precompact if and only if (i) ∃ compact C ⊂ X s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' K ⊂ C ∀K ∈ A and (ii) lim ε→0 sup K∈A mε(K) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18) Recall the definition of the space D[0,1](X) of cadlag functions f : [0, 1] → X in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A cadlag parametrisation of an element K ∈ Ktot(X) is a function γ ∈ D[0,1](X) such that K = � γ(t), γ−(t) : t ∈ [0, 1] � and γ(s) ≺ γ(t) ∀0 ≤ s < t ≤ 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19) 1Indeed, it is well-known that each separable metric space X is homeomorphic to a subset of a compact metric space Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The completion of X in the metric from Y is equal to the closure of X in Y, so unless X is compact, it is not complete in the metric from Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 8 where γ− denotes the caglad modification of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Clearly, not every element of Ktot(X) has a cadlag representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For those that do, the following proposition gives an expression for the metrics dH and dtot that will later help us make the connection between our definitions and the classical definitions of the J1 and M1 topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let Λ be the space of all bijections λ : [0, 1] → [0, 1] and let Λ+ be the subset consisting of all bijections λ that are monotone in the sense that s ≤ t implies λ(s) ≤ λ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Note that each λ ∈ Λ+ is continuous and strictly increasing with λ(0) = 0 and λ(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13 (Distance between cadlag curves) Let (X, d) be a metric space, and as- sume that K1, K2 ∈ Ktot(X) have cadlag parametrisations γ1, γ2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then dH(K1, K2) = inf λ∈Λ sup t∈[0,1] d � γ1(t), γ2 � λ(t) �� , dtot(K1, K2) = inf λ∈Λ+ sup t∈[0,1] d � γ1(t), γ2 � λ(t) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5 Betweenness There are great similarities between Skorohod’s J1 and M1 topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In fact, it turns out to be possible to treat them in a unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To this aim, we introduce a natural concept that we will call “betweenness” and that seems to be new in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It seems quite conceivable it may have been invented in other contexts before, but we have been unable to find a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If X is a set, then we define a betweenness on X to be a function that assigns to each pair x, z of elements of X a subset ⟨x, z⟩ of X, such that the following axioms hold for all x, y, z ∈ X: (i) ⟨x, z⟩ = ⟨z, x⟩, (ii) x ∈ ⟨x, z⟩, (iii) y ∈ ⟨x, z⟩ ⇒ ⟨x, y⟩ ∩ ⟨y, z⟩ = {y}, (iv) y ∈ ⟨x, z⟩ ⇒ ⟨x, y⟩ ∪ ⟨y, z⟩ = ⟨x, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If y ∈ ⟨x, z⟩, then we say that y lies between x and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We call ⟨x, z⟩ the segment with endpoints x and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma lists some elementary consequences of the axioms (i)–(iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14 (Elementary properties) Each betweenness satisfies, for each x, y, y′, z ∈ X: (v) ⟨x, x⟩ = {x}, (vi) y ∈ ⟨x, z⟩ ⇒ ⟨x, y⟩ ⊂ ⟨x, z⟩, (vii) x ∈ ⟨y, z⟩ and y ∈ ⟨x, z⟩ ⇒ x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (viii) y, y′ ∈ ⟨x, z⟩ and y′ ∈ ⟨x, y⟩ ⇒ y ∈ ⟨y′, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For x, z ∈ X and y, y′ ∈ ⟨x, z⟩, one has ⟨x, y⟩ ⊂ ⟨x, y′⟩ ⇔ y ∈ ⟨x, y′⟩ ⇔ y′ ∈ ⟨y, z⟩ ⇔ ⟨y, z⟩ ⊃ ⟨y′, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='21) Setting y ≤x,z y′ if any of these equivalent conditions holds defines a total order on ⟨x, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 9 We next give some examples of betweennesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For any set X, it is straightforward to check that setting ⟨x, z⟩ := {x, z} defines a betweenness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We call this the trivial betweenness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If X is a linear space, then it is easy to see that ⟨x, z⟩ := � (1 − p)x + pz : p ∈ [0, 1] � (x, z ∈ X) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='22) defines a betweenness on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We call this the linear betweenness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If (X, ≤) is a totally ordered space, then one can check that setting ⟨x, z⟩ := � y ∈ X : x ≤ y ≤ z or z ≤ y ≤ x � (x, z ∈ X) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='23) defines a betweenness on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We call this the order betweenness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If (X, d) is a metric space, then we recall that a geodesic in (X, d) is a subset Γ of X that is isometric to a compact real interval, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=', there exists an isometry γ : [s, u] → X (with s, u ∈ R, s ≤ u) such that Γ is the image of [s, u] under γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Clearly, γ is uniquely determined by Γ up to translations and mirror images of the interval [s, u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The points γ(s), γ(u) are called the endpoints of the geodesic Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We say that a metric space has unique geodesics if for each x, z ∈ X, there exists a unique geodesic Γ with endpoints x, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We call the betweenness defined in the following lemma the geodesic betweennesss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15 (Geodesic betweenness) Let (X, d) be a metric space with unique geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then letting ⟨x, z⟩ denote the unique geodesic with endpoints x, z defines a betweenness on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' As an example of spaces without a linear structure where Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15 is applicable we mention real-trees [DT96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We say that a betweenness on a metrisable space X is generated by an interpolation function if there exists a continuous function ϕ : X 2 × [0, 1] → X that satisfies ϕ(x, z, 0) = x, ϕ(x, z, 1) = z, and ⟨x, z⟩ = � ϕ(x, z, p) : p ∈ [0, 1] � (x, z ∈ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='24) A metric space is called proper if for each x ∈ X and r ≥ 0, the closed ball {y ∈ X : d(x, y) ≤ r} is a compact subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We do not know if the properness assumption in the following lemma is needed, but it is certainly sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16 (Interpolation functions) If X is a normed linear space, then the linear be- tweenness is generated by an interpolation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If X is a proper metric space with unique geodesics, then the same is true for the geodesic betweenness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If X is a metrisable space, then we say that a betweenness on X is compatible with the topology if ⟨x, z⟩ is compact for each x, z ∈ X, and the map X 2 ∋ (x, z) �→ ⟨x, z⟩ ∈ K+(X) is continuous with respect to the product topology on X 2 and the Hausdorff topology on K+(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='17 (Compatible betweennesses) If X is a metrisable space, then the trivial be- tweenness is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The same is true for any betweenness that is generated by an interpolation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If X is a closed subset of R, then the order betweenness on X is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 Squeezed space We will need to view graphs of functions as compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This will require us to compactify the real time axis by adding points at ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' At the same time, we want to equip space-time X × R with a metric that cares less about the spatial distance between two points if their time 10 coordinates are very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In the special case when X = R, such a topology has been introduced in [FINR04, formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Here, we generalise this to X being any metrisable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let (X, d) be a metric space and let ∗ be a point not contained in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then we call R(X) := (X × R) ∪ � (∗, −∞), (∗, ∞) � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='25) the squeezed space associated with X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let dR be a metric that generates the topology on the extended real line R and let φ : R → [0, ∞) be a continuous function such that φ(±∞) = 0 and φ(t) > 0 for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We define dsqz : R(X)2 → [0, ∞) by dsqz � (x, s), (y, t) � := � φ(s) ∧ φ(t) �� d(x, y) ∧ 1 � + ��φ(s) − φ(t) �� + dR(s, t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='26) where naturally the first term is zero if (x, s) or (y, t) are elements of � (∗, −∞), (∗, ∞) � (even though d(x, y) is not defined in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18 (Squeezed space) Let (X, d) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then dsqz is a metric on R(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' One has dsqz � (xn, tn), (x, t) � → 0 if and only if: (i) tn → t, (ii) if t ∈ R, then xn → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Usually, we will only be interested in R(X) as a topological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The conditions (i) and (ii) show that the topology on R(X) depends only on the topology on X and not on the choice of the metric d on X, the metric dR on R, and the function φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Condition (ii) is trivially satisfied if tn → −∞ or → +∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=', we have that (xn, tn) → (∗, ±∞) if and only if tn → ±∞, with no conditions on the sequence xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The squeezed space R(R) plays an important role in the theory of the Brownian web, see [SSS17, Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We need some elementary properties of squeezed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma shows that R(X) is Polish if X is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19 (Preservation of Polishness) (a) If (X, d) is separable, then so is (R(X), dsqz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (b) If (X, d) is complete, then so is (R(X), dsqz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma identifies the compact subsets of R(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In particular, the lemma shows that R(X) is compact if X is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20 (Compactness criterion) A set A ⊂ R(X) is precompact if and only if for each T < ∞, there exists a compact set K ⊂ X such that {x ∈ X : ∃t ∈ [−T, T] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (x, t) ∈ A} ⊂ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 3 Topologies on path space 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 Path space For any set I ⊂ R, we let Is denote the subset of the split real line defined as Is := � t−, t+ : t ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let X be a metrisable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By definition, a path in X is an object that consists of two parts: a closed subset I(π) ⊂ R (possibly empty) and a continuous function π : Is(π) → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The path with I(π) = ∅ is called the trivial path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We usually denote a path simply by π, which includes both the function and its domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We let Π(X) denote the space of all paths in X and let Πc(X) := � π ∈ Π(X) : π(t−) = π(t+) ∀t ∈ I(π) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1) 11 denote the subspace consisting of paths without jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For π ∈ Πc(X) and t ∈ I(π) we simply write π(t) := π(t−) = π(t+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then π : I(π) → X is a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For this reason, we call Πc(X) the space of continuous paths, even though using the split real line, we can also view paths with jumps as continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We call σπ := inf I(π) and τπ := sup I(π) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2) the starting time and final time of a path π, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By convention, σπ = ∞ and τπ = −∞ for the trivial path π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We let Π|(X) := � π ∈ Π(X) : t ∈ I(π) ∀s, u ∈ I(π) and t ∈ R s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' s < t < u � , Π↑(X) := � π ∈ Π(X) : t ∈ I(π) ∀s ∈ I(π) and t ∈ R s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' s < t � , Π↓(X) := � π ∈ Π(X) : t ∈ I(π) ∀u ∈ I(π) and t ∈ R s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' t < u � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3) denote the sets of paths π for which I(π) is an interval, an interval that is unbounded from above, and an interval that is unbounded from below, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Note that all these sets contain the trivial path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We also set Π↕(X) := � π ∈ Π(X) : I(π) = R � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4) which is Π↑(X) ∩ Π↓(X) minus the trivial path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We write Π| c(X) := Π|(X) ∩ Πc etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By definition, the closed graph of a path π ∈ Π(X) is the set G(π) ⊂ R(X) defined as G(π) := � (x, t) : t ∈ I(π), x ∈ {π(t−), π(t+)} � ∪ � (∗, −∞), (∗, +∞) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5) Note that G(π) is nonempty, since we always add the points (∗, ±∞), even for the trivial path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If X is equipped with a betweenness (see Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5), then we define the filled-in graph2 of a path π ∈ Π(X) as Gf(π) := � (x, t) : t ∈ I(π), x ∈ ⟨π(t−), π(t+)⟩ � ∪ � (∗, −∞), (∗, +∞) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6) Note that for the trivial betweenness, the filled-in and closed graphs coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This will allow us to treat Skorohod’s J1 and M1 topologies in a unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The filled-in graph Gf(π) is naturally equipped with a total order, which is defined by setting (x, s) ⪯ (y, t) if either s < t and x, y are arbitrary, or s = t and x ≤π(t−),π(t+) y, where ≤π(t−),π(t+) is the total order on the segment ⟨π(t−), π(t+)⟩ defined in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Informally, the total order ⪯ corresponds to the direction of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Recall the definitions of Ktot and R(X) from Subsections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma says that we can view Gf(π) as an element of the space Ktot(R(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 (Filled-in graphs) Assume that X is a metrisable space that is equipped with a betweenness that is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then for any path π, the filled-in graph Gf(π) is a compact subset of the squeezed space R(X), and the total order ⪯ is compatible with the induced topology on Gf(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A path π is uniquely determined by the totally ordered compact set � Gf(π), ⪯ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If π ∈ Πc(X) or if I(π) does not contain any isolated points, then π is uniquely determined by Gf(π) as a set, but if t is an isolated point of I(π) and π(t−) ̸= π(t+), then one needs the order ⪯ to find out which of the two endpoints of the segment ⟨π(t−), π(t+)⟩ is π(t−) and which is π(t+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma gives another indication why it may be useful to view filled-in graphs as totally ordered sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 2Except for the points at infinity, this is what Whitt [Whi02] calls the completed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 12 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 (Characterisation of graphs) Let X be a metrisable space that is equipped with a betweennesss that is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that (G, ⪯) ∈ Ktot(R(X)) and (∗, ±∞) ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then (G, ⪯) is the filled-in graph of a path π ∈ Π(X) if and only if the following conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (i) For each t ∈ R and (x1, t), (x2, t), (x3, t) ∈ G with (x1, t) ⪯ (x2, t) ⪯ (x3, t), one has x2 ∈ ⟨x1, x2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (ii) (x, s) ⪯ (y, t) for all (x, s), (y, t) ∈ G such that s < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 to the trivial betweenness, it is easy to see that a nonempty compact set G ⊂ R(X) with (∗, ±∞) ∈ G is the closed graph of a path if and only if it is possible to equip G with a total order that is compatible with the topology, such that it satisfies (ii) above and (i)’ For each t ∈ R, the set {x ∈ X : (x, t) ∈ G} has at most two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The total order ⪯ is essential here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To see this, assume that x, y ∈ X satisfy x ̸= y, and let G := {x} × [−1, 1] ∪ {(y, 0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then G is not the closed graph of a path π ∈ Π(X), while it satisfies condition (i)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' However, it is not possible to equip G with a total order ⪯ as in (ii) so that ⪯ is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 Metrics on path space Let (X, d) be a metric space that is equipped with a betweennesss that is compatible with the topology and let Π(X) be the path space defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We view the filled-in graph Gf(π) of a path π as an element of the space Ktot(R(X)) of totally ordered compact subsets of the squeezed space R(X) defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let dsqz be any metric that generates the topology on R(X), and let dpart and dtot be the metrics on Ktot(R(X)) defined in terms of dsqz as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since a path is uniquely characterised by its filled-in graph (viewed as an element of Ktot(R(X))), setting dS part(π1, π2) := dpart � Gf(π1), Gf(π2) � and dS tot(π1, π2) := dtot � Gf(π1), Gf(π2) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7) � π1, π2 ∈ Π(X) � defines two metrics on the path space Π(X), that in view of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10 generate the same topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Letting dH denote the Hausdorff metric on K+(R(X)) associated with the metric dsqz on R(X), we moreover define a pseudometric on Π(X) by setting dH(π1, π2) := dH � Gf(π1), Gf(π2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8) Restricted to the space Πc(X) of continuous paths, this is a metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We call the topology on the path space Π(X) generated by the metrics dS part and dS tot the Skorohod topology associated with the given betweenness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In particular, we define the J1 topol- ogy to be the Skorohod topology associated with the trivial betweenness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In the special case when X = R, we define the M1 topology to be the Skorohod topology associated with the linear betweenness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' More generally, Skorohod topologies associated with a betweenness that is gener- ated by an interpolation function may naturally be viewed as generalisations of the classical M1 topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In view of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18, the definition of a Skorohod topology only depends on the topology on X and on the choice of the betweenness, and not on the precise choice of the metrics on X and R(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following proposition says that Skorohod topologies are Polish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 13 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 (Skorohod topologies are Polish) If X is a Polish space, then so is Π(X), equipped with the Skorohod topology, for any choice of the betweennesss that is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Recall from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1) that Πc(X) denotes the space of continuous paths and that dH is a metric on Πc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since paths in Πc(X) make no jumps, the definitions of dS part, dS tot, and dH restricted to Πc(X) do not depend on the choice of the betweenness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' As a result of the following proposition, all these metrics generate the same topology on Πc(X), so we simply call the resulting topology the topology on Πc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The space Π↑ c(R) of half-infinite paths with values in R, equipped with the topology we have just defined, plays an important role in the theory of the Brownian web and net [SSS14, Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 (Space of continuous paths) For paths πn ∈ Π(X) and π ∈ Πc(X), the following statements are equivalent: (i) dS part(πn, π) −→ n→∞ 0, (ii) dS tot(πn, π) −→ n→∞ 0, (iii) dH(πn, π) −→ n→∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In particular, these metrics all generate the same topology on Πc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If X is a Polish space, then so is Πc(X), equipped with this topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The final statement of the following lemma reveals a special property of the J1 topology that does not hold for general Skorohod topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We note that it is not hard to check that Πc(X), contrary to Π| c(X), is in general not closed in the J1 topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5 (Closed subspaces) Let X be a metrisable space that is equipped with a between- ness that is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then Π|(X), Π↑(X), and Π↓(X) are closed subsets of Π(X), equipped with the Skorohod topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If the betweenness is the trivial betweenness, then also Π| c(X) is a closed subset of Π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 Compactness criteria In this subsection, we give criteria for compactness in the spaces Π(X) and Πc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' These criteria are similar to well-known results for spaces of functions defined on a fixed domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let (X, d) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We say that a set A ⊂ Π(X) satisfies the compact containment condition if ∀T < ∞ ∃ compact C ⊂ X s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' π(t±) ∈ C ∀π ∈ A and t ∈ I(π) ∩ [−T, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9) For each 0 < T < ∞ and δ > 0, we define the (traditional) modulus of continuity of a path π ∈ Πc(X) as mT,δ(π) := sup � d � π(t1), π(t2) � : t1, t2 ∈ I(π), −T ≤ t1 < t2 ≤ T, t2 − t1 ≤ δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10) We say that a set A ⊂ Πc(X) is equicontinuous if lim δ→0 sup π∈A mT,δ(π) = 0 ∀T < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11) The following theorem generalises the classical Arzela-Ascoli theorem to sets of functions that are not necessarily all defined on the same domain, which moreover does not need to be an interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 (Arzela-Ascoli) Let X be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then a set A ⊂ Πc(X) is precompact if and only if it is equicontinuous and satisfies the compact containment condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 14 For paths with jumps, it is possible to give a very similar compactness criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that X is equipped with a betweenness that is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For each 0 < T < ∞ and δ > 0, we define the Skorohod modulus of continuity as mS T,δ(π) := sup � d � π(τ2), ⟨π(τ1), π(τ3)⟩ � : τ1, τ2, τ3 ∈ Is(π), τ1 ≤ τ2 ≤ τ2, −T ≤ τ 1, τ 3 ≤ T, τ 3 − τ 1 ≤ δ � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12) where as before d(x, A) denotes the distance of a point x to a set A and ⟨x, y⟩ is the segment with endpoints x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We say that a set A ⊂ Π(X) is Skorohod-equicontinuous if lim δ→0 sup π∈A mS T,δ(π) = 0 ∀T < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13) Specialising these definitions to the trivial betweenness, for which ⟨π(τ1), π(τ3)⟩ = {π(τ1), π(τ3)}, yields the definitions of the J1-modulus of continuity and J1-equicontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If X = R, equip- ped with the linear betweenness, then we speak of the M1-modulus of continuity and M1- equicontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7 (Compactness criterion) Let (X, d) be a metric space that is equipped with a betweenness that is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then a set A ⊂ Π(X) is precompact in the Skorohod topology if and only if it is Skorohod-equicontinuous and satisfies the compact containment condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 Paths on fixed domains Let X be a metrisable space that is equipped with a betweenness that is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let I be a closed real interval of positive length, let int(I) denote its interior and let ∂I := I\\int(I) denote its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let DI(X) be the set of cadlag functions f : I → X defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We have seen in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 that we may identify DI(X) with the set of paths � π ∈ Π(X) : I(π) = I, π(t−) = π(t+) if t ∈ ∂I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14) In this identification, dS part, dS tot, and dH are metrics3 on DI(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We have already seen that dS part and dS tot generate the same topology on the larger space Π(X) and hence the same is true on DI(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If X is equipped with the trivial betweenness, then we call the topology on DI(X) generated by the metrics dS part and dS tot the J1 topology, and we call the topology generated by dH the J2 topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If X = R, equipped with the linear betweenness, then we call these the M1 topology and M2 topology, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Skorohod [Sko56] only considered compact time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It is easy to see that his definition of the M2 topology [Sko56, Def 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6] coincides with our definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For the J1, J2, and M1 topologies, the equivalence of [Sko56, Defs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4] with our definitions follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It is interesting to note that all previous treatments of the Skorohod topology seem to have been based on variants of the metric dS tot, while te fact that the metric dS part generates the same topology seems to have been overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Skorohod [Sko56] did not consider unbounded time intervals but other authors such as [EK86, Whi02] have done so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To see that our definitions agree with their definitions, one can use the following simple lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We note that for the Skorohod topologies, the restriction map that restricts a function to a smaller time interval is in general not a continuous map, which is why in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15) we have to restrict ourselves to continuity points of the limit function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 3For the metric dH, the assumption that I has positive length is essential, since otherwise this would only be a pseudometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 15 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 (Convergence of restricted functions) Let (X, d) be a metric space that is equipped with a betweenness that is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let g �� [0,t] denote the restriction of a function g to the interval [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then for all fn, f ∈ D[0,∞)(X), one has dH(fn, f) −→ n→∞ 0 ⇔ dH� fn �� [0,t], f �� [0,t] � −→ n→∞ 0 ∀t > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' f(t−) = f(t), dS tot(fn, f) −→ n→∞ 0 ⇔ dS tot � fn �� [0,t], f �� [0,t] � −→ n→∞ 0 ∀t > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' f(t−) = f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15) It is not hard to see that for f ∈ DI(X), the Skorohod modulus of continuity defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12) can alternatively be written as mS T,δ(f) = sup � d � f(t2), ⟨f(t1), f(t3)⟩ � : t1, t2, t3 ∈ I −T ≤ t1 < t2 < t3 ≤ T, t3 − t1 ≤ δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16) Moreover, a set F ⊂ DI(X) satisfies the compact containment condition if and only if ∀T < ∞ ∃ compact C ⊂ X s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' f(t) ∈ C ∀f ∈ F and t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='17) In other words, these last two formulas say that in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12), it suffices to consider π(t+) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' As a straightforward application of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7, we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9 (Compactness criterion) Let (X, d) be a metric space that is equipped with a betweenness that is compatible with the topology and let I be a closed real interval of positive length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then a set F ⊂ DI(X) is precompact in the Skorohod topology if and only if: (i) the compact containment condition holds, (ii) lim δ→0 sup f∈F mS T,δ(f) = 0 for all T < ∞, (iii) lim δ→0 sup f∈F sup � d � f(s), f(t) � : s ∈ I, |s − t| ≤ δ � = 0 for all t ∈ ∂I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Note that compared to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7, we need the extra condition (iii) to guarantee that a sequence of functions in F cannot converge to a function with a discontinuity at a time t ∈ ∂I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14) we drop the condition that π(t−) = π(t+) for t ∈ ∂I, then condition (iii) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9 can be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For the J1 topology on D[0,1], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9 was first proved by Kolmogorov in [Kol56, Thm IV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The analogue statement for the M1 topology was proved by Skorohod in [Sko56, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5 Interpolation It often happens that a sequence of functions that are defined on a countable subset of R converge to a limit that is defined on a subinterval of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In such situations, to formulate what convergence means, it is common practise to interpolate the approximating functions, so that all functions are defined on the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' With the use of path space, one can compare functions that are defined on different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In the present subsection, we show that in such situations, for the J1 topology, there is no need to interpolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For the M1 topology, on the other hand, it still makes sense to interpolate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let ˆI denote the convex hull of a closed set I ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Fix π ∈ Πc(X) and for each t ∈ ˆI(π)\\I(π), let tl := sup{s ∈ I : s < t} and tr := inf{s ∈ I : s > t}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18) 16 where the subscripts l and r stand for “left” and “right”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then we can uniquely define interpolated paths πl ∈ DˆI(π)(X) and πr ∈ D− ˆI(π)(X) by πl(t+) := � π(t) if t ∈ I(π), π(tl) if t ∈ ˆI(π)\\I(π), πr(t−) := � π(t) if t ∈ I(π), π(tr) if t ∈ ˆI(π)\\I(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19) As in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14), we can identify DˆI(π)(X) and D− ˆI(π)(X) with subsets of Π(X) and hence view πl and πr as paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let the metric on the squeezed space R(X) be defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='26) in terms of a metric dR generating the topology on R and a function φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By setting up a monotone correspondence, it is easy to see that for the J1 topology dS tot(π, πl) ≤ εl(π) := sup t∈ˆI(π)\\I(π) � dR(t, tl) + |φ(t) − φ(tl)| � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20) and similarly for πr (with a similar εr(π)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In particular, when πn ∈ Πc(X) and π ∈ Π|(X) satisfy εl(πn) → 0, then with respect to the J1 topology one has πn → π if and only if πl n → π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In other words, no interpolation is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Indeed, convergence of the uninterpolated paths πn → π gives more information since this also implies εl(πn) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We next consider Skorohod topologies associated with a betweenness that is generated by an interpolation function ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In this case, for any π ∈ Πc(X), we can define a continuously interpolated path πϕ by I(πφ) := ˆI(π) and πϕ(t+) := � π(t) if t ∈ I(π), ϕ � π(tl), π(tr), p(t) � if t ∈ ˆI(π)\\I(π), where p(t) := t − tl tr − tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='21) By setting up a monotone correspondence, it is easy to see that dS tot(πϕ, πl), dS tot(πϕ, πr) ≤ ε(π) := εl(π) ∨ εr(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='22) Thus, when πn ∈ Πc(X) and π ∈ Π|(X) satisfy ε(πn) → 0, then with respect to the M1 topology, the conditions πϕ n → π, πl n → π, and πr n → π are all equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In other words, for the M1 topology, it makes sense to interpolate, but it does not matter if we interpolate in a continuous way or in a piecewise constant manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 Open problems By definition, a laglad function (from the French “limite `a gauche, limite `a droit”) is a function that has both left- and right- limits in each point, but whose value in a point does not need to be equal to either the left or right limit at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Whitt [Whi02, Chapter 15] introduces a topology on spaces of laglad functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It seems it should be possible to develop the theory of laglad functions very much in parallel to the theory of cadlag functions, except that instead of splitting each point of the real line into two points, as we did in the split real line, one now needs a topological space where each point of the real line is replaced by three points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A slight complication is that the closed graph of a laglad function cannot always be equipped with a total order that is compatible with the topology, as pointed out below Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' However, it seems likely this can be overcome and it should be possible to prove a compactnes criterion similar to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7, this time involving a modulus of continuity that compares the function values at four consecutive times, rather than three as for the Skorohod modulus of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let X be a metrisable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Recall from Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 that Kpart(X) denotes the space of all compact subsets K ⊂ X that are equipped with a partial order ⪯ that is compatible with 17 the (induced) topology on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For each finite partially ordered set (S, ≤), let KS denote the space of all monotone functions f : S → K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=', functions such that i ≤ j implies f(i) ⪯ f(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then K⟨2⟩, defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10), is the same as KS where S is the totally ordered space {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 below, we more generally define K⟨m⟩ := KS with S the totally ordeed set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For each partially ordered set (S, ≤), similar to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13), we can define a pseudometric dS by dS(K1, K2) := dH(KS 1 , KS 2 ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='23) where dH is the Hausdorff metric on the product space X S, equipped with a product metric as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4, we show that the metrics dS with S a finite totally ordered set with at least two elements all generate the same topology on the space Ktot(X) of totally ordered compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It seems this result does not generalise to partially ordered sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A natural idea is therefore to equip the larger space Kpart(X) with a topology such that Kn → K in the topology on Kpart(X) if and only if dS(Kn, K) → 0 for every finite partially ordered set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Such a topology can be generated by a metric, for example by setting d(K1, K2) := � S rSdS(K1, K2) where rS are positive weights such that the sum over all partially ordered finite sets � S rS is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It then seems interesting to study the associated “ordered” Gromov-Hausdorff distance between two partially ordered sets K1, K2, which is the infimum of d(K1, K2) over all isometric embeddings of K1 and K2 into a common metric space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' As a final, minor open problem, we ask whether the properness assumption in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16 can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This may be known;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' we have just not managed to find this in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7 Outline of the proofs The results from Section 2 are proved in Section 4 and the from Section 3 are proved in Sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' More precisely, Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3, and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 are proved in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9 are proved in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We cited Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 from [SSS14, Appendix B], so these don’t need proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10 is proved in Subsec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11 is proved in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12 is proved in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13 is proved in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='17 are proved in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20 are proved in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 are proved in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 are proved in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2, which also contains the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7 are proved in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9 are proved in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 4 Proofs of the preliminary results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 The split real line In this subsection, we prove Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3, and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4, as well as one more lemma that will be needed in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We recall some basic definitions from topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A topology on a set X is a collection O of subsets of X that are called open and that have the properties that ∅, X ∈ O and O is closed under finite intersections and arbitrary unions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If Y is a subset of X, then the induced topology is defined as {O ∩Y : O ∈ O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A basis for the topology on X is a subset O′ ⊂ O such that each element of O can be written as the union of elements of O′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The set Vx of neighbourhoods of a point x ∈ X is Vx := {V ⊂ X : x ∈ O ⊂ V for some O ∈ O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A fundamental system of neighbourhoods is a set V′ x ⊂ Vx such that ∀V ∈ Vx ∃V ′ ∈ V′ x s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' V ′ ⊂ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A Hausdorff topology is a topology that has the Hausdorff property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=', for all x1, x2 ∈ X with x1 ̸= x2 there exist disjoint O1, O2 ∈ O such that x1 ∈ O1, x2 ∈ O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A topology is 18 first countable if each point has a countable fundamental system of neighbourhoods and second countable if there exists a countable basis for the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A sequence converges to a limit, denoted xn → x, if for each V ∈ Vx, there exists an m such that xn ∈ V for all n ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It suffices to check this for a fundamental system of neighbourhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In a Hausdorff space, a sequence can have at most one limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A set is closed if its complement is open and sequentially closed if it contains the limits of all convergent sequences that lie inside it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' in first countable spaces, the concepts are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The closure of a set is the smallest closed set that contains it and a dense set is a set whose closure is the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A topological space X is separable if it contains a countable dense set and connected if ∅, X are the only sets that are both open and closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A set C ⊂ X is compact if each covering with open sets has a finite subcover and sequentially compact if each sequence in C has a subsequence that converges to a limit in C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' in second countable spaces, the concepts are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A metric defines a topology in the usual way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' a topology that is generated by a metric is called metrisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 By symmetry, it suffices to prove (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By definition, a basis for the topology is formed by all intervals of the form (σ, ρ) with σ, ρ ∈ Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If t+ ∈ (σ, ρ), then σ < t+ < ρ and hence (t−, u+) ⊂ (σ, ρ) for some u > t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It follows that the sets of the form (t−, u+) = [t+, u−] with u ∈ {t + n−1 : n ≥ 1} form a fundamental system of neighbourhoods of t, which is easily seen to imply the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 It is easy to see that Rs has the Hausdorff property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1, we have already seen that each point has a countable fundamental system of neighbourhoods, so Rs is first countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' On the other hand, each basis of the topology must for each t ∈ R contain an open set O such that t ∈ O ⊂ (t−, (t + 1)+) = [t+, (t + 1)−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' These open sets are all distinct, so Rs is not second countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1, the set {t+ : t ∈ Q} is dense so Rs is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since in metric spaces, separability implies second countability, we conclude that Rs is not metrisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since for each t ∈ R, we can write Rs as the union of two disjoint closed sets as Rs = (−∞, t−] ∪ [t+, ∞), we see that Rs is totally disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The next lemma prepares for the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Even though Rs is not second countable, it has a property that is almost as good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 (Strong Lindel¨of property) Every open cover of a subset of Rs has a countable subcover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof This is proved in [AU29], but since the latter reference is not readily available to everyone, including the present authors, we provide our own proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The Sorgenfrey line is the set of real numbers equipped with the lower limit topology that is generated by intervals of the form [a, b) [SS95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Similarly, the upper limit topology on R is generated by intervals of the form (a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' One can check that the topology on Rs induces the lower limit topology on its subspace {t+ : t ∈ R} and the upper limit topology on its subspace {t− : t ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In view of this, it suffices to prove that the Sorgenfrey line has the strong Lindel¨of property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This is well-known [Sor47] but for completeness we provide a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since sets of the form [a, b) form a basis for the topology, it suffices to prove that if A ⊂ R and I is a collection of intervals of the form [a, b) that covers A, then a countable subset of I covers A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Fix n ≥ 1, let I′ denote the subset of I consisting of all intervals [a, b) ∈ I with b − a ≥ 2/n, and let A′ be the union of all elements of I′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It suffices to show that for each n ≥ 1, a countable subset of I′ already covers A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Fix k ∈ Z, let A′′ := A′ ∩ [k/n, (k + 1)/n], and let I′′ denote the subset of I′ consisting of all intervals [a, b) ∈ I′ with [a, b) ∩ [k/n, (k + 1)/n] ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It suffices to show that for each k ∈ Z, a countable subset of I′′ already covers A′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since b − a ≥ 2/n 19 for all [b, a) ∈ I′′, each element of I′′ must contain either k/n or (k + 1)/n, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If I′′ contains an element [a, b) that contains both k/n and (k + 1)/n we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Otherwise, we can select countable subsets of {[a, b) ∈ I′′ : k/n ∈ [a, b)} and {[a, b) ∈ I′′ : (k + 1)/n ∈ [a, b)} that cover A′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We note that the product space Rs × Rs, equipped with the product topology, does not have the strong Lindel¨of property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Indeed, the collection of open sets � [t+, ∞−) × [−t+, ∞−) : t ∈ R � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1) covers the set {(t+, −t+) : t ∈ R}, but no countable subset of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1) has this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The set {(s+, t+) : s, t ∈ R} with the induced topology from R2 s is a well-known counterexample in topology, known as the Sorgenfrey plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 We first prove the statement for Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In any first countable space, being sequentially compact is equivalent to being countably compact, which means that every countable open covering has a finite subcovering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By the strong Lindel¨of property, a subset of Rs is compact if and only if it is countably compact, proving that (i) and (ii) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since the map τ �→ τ, that assigns to a split real number τ its real part, is continuous, and since the continuous image of a compact set is compact, (i) implies that C := {τ : τ ∈ C} is closed and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since moreover a compact subset of a Hausdorff space is closed, (i) implies (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Property (iii) implies that each sequence τn ∈ C has a subsequence τ ′ n such that τ ′ n converges to a limit t ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The τ ′ n must then contain a further subsequence τ ′′ n such that one of the following three cases occurs: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' τ ′′ n < t for all n, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' τ ′′ n > t for all n, or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' τ ′′ n is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In either case, the fact that C is closed implies that τ ′′ n converges to a limit in C, proving the implication (iii)⇒(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This completes the proof for Rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We saw before that the strong Lindel¨of property does not hold for Rd s in dimensions d ≥ 2, so to prove the statement for these spaces we have to proceed differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Property (i) implies countable compactness which by the fact that Rs and hence also Rd s are first countable is equivalent to (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The continuous image of a countably compact set is countably compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Applying this to the coordinate projections and using what we already know for Rs, we see that (ii) implies that C is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since, moreover, in any first countable Hausdorff space, being sequentially compact implies being closed, we see that (ii) implies (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Tychonoff’s theorem and what we already know for Rs, the set [s−, t+]d is compact for each −∞ < s < t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since a closed subset of a compact set is compact, (iii) implies (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 A function f : Iin → X is continuous if and only if (i) f(τn) → f(t+) for all t+ ∈ Iin and τn ∈ Iin such that τ n → t and τn ≥ t+, (ii) f(τn) → f(t−) for all t− ∈ Iin and τn ∈ Iin such that τ n → t and τn ≤ t−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We see from this that for a given f ∈ CIin(X), setting f ±(t) := f(t±) if t± ∈ Iin, f +(t) := f(t−) if t = sup I < ∞, f −(t) := f(t+) if t = inf I > −∞ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2) defines f + ∈ DI(X) and f − ∈ D− I (X) such that f − is the left-continuous modification of f + and f is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 20 Conversely, if such f ± are given, then to see that f defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5) satisfies f ∈ CIin(X), by symmetry, it suffices to check only condition (i) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To show that f(τn) → f(t+), it suffices to prove that each subsequence τ ′ n contains a further subsequence τ ′′ n such that f(τ ′′ n) → f(t+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In view of this, without loss of generality, we may assume that either s(τn) = + for all n or s(τn) = − for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In the first case, we have f(τn) → f(t+) by the right-continuity of f +, while in the second case we have f(τn) → f(t+) by the fact that f + is the right-continuous modification of f −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following fact is well-known, but since we need this in what follows, for completeness we include the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 (Countably many discontinuities) Let (X, d) be a metric space, let I be a closed real interval, and let f ∈ DI(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then the set {t ∈ I : f(t−) ̸= f(t)} is countable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof For each ε > 0 and T < ∞, the set J := {t ∈ I ∩ [−T, T] : d � f(t−), f(t) � ≥ ε} must be finite, since otherwise there exist a strictly increasing or decreasing sequence tn ∈ J, which is easily seen to contradict the cadlag property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 The Hausdorff metric In this subsection, we prove Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9, which are the only results from Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 for which we did not give a reference, as well as Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 below that will be needed in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5 Let R ∈ Cor(K1, K2) and let D := sup(x1,x2)∈R d(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then d(x1, K2) ≤ D and d(x2, K1) ≤ D for each x1 ∈ K1, x2 ∈ K2, and hence dH(K1, K2) ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' On the other hand, by the compactness of K2 and the continuity of the function d(x1, · ), for each x1 ∈ K1, there exists an x2 ∈ K2 such that d(x1, K2) = d(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The same statement holds with the roles of K1 and K2 interchanged, so setting R := � (x1, x2) ∈ K1 × K2 : d(x1, x2) ∈ {d(x1, K2), d(x2, K1)} � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3) defines a correspondence between K1 and K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By the compactness of K1 and the continuity of the map d( · , K2), there exists an x′ 1 ∈ K1 such that d(x′ 1, K2) = maxx1∈K1 d(x1, K2), and similarly there exists an x′′ 2 ∈ K2 such that d(x′′ 2, K1) = maxx2∈K2 d(x2, K1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By our earlier arguments, there exist x′ 2 ∈ K2 and x′′ 1 ∈ K1 such that d(x′ 1, K2) = d(x′ 1, x′ 2) and d(x′′ 2, K1) = d(x′′ 1, x′′ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then dH(K1, K2) = d(x′ 1, x′ 2) ∨ d(x′′ 1, x′′ 2) = max (x1,x2)∈R d(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4) Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9 Imagine that Kn, K ∈ K+(X) satisfy Kn → K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If K is not connected, then there exist disjoint nonempty compact sets C1, C2 such that K = C1 ∪ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let ε := d(C1, C2) = inf{d(x1, x2) : x1 ∈ C1, x2 ∈ C2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By the compactness of C1 and C2, the infimum is attained and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let Ui := {x ∈ X : d(x, Ci) ≤ ε/3} (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then U1, U2 are disjoint closed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For all n large enough such that dH(Kn, K) ≤ ε/3, one has Kn ⊂ U1 ∪ U2 while Kn ∩ U1 and Kn ∩ U2 are both nonempty, which proves that Kn is not connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We recall that the image of a compact set under a continuous map is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In what follows, we will need the following simple observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 (Continuous image) Let X, Y be metrisable spaces and let ψ : X → Y be con- tinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If Kn, K ∈ K+(X) satisfy Kn → K, then their images under ψ satisfy ψ(Kn) → ψ(K) in K+(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 21 Proof This follows easily from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since Kn → K, there exists a compact set C ⊂ X such that Kn ⊂ C for all n, and now ψ(C) ⊂ Y is a compact set such that ψ(Kn) ⊂ ψ(C) for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9), it now suffices to check that: (i) ψ(K) ⊂ {y ∈ Y : ∃xn ∈ Kn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' ψ(xn) → y}, (ii) {y ∈ Y : ∃xn ∈ Kn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' y is a subsequential limit of (ψ(xn))n∈N} ⊂ ψ(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5) Here (i) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9) and the continuity of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove (ii), if ψ(x′ n) → y for some subsequence x′ n, then since Kn ⊂ C for all n there exists a further subsequence x′′ n such that x′′ n → x for some x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then x ∈ K by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9) and hence ψ(x) = y by the continuity of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 The ordered Hausdorff metric In this subsection, we study the metrics dpart and dtot defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15), preparing for the proofs of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11, and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12, which will be given in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let (X, d) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Generalising the definition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10), for each m ≥ 1 and K ∈ Kpart(X), we set K⟨m⟩ := � (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xm) ∈ Km : x1 ⪯ · · · ⪯ xm � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6) It is straightforward to check that K⟨m⟩ is a closed subset of Km and hence a compact subset of X m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Generalising the definitions in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12), we equip X m with the metric dm� (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xm), (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , ym) � := m � k=1 d(xk, yk), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7) and we equip K+(X m) with the associated Hausdorff metric dm H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Generalising the definition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13), for each m ≥ 1, we define a function d⟨m⟩ on Kpart(X)2 by d⟨m⟩(K1, K2) := dm H (K⟨m⟩ 1 , K⟨m⟩ 2 ) � K1, K2 ∈ Kpart(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8) In particular, when m ≥ 2, this is a metric on Kpart(X) since (K, ⪯) is uniquely characterised by K⟨m⟩ for m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' On the other hand, d⟨1⟩(K1, K2) is simply the Hausdorff distance between K1 and K2 as sets, which gives no information about the partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma describes a simple property of the metric d⟨2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 (Ordered limit) Let X be a metrisable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that Kn, K ∈ Kpart(X) satisfy d⟨2⟩(Kn, K) → 0 and that xn, yn ∈ Kn satisfy xn → x, yn → y, and xn ⪯ yn for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then x, y ∈ K satisfy x ⪯ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof Since d⟨2⟩(Kn, K) → 0, we have K⟨2⟩ n → K⟨2⟩ and hence by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 x, y ∈ K⟨2⟩, which proves that x ⪯ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The folling lemma gives a one-sided bound between metrics of the form d⟨m⟩ for different values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5 (One-sided bound) One has d⟨m⟩(K1, K2) ≤ d⟨m+1⟩(K1, K2) � m ≥ 1, K1, K2 ∈ Kpart(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9) 22 Proof By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5, there exists a correspondence R between K⟨m+1⟩ 1 and K⟨m+1⟩ 2 such that dm+1(x, y) ≤ dm+1 H (K⟨m+1⟩ 1 , K⟨m+1⟩ 2 ) for all (x, y) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let ψ : X m+1 → X denote the projection ψ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xm+1) := (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7) implies that dm� ψ(x), ψ(y) � ≤ dm+1(x, y) (x, y ∈ X m+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10) Since ψ(K⟨m+1⟩ i ) = K⟨m⟩ i (i = 1, 2), it follows that R′ := �� ψ(x), ψ(y) � : (x, y) ∈ R � is a correspondence between K⟨m⟩ 1 and K⟨m⟩ 2 such that dm(x′, y′) ≤ dm+1 H (K⟨m+1⟩ 1 , K⟨m+1⟩ 2 ) for all (x′, y′) ∈ R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5, this proves that dm H (K⟨m⟩ 1 , K⟨m⟩ 2 ) ≤ dm+1 H (K⟨m+1⟩ 1 , K⟨m+1⟩ 2 ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11) which in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8) implies the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemmas show that in general, the metrics d⟨m⟩ for different values of m are not comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' More precisely, the one-sided bound in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5 is not matched by an opposite inequality of the form d⟨m+1⟩(K1, K2) ≤ Cd⟨m⟩(K1, K2) for any finite constant C, and convergence in d⟨m⟩ does not imply convergence in d⟨m+1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 (No opposite inequality) Let X = [0, 1], equipped with the usual distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then for each m ≥ 1 and 0 < ε ≤ 1/4, there exist K1, K2 ∈ Ktot(X) such that d⟨m⟩(K1, K2) ≤ ε and d⟨m+1⟩(K1, K2) ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof We choose K1 = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xm+1} with xk ∈ [0, ε] when k is even and xk ∈ [1 − ε, 1] if k is odd, and we choose K2 = {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , ym+1} with yk ∈ [0, ε] when k is odd and yk ∈ [1 − ε, 1] if k is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We equip K1 and K2 with total orders such that x1 ≺ · · · ≺ xm+1 and y1 ≺ · · · ≺ ym+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It is easy to see that d � (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xm+1), K⟨m+1⟩ 2 � ≥ 1/2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12) and hence d⟨m+1⟩(K1, K2) ≥ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' On the other hand, it is easy to see that for each (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , zm) ∈ K⟨m⟩ 1 , there exists a (z′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , z′ m) ∈ K⟨m⟩ 2 such that |zk − z′ k| ≤ ε for all k, and vice versa, so d⟨m⟩(K1, K2) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7 (Different topologies) Let X = [0, 1], equipped with the usual distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then for each m ≥ 1, there exist Kn ∈ Kpart(X) and K ∈ Ktot(X) such that d⟨m⟩(Kn, K) → 0 as n → ∞ but d⟨m+1⟩(Kn, K) ≥ 1/2 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof It will be convenient to use the notation [m] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , m} (m ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We choose x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xm+1, all different, with xk ∈ [0, 1/4] when k is even and xk ∈ [3/4, 1] if k is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We set K = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xm+1} and we equip K with a total order by setting x1 ≺ · · · ≺ xm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We choose points � xl k(n) �l∈[m+1] k∈[m+1] (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13) in [0, 1], all different, such that xl k(n) → xk as n → ∞ for all k, l ∈ [m + 1], and we set Kn := � xl k : k, l ∈ [m + 1], k ̸= l � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14) We equip Kn with a partial order such that xl k ⪯ xl′ k′ ⇔ k ⪯ k′ and l = l′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15) 23 Then it is easy to check that d⟨m⟩(Kn, K) → 0 as n → ∞ but d⟨m+1⟩(Kn, K) ≥ 1/2 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We note that d⟨m⟩(K1, K2) ≤ sup(x1,x2)∈K1×K2 d(x1, x2), which is finite by the continuity of d and the compactness of K1 × K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We use this and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5 to define d⟨∞⟩ as the increasing limit d⟨∞⟩(K1, K2) := lim m→∞ d⟨m⟩(K1, K2) � K1, K2 ∈ Kpart(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16) It is straightforward to check that d⟨∞⟩ is a metric on Kpart(X): symmetry and the triangle inequality follow by taking the limit in the corresponding properties of the metrics d⟨m⟩, and d⟨∞⟩(K1, K2) = 0 clearly implies d⟨m⟩(K1, K2) = 0 for all m ≥ 1 and hence equality of K1 and K2 as partially ordered spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In the special case that K1 and K2 are totally ordered, the following proposition identifies d⟨∞⟩(K1, K2) as the metric dtot defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 (Monotone correspondences) Let (X, d) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then one has d⟨∞⟩(K1, K2) = dtot(K1, K2) for all K1, K2 ∈ Ktot(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 uses the following simple lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9 (Eventually ordered sequences) Let K be a compact metrisable set that is equipped with a total order that is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that x ≺ y and that xn, yn ∈ K satisfy xn → x and yn → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then xn ≺ yn for all n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof Since ⪯ is a total order, if the statement is not true, then yn ⪯ xn for infinitely many n, so we can select a subsequence such that yn ⪯ xn for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Taking the limit, using the fact that the total order that is compatible with the topology, we find that y ⪯ x, which contradicts x ≺ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 We first prove the inequality d⟨∞⟩(K1, K2) ≤ dtot(K1, K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let R be a monotone correspondence between K1 and K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xm ∈ K1 satisfy x1 ⪯ · · · ⪯ xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then we can choose x′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , x′ m ∈ K2 such that (xk, x′ k) ∈ R for all 1 ≤ k ≤ m, and moreover x′ k = x′ k+1 whenever xk = xk+1 (1 ≤ k < m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since R is monotone and K2 is totally ordered, we must have x′ 1 ⪯ · · · ⪯ x′ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This shows that dm� (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xm), K⟨m⟩ 2 � ≤ sup (x,x′)∈R d(x, x′) � (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xm) ∈ K⟨m⟩ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='17) The same is true with the roles of K1 and K2 interchanged, so we conclude that d⟨m⟩(K1, K2) = dm H (K⟨m⟩ 1 , K⟨m⟩ 2 ) ≤ sup (x,x′)∈R d(x, x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18) Taking the infimum over all monotone correspondences between K1 and K2 and letting m → ∞ we see that d⟨∞⟩(K1, K2) ≤ dtot(K1, K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove the opposite inequality, let εn be positive constants, tending to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since K1 is totally bounded, for each n, we can find an m(n) ≥ 1 and xn 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xn m(n) ∈ K1 such that d � x, {xn 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xn m(n)} � ≤ εn ∀x ∈ K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19) Since K1 is totally ordered, we can assume without loss of generality that xn 1 ⪯ · · · ⪯ xn m(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since dm H (K⟨m⟩ 1 , K⟨m⟩ 2 ) = d⟨m⟩(K1, K2) ≤ d⟨∞⟩(K1, K2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20) we can find yn 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , yn m(n) ∈ K2 with yn 1 ⪯ · · · ⪯ yn m(n) such that d(xn k, yn k) ≤ d⟨∞⟩(K1, K2) (1 ≤ k ≤ m(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='21) 24 Using the fact that K2 is totally bounded, adding points to {yn 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , yn m(n)} and making m(n) larger if necessary, we can arrange things so that also d � y, {yn 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , yn m(n)} � ≤ ε ∀y ∈ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='22) Now using again (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20) we can add corresponding points in K1 for the new points we have added to K2 so that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='21) remains true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Adding points will not spoil (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19) so we can arrange things such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='22) are satisfied simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let Rn ⊂ K1 × K2 be the set Rn := � (xn k, yn k) : 1 ≤ k ≤ m(n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='23) We claim that Rn is monotone in the sense that there are no (xn k, yn k), (xn l , xn l ) ∈ Rn such that xn k ≺ xn l and yn l ≺ yn k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='24) Indeed, xn k ≺ xn l implies k < l and yn l ≺ yn k implies l < k, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since K1 × K2 is compact, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8, we can select a subsequence such that Rn → R in the Hausdorff topology on K+(K1 × K2), for some compact set R ⊂ K1 × K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We claim that R is a correspondence between K1 and K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Indeed, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19), for each x ∈ K1, we can choose k(n) such that xn k(n) → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By the compactness of K1 × K2, the sequence (xn k(n), yn k(n)) has at least one cluster point (x, y), and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 (x, y) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Similarly, for each y ∈ K2 there exists an x ∈ K1 such that (x, y) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We next claim that R is monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that conversely, there exist (x, y), (x′, y′) ∈ R such that x ≺ x′ and y′ ≺ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, there exist k(n), k′(n) such that (xn k(n), yn k(n)) → (x, y) and (xn k′(n), yn k′(n)) → (x′, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9, xn k(n) ≺ yn k(n) and yn k′(n) ≺ xn k′(n) for all n large enough, which contradicts (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Taking the limit in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='21), using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, we see that d(x, y) ≤ d⟨∞⟩(K1, K2) ∀(x, y) ∈ R, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='25) and hence by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15) dtot(K1, K2) ≤ d⟨∞⟩(K1, K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 The mismatch modulus In this subsection, we prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Generalising the definition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='17) for any K1, K2 ∈ Ktot(X) and ε > 0, we define the mismatch modulus mε(K1, K2) by mε(K1, K2) := sup � d(x1, y1) ∨ d(x2, y2) : x1, y1 ∈ K1, x2, y2 ∈ K2, d(x1, x2) ∨ d(y1, y2) ≤ ε, x1 ⪯ y1, y2 ⪯ x2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='26) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10 (Convergence of the mismatch modulus) Let X be a metrisable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' As- sume that Kn, K ∈ Kpart(X) satisfy d⟨2⟩(Kn, K) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then mεn(Kn, K) −→ n→∞ 0 with εn := d⟨1⟩(Kn, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='27) Proof If (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='27) does not hold, then, by going to a subsequence, we can assume that there exists a δ > 0 such that mεn(Kn, K) ≥ δ for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It follows that for each n, we can find x1(n), y1(n) ∈ Kn and x2(n), y2(n) ∈ K with d(x1(n), y1(n)) ∨ d(x2(n), y2(n)) ≥ δ and d(x1(n), x2(n))∨d(y1(n), y2(n)) ≤ εn such that x1(n) ⪯ y1(n) and y2(n) ⪯ x2(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5, our assumption d⟨2⟩(Kn, K) → 0 implies εn = d⟨1⟩(Kn, K) → 0 and hence Kn → K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Therefore, 25 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, there exists a compact set C ⊂ X such that Kn ⊂ C for all n, so by going to a subsequence, we can assume that x1(n), x2(n), y1(n), y2(n) converge to limits x1, x2, y1, y2 in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since d(x1(n), x2(n))∨d(y1(n), y2(n)) ≤ εn → 0, we have x := x1 = x2 and y := y1 = y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' On the other hand, our assumption that d(x1(n), y1(n)) ∨ d(y2(n), x2(n)) ≥ δ implies that d(x, y) ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This leads to a contradiction, since by the assumption that d⟨2⟩(Kn, K) → 0 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4, x1(n) ⪯ y1(n) and y2(n) ⪯ x2(n) imply x ⪯ y and y ⪯ x and hence x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following estimate essentially uses that the spaces are totally ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11 (Estimate in terms of mismatch modulus) Let X be a metrisable space and let K1, K2 ∈ Ktot(X) satisfy d⟨1⟩(K1, K2) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then d⟨m⟩(K1, K2) ≤ mε(K1, K2) + ε (m ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='28) Proof By symmetry, it suffices to show that for each x1 = (x1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xm 1 ) ∈ K⟨m⟩ 1 , there exists an x2 = (x1 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xm 2 ) ∈ K⟨m⟩ 2 such that dm(x1, x2) ≤ mε(K1, K2) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7), the latter means that d(xk 1, xk 2) ≤ mε(K1, K2) + ε for all 1 ≤ k ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since d⟨1⟩(K1, K2) ≤ ε, there exists a z(1) ∈ K2 such that d(x1 1, z(1)) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We set xi 2 = z(1) for all 1 ≤ i < I(1), where I(1) := inf{i > 1 : d(xi 1, z(1)) > mε(K1, K2) + ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Using again that d⟨1⟩(K1, K2) ≤ ε, there exists a z(2) ∈ K2 such that d(xI(1) 1 , z(2)) ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then d � z(1), z(2) � > d � xI(1) 1 , z(2) � − ε ≥ mε(K1, K2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='29) and hence by the definition of mε(K1, K2) and the fact that x1 1 ⪯ xI(1) 1 we cannot have z(2) ⪯ z(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since K2 is totally ordered, we conclude that z(1) ≺ z(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This allows us to set xi 2 = z(2) for all I(1) ≤ i < I(2), where I(2) := inf{i > I(1) : d(xi 1, z(2)) > mε(K1, K2) + ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Continuing inductively, we obtain (x1 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , xm 2 ) ∈ K⟨m⟩ 2 such that d(xk 1, xk 2) ≤ mε(K1, K2)+ε for all 1 ≤ k ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' As a consequence of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11, we can prove that the metrics d⟨m⟩ with 2 ≤ m ≤ ∞ all generate the same topology on Ktot(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This may be a bit surprising in view of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' As the latter shows, the restriction to totally ordered sets is essential in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12 (Convergence of totally ordered sets) Let X be a metrisable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then Kn, K ∈ Ktot(X) satisfy d⟨2⟩(Kn, K) → 0 if and only if d⟨∞⟩(Kn, K) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof Since d⟨m⟩(Kn, K) ≤ d⟨m+1⟩(Kn, K) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5 and since d⟨∞⟩(Kn, K) is defined as the increasing limit of d⟨m⟩(Kn, K) as m → ∞, it is clear that d⟨∞⟩(Kn, K) → 0 implies d⟨2⟩(Kn, K) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove the opposite implication, it suffices to show that d⟨2⟩(Kn, K) → 0 implies sup m≥1 d⟨m⟩(Kn, K) −→ n→∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='30) By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5, d⟨2⟩(Kn, K) → 0 implies εn := d⟨1⟩(Kn, K) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11 now imply that sup m≥1 d⟨m⟩(Kn, K) ≤ mεn(Kn, K) + εn −→ n→∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='31) Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10 By the definition of dpart and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 we have dpart = d⟨2⟩ and dtot = d⟨∞⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12 both metrics generate the same topology on Ktot(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If d 26 and d′ generate the same topology on X and dpart and d′ part are defined in terms of d and d′ as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13), then by Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8, dpart and d′ part generate the same topology on Ktot(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The inequalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16) follow from the fact that d⟨1⟩(K1, K2) ≤ d⟨2⟩(K1, K2) ≤ d⟨∞⟩(K1, K2) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' On the other hand, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 shows that if X = [0, 1], then for each ε > 0 we can find K1, K2 ∈ Ktot(X) such that d⟨2⟩(K1, K2) ≤ ε while 1/2 ≤ d⟨3⟩(K1, K2) ≤ d⟨∞⟩(K1, K2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='32) proving the final claim of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5 Polishness In this subsection, we prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We start with the following lemma, announced in the introduction, that shows that even when (X, d) is complete, it is in general not true that the metrics dpart and dtot are complete on Ktot(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13 (Metric not complete) Let X = [0, 1], equipped with the usual distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then the metrics d⟨m⟩ with 2 ≤ m ≤ ∞ are not complete on Ktot(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof It suffices to construct a Cauchy sequence that does not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In view of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5, it suffices to construct a Cauchy sequence in the metric d⟨∞⟩, which by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 equals dtot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let εn be positive constants converging to zero, and let Kn := {0, 1, εn} equipped with a total order such that 0 ≺ 1 ≺ εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For each n, m, we define a monotone correspondence Rn,m between Kn and Km by Rn,m := {(0, 0), (1, 1), (εn, εm)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then dtot(Kn, Km) ≤ sup (x1,x2)∈Rn,m |x1 − x2| = |εn − εm|, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='33) so Kn is clearly a Cauchy sequence in dtot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' However, the sequence Kn does not converge in the ordered Hausdorff topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If it had a limit K, then (in view of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4) this would have to be the set K = {0, 1} equipped with a total order such that 0 ⪯ 1 and 1 ⪯ 0, but such a totally ordered set does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11 needs some preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For each L ∈ K+(X 2) and ε ≥ 0, we set m⟨2⟩ ε (L) := sup � d(x1, y1) ∨ d(x2, y2) : (x1, y1), (y2, x2) ∈ L, d(x1, x2) ∨ d(y1, y2) ≤ ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='34) In particular, this implies m⟨2⟩ ε (K⟨2⟩) = mε(K) (K ∈ Ktot(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14 (Right continuity) For any metric space X and L ∈ K+(X 2), the function [0, ∞) ∋ ε → m⟨2⟩ ε (L) is right-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof The function ε �→ m⟨2⟩ ε (L) is clearly nondecreasing, so it suffices to prove that m⟨2⟩ ε (L) ≥ lim η↓ε m⟨2⟩ η (L) (ε ≥ 0) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='35) where the limit exist by monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Fix εn > ε such that εn → ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then for each δ > 0 and for each n, we can choose (xn 1, yn 1 ), (yn 2 , xn 2) ∈ L such that d(xn 1, xn 2) ∨ d(yn 1 , yn 2 ) ≤ εn and d(xn 1, yn 1 ) ∨ d(xn 2, yn 2 ) ≥ m⟨2⟩ ηn (L) − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since L is compact, by going to a subsequence, we can assume that (xn 1, yn 1 ) → (x1, y1) and (yn 2 , xn 2) → (y2, x2) for some (x1, y1), (y2, x2) ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then d(x1, x2)∨d(y1, y2) ≤ ε and d(x1, y1)∨d(x2, y2) ≥ limη↓ε m⟨2⟩ η (L)−δ, which proves that m⟨2⟩ ε (L) ≥ limη↓ε m⟨2⟩ η (L) − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since δ > 0 is arbitrary, this implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 27 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15 (Upper semi-continuity) Let X be a metric space and let Ln, L ∈ K+(X 2) satisfy Ln → L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then m⟨2⟩ ε (L) ≥ lim sup n→∞ m⟨2⟩ ε (Ln) (ε ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='36) Proof By the compactness of [0, ∞] we can select a subsequence for which limn→∞ m⟨2⟩ ε (Ln) exists and is equal to the limit superior of the original sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let δn > 0 converge to zero and pick (xn 1, yn 1 ), (yn 2 , xn 2) ∈ Ln such that d(xn 1, xn 2) ∨ d(yn 1 , yn 2 ) ≤ ε and d(xn 1, yn 1 ) ∨ d(xn 2, yn 2 ) ≥ m⟨2⟩ ε (Ln) − δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, there exists a compact C ⊂ X 2 such that Ln ⊂ C for all n, so by going to a further subsequence we can assume that (xn 1, yn 1 ) → (x1, y1) and (yn 2 , xn 2) → (y2, x2) for some (x1, y1), (y2, x2) ∈ X 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then (x1, y1), (y2, x2) ∈ L by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, d(x1, x2) ∨ d(y1, y2) ≤ ε, and hence m⟨2⟩ ε (L) ≥ d(x1, y1) ∨ d(x2, y2) ≥ lim n→∞ � m⟨2⟩ ε (Ln) − δn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='37) Since δn → 0, this proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Before we can prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11 we need one more lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For any metric space (X, d), we define L(X) ⊂ K+(X 2) by L(X) := � K⟨2⟩ : K ∈ Ktot(X) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='38) and we let L(X) denote the closure of L(X) in the metric space � K+(X 2), d2 H � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16 (Totally ordered sets) For any metric space X, one has L(X) = � L ∈ L(X) : m⟨2⟩ 0 (L) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='39) Proof To prove the inclusion ⊂ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='39), it suffices to observe that m⟨2⟩ 0 (K⟨2⟩) = sup � d(x, y) : (x, y), (y, x) ∈ K⟨2⟩� = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='40) for all K ∈ Ktot(X), since x ⪯ y and y ⪯ x imply x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We next prove the inclusion ⊃ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that L ∈ L(X) satisfies m⟨2⟩ 0 (L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since L ∈ L(X), there exist Kn ∈ Ktot(X) such that K⟨2⟩ n → L in the topology on K+(X 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let πi(x1, x2) := xi (i = 1, 2) denote the coordinate projections πi : X 2 → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since π1(K⟨2⟩ n ) = Kn = π2(K⟨2⟩ n ) for each n, using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3, we see that Kn → K in the Hausdorff topology on K+(X), where K := π1(L) = π2(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We define a relation ⪯ on K by setting x ⪯ y if and only if (x, y) ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To complete the proof, it suffices to show that ⪯ is a total order on K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=', (i) for each x, y ∈ K, either x ⪯ y or y ⪯ x, or both, (ii) x ⪯ y and y ⪯ x imply x = y, (iii) x ⪯ y ⪯ z imply x ⪯ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove (i), let x, y ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since Kn → K in the Hausdorff topology on K+(X), by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, there exist xn, yn ∈ Kn such that xn → x and yn → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since Kn is totally ordered, either xn ⪯ yn happens for infinitely many n, or yn ⪯ xn happens for infinitely many n, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since K⟨2⟩ n → L in the Hausdorff topology on K+(X 2), by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, it folows that either x ⪯ y or y ⪯ x, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Property (ii) follows immediately from the fact that sup{d(x, y) : (x, y), (y, x) ∈ L} = m⟨2⟩ 0 (L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove (iii), assume that x, y, z ∈ K satisfy x ⪯ y ⪯ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If x = y or y = z then trivially x ⪯ z, so without loss of generality we may assume that x ̸= y ̸= z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since 28 Kn → K in the Hausdorff topology on K+(X), by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, there exist xn, yn, zn ∈ Kn such that xn → x, yn → y, and zn → z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since Kn is totally ordered, for each n either xn ⪯ yn, or yn ⪯ xn, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' But yn ⪯ xn can happen only for finitely many n since otherwise the fact that K⟨2⟩ n → L in the Hausdorff topology on K+(X 2) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 would imply that y ⪯ x, which together with our assumptions x ⪯ y and x ̸= y contradicts (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We conclude that xn ⪯ yn for all n sufficiently large and by the same argument also yn ⪯ zn for all n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It follows that xn ⪯ zn for all n sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since K⟨2⟩ n → L in the Hausdorff topology on K+(X 2), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 shows that (x, z) ∈ L and hence x ⪯ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We are now ready to prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We need to recall one well-known fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' A subset A ⊂ X of a topological space X is called a Gδ-set if A is a countable intersection of open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Our proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11 makes use of the following fact, that we cite from [Bou58, §6 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 1, Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 1] (see also [Oxt80, Thms 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='17 (Subsets of Polish spaces) A subset Y ⊂ X of a Polish space X is Polish in the induced topology if and only if Y is a Gδ-subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11 We first observe that if X is Polish, then so is X 2, equipped with the product topology, and hence, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7, also K+(X 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Therefore, since � Ktot(X), dpart � is isometric to � L(X), d2 H � defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='38), in view of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='17, it suffices to show that L(X) is a Gδ-subset of K+(X 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15 that for each ε, δ > 0, the set Aδ,ε := � L ∈ K+(X 2) : m⟨2⟩ ε (L) ≥ δ � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='41) is a closed subset of K+(X 2) and hence its complement Ac ε,δ is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' As a consequence, G := ∞ � n=1 ∞ � m=1 Ac 1/n,1/m (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='42) is a Gδ-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since each closed set is a Gδ-set, and the intersection of two Gδ-sets is a Gδ-set, using Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16, we conclude that L(X) ∩ G = � L ∈ L(X) : ∀δ > 0 ∃ε > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' m⟨2⟩ ε (L) < δ � = � L ∈ L(X) : lim ε→0 m⟨2⟩ ε (L) = 0 � = � L ∈ L(X) : m⟨2⟩ 0 (L) = 0 � = L(X) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='43) is a Gδ-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 Compactness criterion In this subsection, we prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12 Since the map K �→ K⟨2⟩ is a homeomorphism from Ktot(X) to L(X), equipped with the induced topology from K+(X 2), a set A ⊂ K⟨2⟩ is precompact if and only if B := {K⟨2⟩ : K ∈ A} is a precompact subset of K+(X 2) and its closure B is contained in L(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We will show that B is a precompact subset of K+(X 2) if and only if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18 (i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Moreover, if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18 (i) holds, then B is contained in L(X) if and only if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18 (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18 (i) holds, then C2 is a compact subset of X 2 and K⟨2⟩ ⊂ C2 for all K ∈ A, so B is a precompact subset of K+(X 2) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Conversely, if B is a precompact subset of K+(X 2), then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 there exists a compact subset D ⊂ X 2 such that K⟨2⟩ ⊂ D 29 for all K ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Without loss of generality, we may assume that D is of the form D = C2 for some compact subset C of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' for example, we may take for C the union of the two coordinate projections of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then K ⊂ C for all K ∈ A, proving that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18 (i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To complete the proof, assume that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18 (i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We must show that B is contained in L(X) if and only if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18 (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume, first, that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18 (ii) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then we can find a δ > 0 and εn > 0 tending to zero, as well as Kn ∈ A, such that m⟨2⟩ εn (Kn) ≥ δ for each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18 (i), going to a subsequence if necessary, we can assume that K⟨2⟩ n → L for some L ∈ K+(X 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Now Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15 implies that m⟨2⟩ ε (L) ≥ lim sup n→∞ m⟨2⟩ ε (K⟨2⟩ n ) ≥ lim sup n→∞ m⟨2⟩ εn (K⟨2⟩ n ) ≥ δ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='44) for each ε > 0, so letting ε ↓ 0, using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14, we conclude that m⟨2⟩ 0 (L) ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16, we conclude that L ̸∈ L(X) and hence B is not contained in L(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume now that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18) (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We must show that B is contained in L(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that K⟨2⟩ n → L for some Kn ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then clearly L ∈ L(X) so by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16 it suffices to prove that m⟨2⟩ 0 (L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that, conversely, there exist x, y ∈ X with x ̸= y such that (x, y), (y, x) ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, there exist xn 1, xn 2, yn 1 , yn 2 ∈ Kn with xn 1 ⪯ yn 1 and yn 2 ⪯ xn 2 such that xn i → x and yn i → y as n → ∞ (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then for each ε > 0, we can choose n large enough such that d(xn i , x) ≤ ε/2 (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It follows that d(xn 1, yn 1 ) ∨ d(xn 2, yn 2 ) ≥ d(x, y) − ε and d(xn 1, xn 2) ∨ d(yn 1 , yn 2 ) ≥ ε so that mε(Kn) ≥ d(x, y) − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This clearly contradicts (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18) (ii), so we conclude that m⟨2⟩ 0 (L) = 0 as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7 Cadlag curves In this subsection, we prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let (X, d) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If R is any subset of X 2, then let us call dist(R) := sup (x1,x2)∈R dsqz(x1, x2) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='45) the distortion of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15) say that dH(K1, K2) = inf R∈Corr(K1,K2) dist(R) and dtot(K1, K2) = inf R∈Corr+(K1,K2) dist(R), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='46) where Corr(K1, K2) and Corr+(K1, K2) denote the sets of all (monotone) correspondences be- tween K1 and K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let R denote the closure of a set R ⊂ X 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then dist(R) = dist(R) (R ⊂ X 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='47) Indeed, the inequality ≤ is trivial, while the opposite inequality follows from the fact that for each (x1, x2) ∈ R, there exist (xn 1, xn 2) ∈ R such that (xn 1, xn 2) → (x1, x2) and hence d(xn 1, xn 2) → d(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We need one preparatory lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18 (Fine partition) Let (X, d) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then for each γ ∈ D[0,1](X) and ε > 0, there exist t0 < 0 < t1 < · · · < tn−1 < 1 < tn such that sup � d � γ(s), γ(s′) � : 1 ≤ k ≤ n, s, t ∈ [0, 1] ∩ [tk−1, tk) � < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='48) 30 Proof By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4, writing γ(t+) := γ(t) and γ(t−) := γ−(t), where γ− is the caglad modification of γ, we can view γ as a continuous function on the split real interval [0−, 1+], that moreover satisfies γ(0−) = γ(0+) and γ(1−) = γ(1+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Fix ε > 0 and let R := � (s, t) ∈ R2 : s < t, s ̸= 0, t ̸= 1, d � γ(σ), γ(τ) � < ε ∀σ, τ ∈ [s+, t−] ∩ [0−, 1+] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='49) Using the properties of γ, it is easy to see that � (s,t)∈R [s+, t−] ⊃ [0−, 1+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='50) Since the intervals [s+, t−] are open in the topology of the split real line and since [0−, 1+] is compact by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3, there exists a finite subcover, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=', there exists a finite set S ⊂ R such that � (s,t)∈S [s+, t−] ⊃ [0−, 1+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='51) Let T := {s : (s, t) ∈ S} ∪ {t : (s, t) ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then, letting t0 denote the largest element of T ∩ (−∞, 0), ordering the elements of T ∩ (0, 1) as t1 < · · · < tn−1, and letting tn denote the smallest element of T ∩ (1, ∞), we obtain times t0 < · · · < tn as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13 If γ1, γ2 are cadlag parametrisations of K1, K2, and λ ∈ Λ, then let us set Rλ := �� γ1(t), γ2 � λ(t) �� : t ∈ [0, 1] � = �� γ1 � λ−1(t) � , γ2(t) � : t ∈ [0, 1] � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='52) and let Rλ denote its closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We claim that Rλ is a correspondence between K1 and K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To see this, let γ− i denote the caglad modification of γi (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By the definition of a cadlag parametrisation, each element x1 ∈ K1 is of the form x1 = γ1(t) or = γ− 1 (t) for some t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If x1 = γ1(t), then clearly there exists an x2 ∈ K2 such that (x1, x2) ∈ Rλ, namely x2 := γ2(λ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If x1 = γ− 1 (t), then we can choose tn ↑ t and set xn 1 := γ1(tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then xn 1 → x1 by the left continuity of γ− 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We have already seen that there exist xn 2 ∈ K2 such that (xn 1, xn 2) ∈ Rλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since K2 is compact, by going to a subsequence, we can assume that xn 2 → x2 for some x2 ∈ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then (x1, x2) ∈ Rλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In the same way, we see that for each x2 ∈ K2, there exists an x1 ∈ K1 such that (x1, x2) ∈ Rλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This completes the proof that Rλ is a correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Using the fact that the total orders on K1 and K2 are compatible with the topology, it is easy to see that Rλ is monotone if λ ∈ Λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Using these facts as well as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='46) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='47), we see that dH(K1, K2) ≤ inf λ∈Λ dist(Rλ) = inf λ∈Λ dist(Rλ) = inf λ∈Λ sup t∈[0,1] d � γ1(t), γ2 � λ(t) �� , dtot(K1, K2) ≤ inf λ∈Λ+dist(Rλ) = inf λ∈Λ+dist(Rλ) = inf λ∈Λ+ sup t∈[0,1] d � γ1(t), γ2 � λ(t) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='53) To complete the proof, we must show that: (i) For each R ∈ Corr(K1, K2) and ε > 0, there exists a λ ∈ Λ such that dist(Rλ) ≤ dist(R)+ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (ii) For each R ∈ Corr+(K1, K2) and ε > 0, there exists a λ ∈ Λ+ such that dist(Rλ) ≤ dist(R) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We first prove (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Fix R ∈ Corr(K1, K2) and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18, for i = 1, 2, there exist ti 0 < 0 < ti 1 < · · · < ti ni−1 < 1 < ti ni such that sup � d � γi(s), γi(s′) � : 1 ≤ k ≤ ni, s, t ∈ [0, 1] ∩ [ti k−1, ti k) � < ε/2 (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='54) 31 For 1 ≤ k ≤ ni, let us write Ki k := {γi(t) : t ∈ [0, 1] ∩ [ti k−1, ti k)} (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We can choose a correspondence S between {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , n1} and {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , n2} such that for each (k1, k2) ∈ S, there exists an (x1, x2) ∈ R with xi ∈ Ki ki (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then sup (k1,k2)∈S sup x1∈K1 x2∈K2 d(x1, x2) ≤ dist(R) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='55) By refining the partitions ti 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , ti ni, we can make sure that for each k1 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , n1}, there is a unique k2 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , n2} such that (k1, k2) ∈ S, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We can then construct a bijection λ : [0, 1] → [0, 1] such that for each (k1, k2) ∈ S, the restriction of λ to [0, 1] ∩ [t1 k1−1, t1 k1) is a bijection to [0, 1] ∩ [t2 k2−1, t2 k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='55) implies that dist(Rλ) ≤ dist(R) + ε, completing the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To also prove (ii), we observe that if R is a monotone correspondence, then S as we initially constructed it will be a monotone correspondence between {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , n1} and {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' , n2}, and monotonicity will be preserved after we refine the partitions so that they have the same size and S is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Now λ can be chosen monotone too, completing the proof of (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 Betweenness In this subsection, we prove Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='17, as well as Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19 below that will be used in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14 Clearly, (ii) and (iii) imply (v) and (iv) implies (vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove (vii), we first observe that x ∈ ⟨y, z⟩ and y ∈ ⟨x, z⟩ imply by (vi) ⟨x, z⟩ ⊂ ⟨y, z⟩ ⊂ ⟨x, z⟩ and hence ⟨x, z⟩ = ⟨y, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Using this as well as the assumptions y ∈ ⟨x, z⟩ and x ∈ ⟨y, z⟩ we can conclude by (i) and (iii) that {y} = ⟨x, y⟩ ∩ ⟨y, z⟩ = ⟨y, x⟩ ∩ ⟨x, z⟩ = {x} and hence x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove (viii), assume that y, y′ ∈ ⟨x, z⟩ and y′ ∈ ⟨x, y⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The statement is trivial if y = y′ so without loss of generality we assume that y ̸= y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since y′ ∈ ⟨x, z⟩ we have by (iv) that ⟨x, z⟩ = ⟨x, y′⟩ ∪ ⟨y′, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since also y ∈ ⟨x, z⟩ we must have either y ∈ ⟨x, y′⟩, or y ∈ ⟨y′, z⟩, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The first possibility would by (i) and (vii) and the fact that y′ ∈ ⟨x, y⟩ imply that y = y′, which contradicts our assumptions, so we conclude that y ∈ ⟨y′, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The first implication ⇒ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='21) follow from the fact that y ∈ ⟨x, y⟩ by (i) and (ii), while the reverse implication follows from (vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The second equivalence in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='21) follows from (viii) and the third equivalence follows from the first one, by the symmetry (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It is clear that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='21) defines a partial order ≤x,z on ⟨x, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By (iv), if y, y′ ∈ ⟨x, z⟩, then at least one of the conditions y′ ∈ ⟨x, y⟩ and y′ ∈ ⟨y, z⟩ must hold, which shows that ≤x,z is a total order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15 We need to check that our definition satisfies axioms (i)–(iv) of a betweenness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Axioms (i) and (ii) are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove (iii) and (iv), set r := d(x, z) and let γ : [0, r] → X be the unique isometry such that γ(0) = x and γ(r) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since an isometry is one-to-one, there exists a unique p ∈ [0, r] such that γ(p) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Clearly, the restrictions of γ to [0, p] and [p, r] are isometries, so ⟨x, y⟩ = {γ(t) : 0 ≤ t ≤ p} and ⟨y, z⟩ = {γ(t) : p ≤ t ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' From these observations, axioms (iii) and (iv) follow immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We skip the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16 for the moment and first prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='17 and the already announced Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='17 For the trivial betweenness, ⟨x, z⟩ = {x, z} is clearly compact for each x, z ∈ X, and the continuity of the map (x, z) �→ {x, z} with respect to the Hausdorff topology follows immediately from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 32 If a betweenness is generated by an interpolation function, then ⟨x, z⟩, being the image of [0, 1] under the continuous map p �→ ϕ(x, z, p), is clearly compact for all x, z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let xn → x and zn → z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To show that ⟨xn, zn⟩ → ⟨x, z⟩ in the Hausdorff topology, we check the conditions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since xn → x and zn → z, the sets A := {x} ∪ {xn : n ∈ N} and B := {z} ∪ {zn : n ∈ N} are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let ϕ(A × B × [0, 1]) denote the image of A × B × [0, 1] under ϕ, which is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Clearly ⟨xn, zn⟩ ⊂ ϕ(A × B × [0, 1]) for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To complete the argument, it suffices to show that � y ∈ X : ∃yn ∈ ⟨xn, zn⟩ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' y is a cluster point of (yn)n∈N � ⊂ ⟨x, z⟩ ⊂ � y ∈ X : ∃yn ∈ ⟨xn, zn⟩ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' yn → y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='56) For the first inclusion, assume that y is a cluster point of yn = ϕ(xn, zn, pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By going to a subsequence, we can assume that pn → p for some p ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then y = ϕ(x, z, p) ∈ ⟨x, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For the second inclusion, assume that y = ϕ(x, z, p) for some p ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then yn := ϕ(xn, zn, p) ∈ ⟨xn, zn⟩ converge to y, completing the proof that each betweenness that is generated by an interpolation function is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For the final statement of the lemma, assume that X is a closed subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then clearly ⟨x, z⟩ := [x, z] ∩ X is compact for each x, z ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let xn → x and zn → z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To show that ⟨xn, zn⟩ → ⟨x, z⟩ in the Hausdorff topology, we again check the conditions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Clearly, ⟨xn, zn⟩ ⊂ [S, T] ∩ X for each n, where S := infn xn and T := supn zn, so to complete the argument, it again suffices to check (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For the first inclusion, assume that y ∈ X is a cluster point of yn ∈ ⟨xn, zn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since xn ≤ yn ≤ zn for each n, taking the limit, we see that x ≤ y ≤ z and hence y ∈ ⟨x, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For the second inclusion, assume that y ∈ ⟨x, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If x < y < z, then xn < y < zn for all n large enough, so setting yn := y for n large enough and yn := xn otherwise proves that y is al element of the set on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If y ∈ {x, z}, then setting yn := xn or := zn proves the same claim, so the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma will be used in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1, and is also of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19 (Segments as ordered sets) Let X be a metrisable space that is equipped with a betweenness that is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then for each x, z ∈ X, the segment ⟨x, z⟩ equipped with the total order ≤x,z is an element of Ktot(X), and the map (x, z) �→ ⟨x, z⟩ is continuous with respect to the product topology on X 2 and the topology on Ktot(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof To show that ⟨x, z⟩ is an element of Ktot(X), we must show that the total order ≤x,z is compatible with the induced topology on ⟨x, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that yn, y′ n, y, y′ ∈ ⟨x, z⟩ satisfy yn → y, y′ n → y′, and yn ≤x,z y′ n for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then yn ∈ ⟨x, y′ n⟩ for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since the betweenness is compatible with the topology, ⟨x, y′ n⟩ → ⟨x, y′⟩ in the Hausdorff topology, which by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 implies that y ∈ ⟨x, y′⟩ and hence y ≤x,z y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This shows that the total order ≤x,z is compatible with the induced topology on ⟨x, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To show that the map (x, z) �→ ⟨x, z⟩ is continuous with respect to the topology on Ktot(X), assume that xn → x, zn → z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We will show that dpart � ⟨xn, zn⟩, ⟨x, z⟩ � −→ n→∞ 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='57) which is equivalent to the statement that ⟨xn, zn⟩⟨2⟩ converges to ⟨x, z⟩⟨2⟩ in the Hausdorff topology on K+(X 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since ⟨xn, zn⟩ → ⟨x, z⟩, there exists a compact C ⊂ X such that ⟨xn, zn⟩ ⊂ C for all n and hence ⟨xn, zn⟩⟨2⟩ ⊂ C2 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Thus, it suffices to 33 check that (compare (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='56)) � (y, y′)∈X 2 : ∃yn, y′ n ∈ ⟨xn, zn⟩ with yn ≤xn,zn y′ n s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (y, y′) is a cluster point of (yn, y′ n)n∈N � ⊂ ⟨x, z⟩⟨2⟩ ⊂ � (y, y′) ∈ X : ∃yn, y′ n ∈ ⟨xn, zn⟩ with yn ≤xn,zn y′ n s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (yn, y′ n) → (y, y′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='58) If (y, y′) is an element of the set on the left-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='58) and (yn, y′ n) fulfill the conditions of the definition of this set, then by going to a subsequence we may assume that (yn, y′ n) → (y, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then y, y′ ∈ ⟨x, z⟩ since ⟨xn, zn⟩ → ⟨x, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Moreover yn ≤xn,zn y′ n means yn ∈ ⟨xn, y′ n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since yn → y and ⟨xn, y′ n⟩ → ⟨x, y′⟩, this implies y ∈ ⟨x, y′⟩ and hence y ≤x,z y′, proving that (y, y′) ∈ ⟨x, z⟩⟨2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove the second inclusion in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='58), assume that (y, y′) ∈ ⟨x, z⟩⟨2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since ⟨xn, zn⟩ → ⟨x, z⟩, there exist yn, y′ n ∈ ⟨xn, zn⟩ such that yn → y and y′ n → y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We now distinguish two cases: y ̸= y′ and y = y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If y ̸= y′, then we claim that yn ≤xn,zn y′ n for all n large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Indeed, in the opposite case, since ≤xn,zn is a total order, by going to a subsequence, we can assume that y′ n ≤xn,zn yn for all n, which by the arguments we have already seen implies y′ ≤x,z y, so that by the fact that (y, y′) ∈ ⟨x, z⟩⟨2⟩ we must have y = y′, contradicting our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since yn ≤xn,zn y′ n for all n large enough, changing the definitions of yn, y′ n for finitely many n, we see that there exist yn, y′ n ∈ ⟨xn, zn⟩ with yn ≤xn,zn y′ n such that s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (yn, y′ n) → (y, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In the case y = y′ the argument is even simpler, since now (yn, yn) → (y, y′) while obviously yn ≤xn,zn yn for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In this subsection, it only remains to prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The statement about the linear betweenness is trivial, but before we can prove the statement about the geodesic betweenness, we first need a better understanding of metric spaces with unique geodesics, which is provided by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In any metric space (X, d), for all x, z ∈ X and ε ≥ 0, we define ηx,z(ε) := sup � d(y1, y2) : � d(x, y1) ∧ d(x, y2) � + � d(y1, z) ∧ d(y2, z) � ≤ d(x, z) + ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='59) In other words, this is the largest distance between two points y1, y2 ∈ X for which there exist constants r, r′ ≥ 0 with r + r′ ≤ d(x, z) + ε such that d(x, yi) ≤ r and d(yi, z) ≤ r′ (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20 (Unique geodesics) Let (X, d) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Consider the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (i) For all x, z ∈ X and r, r′ ≥ 0 with r + r′ = d(x, z), there exists an y ∈ X such that d(x, y) = r and d(y, z) = r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (ii) ηx,z(0) = 0 for all x, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (ii)’ lim ε→0 ηx,z(ε) = 0 for all x, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then (X, d) has unique geodesics if and only if (i) and (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Moreover, (ii)’ implies (ii), and if (X, d) is a proper metric space, then (ii)’ implies (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof In any metric space (X, d), let us introduce the notation ⟨x, z⟩ := � y ∈ X : d(x, y) + d(y, z) = d(x, z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='60) We claim that y ∈ ⟨x, z⟩, y′ ∈ ⟨x, y⟩, y′′ ∈ ⟨y, z⟩ ⇒ y ∈ ⟨y′, y′′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='61) 34 To see this, we note that if the assumptions in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='61) hold but the conclusion does not, then by the triangle inequality d(x, z) = d(x, y) + d(y, z) = d(x, y′) + d(y′, y) + d(y, y′′) + d(y′′, z) > d(x, y′) + d(y′, y′′) + d(y′′, z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='62) which contradicts the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Now let (X, d) be a metric space with unique geodesics and let ⟨⟨x, z⟩⟩ denote the unique geodesic with endpoints x, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Clearly ⟨⟨x, z⟩⟩ ⊂ ⟨x, z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We claim that y ∈ ⟨x, z⟩ ⇒ ⟨⟨x, y⟩⟩ ∪ ⟨⟨y, z⟩⟩ = ⟨⟨x, z⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='63) To see this, let r := d(x, y), r′ := d(y, z), and let γ : [0, r] → X and γ′′ : [r, r + r′] → X be the unique isometries with γ(0) = x, γ(r) = γ′(r) = y, and γ′(r + r′) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We claim that γ′′ : [0, r + r′] → X defined as γ′′(t) = γ(t) for t ∈ [0, r] and := γ′(t) for t ∈ [r, r + r′] is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' So see this, let 0 ≤ t′ < t′′ ≤ r + r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We need to show that d � γ′′(t′), γ′′(t′′) � = t′′ − t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This is clear when t′′ ≤ r or r ≤ t′′, while in the remaining case t′ < r < t′′ the claim follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We now prove that if (X, d) is a metric space with unique geodesics, then conditions (i) and (ii) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Condition (i) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove (ii), let x, z ∈ X, let r, r′ ≥ 0 satisfy r + r′ := d(x, z), and assume that y1, y2 ∈ X satisfy d(x, yi) = r, d(yi, z) = r′ (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='63), there exist isometries γi : [0, r + r′] → X with γi(0) = x, γi(r) = yi, and γi(r + r′) = z, so by the assumption that (X, d) has unique geodesics we conclude that y1 = y2, proving (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Conversely, if (X, d) is a metric space for which (i) and (ii) hold, then for each x, z ∈ X with r := d(x, z), we can uniquely define γx,z : [0, r] → X by γx,z(t) := y with d(x, y) = t, d(y, z) = r − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='64) Clearly, if γ : [0, r] → X is an isometry with γ(0) = x and γ(r) = z, then we must have γ = γx,z, so to prove that (X, d) has unique geodesics, it suffices to show that γx,z is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let 0 ≤ t1 ≤ t2 ≤ r and let yi := γx,z(ti) (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Set y′ 1 := γx,y2(t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then d(x, y′ 1) = t1 and d(y′ 1, z) ≤ d(y′ 1, y2) + d(y2, z) = r − t1, which by the assumption (ii) implies y′ 1 = y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since d(y′ 1, y2) = t2 − t1, this proves that γx,z is an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This completes the proof that a metric space (X, d) has unique geodesics if and only if (i) and (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Trivially, (ii)’ implies (ii), so to complete the proof of the proposition, it suffices to prove that for proper metric spaces, (ii) implies (ii)’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that (ii)’ does not hold for some x, z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let 0 < εn ≤ 1 satisfy εn → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then for some δ > 0, we can find yn 1 , yn 2 ∈ X with d(yn 1 , yn 2 ) ≥ δ, as well as rn, r′ n ≥ 0 with rn+r′ n ≤ d(x, z)+εn, such that d(x, yn i ) ≤ rn and d(yn i , z) ≤ r′ n (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since d(x, yn i ) ≤ d(x, z) + 1 for all n, by the properness assumption, we can select a subsequence such that yn i → yi for some y1, y2 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since rn + r′ n ≤ d(x, z) + 1, by going to a further subsequence, we can assume that rn → r and r′ n → r′ for some r, r′ ≥ 0 with r + r′ = d(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then d(x, yi) ≤ r and d(yi, z) ≤ r′ while d(y1, y2) ≥ δ which shows that ηx,z(0) ≥ δ, violating (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If a metric space (X, d) has unique geodesics, then by conditions (i) and (ii) of Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20, we can uniquely define a function ϕ : X 2 × [0, 1] → X by ϕ(x, z, p) := y with d(x, y) = pd(x, z) and d(y, z) = (1 − p)d(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='65) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16 is now implied by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20 and the following lemma (the statement in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16 about normed linear spaces being trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 35 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='21 (Geodesic interpolation function) Let (X, d) be a metric space with unique geodesics and let ϕ be defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then for each x, z ∈ X, the unique geodesic with endpoints x, z is given by {ϕ(x, z, p) : p ∈ [0, 1]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If condition (ii)’ of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20 is satisfied, then ϕ : X 2 × [0, 1] → X is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof Let x, z ∈ X and r := d(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We observe that {ϕ(x, z, p) : p ∈ [0, 1]} = � γx,z(t) : t ∈ [0, r]} where γx,z is defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It has already been shown in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20 that this is the unique geodesic with endpoints x, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Therefore, to complete the proof, it suffices to show that condition (ii)’ of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20 implies that ϕ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that xn, x, zn, z ∈ X and pn, p ∈ [0, 1] satisfy xn → x, zn → z, and pn → p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Set yn := ϕ(xn, zn, pn) and y := ϕ(x, z, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We have to show that yn → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We observe that d(x, yn) + d(yn, z) ≤ d(xn, yn) + d(yn, zn) + d(x, xn) + d(z, zn) = d(xn, zn) + d(x, xn) + d(z, zn) −→ n→∞ d(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='66) Thus, for each ε > 0, we can find an m such that d(x, yn) + d(yn, z) ≤ d(x, z) + ε for all n ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since moreover d(x, y) + d(y, z) = d(x, z), it follows that d(yn, y) ≤ ηx,z(ε) for all n ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since ε > 0 is arbitrary, by (ii)’, this implies d(yn, y) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9 Squeezed space In this subsection, we prove Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18 We first prove that dsqz is a metric on R(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For brevity, we write d′(x, y) := d(x, y) ∧ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then d′ is a metric on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The only nontrivial statement that we have to prove is the triangle inequality, and it suffices to prove this for the function ρ � (x, s), (y, t) � := � φ(s) ∧ φ(t) � d′(x, y) + ��φ(s) − φ(t) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We estimate ρ � (x, s), (z, u) � ≤ � φ(s) ∧ φ(u) �� d′(x, y) + d′(y, z) � + ��φ(s) − φ(u) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='67) If φ(t) ≥ φ(s) ∧ φ(u), then φ(s) ∧ φ(u) is less than φ(s) ∧ φ(t) and also less than φ(t) ∧ φ(u), so we can simply estimate the expression in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='67) from above by � φ(s) ∧ φ(t) � d′(x, y) + � φ(t) ∧ φ(u) � d′(y, z) � + ��φ(s) − φ(t) �� + ��φ(t) − φ(u) �� and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' On the other hand, if φ(t) < φ(s) ∧ φ(u), then ��φ(s) − φ(t) �� + ��φ(t) − φ(u) �� = ��φ(s) − φ(u) �� + 2 � φ(s) ∧ φ(u) − φ(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Using the fact that d′ ≤ 1, we can now estimate the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='67) from above by φ(t) � d′(x, y) + d′(y, z) � + 2 � φ(s) ∧ φ(u) − φ(t) � + ��φ(s) − φ(u) �� = � φ(s) ∧ φ(t) � d′(x, y) + � φ(t) ∧ φ(u) � d′(y, z) + ��φ(s) − φ(t) �� + ��φ(t) − φ(u) ��, and again we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This completes the proof that dsqz is a metric on R(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It remains to prove that � φ(tn) ∧ φ(t) �� d(xn, x) ∧ 1 � + ��φ(tn) − φ(t) �� + dR(tn, t) −→ n→∞ 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='68) 36 if and only if conditions (i) and (ii) of the lemma are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Because of the third term on the left-hand side, a necessary condition for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='26) is that tn → t, and this condition also guarantees that the second term tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If t ∈ {−∞, +∞}, then this is all one needs since the first term now tends to zero regardless of the values of xn and x, but if t ∈ R, then one needs in addition that d(xn, x) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19 If D is a countable dense subset of (X, d), then D × Q is a countable dense subset of (R(X), dsqz), proving (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove (b), let (xn, tn) be a Cauchy sequence in (R(X), dsqz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='26) tn is a Cauchy sequence in R and hence tn → t for some t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If t ∈ R, then by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='26) xn is a Cauchy sequence in (X, d) so by the completeness of the latter, xn → x for some x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18, it follows that (xn, tn) converges, proving the completeness of (R(X), dsqz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20 Assume that A ⊂ R(X) has the property that for each T < ∞, there exists a compact set K ⊂ X such that {x ∈ X : (x, t) ∈ A, t ∈ [−T, T]} ⊂ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To show that A is precompact, we will show that each sequence (xn, tn) ∈ A has a convergent subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By the compactness of R, we can select a subsequence (x′ n, t′ n) such that t′ n → t for some t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If t = ±∞, then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18 (x′ n, t′ n) → (∗, ±∞) and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Otherwise, there exists a T < ∞ such that t′ n ∈ [−T, T] for all n large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By assumption, there then exists a compact set K ⊂ X such that x′ n ∈ K for all n large enough, so we can select a further subsequence such that (x′′ n, t′′ n) converges to a limit (x, t) ∈ X × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume, on the other hand, that A ⊂ R(X) has the property that for some T < ∞, there does not exist a compact set K ⊂ X such that {x ∈ X : (x, t) ∈ A, t ∈ [−T, T]} ⊂ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Set B := � x ∈ X : (x, t) ∈ A for some t ∈ [−T, T] � The closure of B cannot be compact, since this would contradict our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It follows that there exists a sequence xn ∈ B that does not contain a convergent subsequence, and there exist tn ∈ [−T, T] such that (xn, tn) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' But then, in view of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18, the sequence (xn, tn) cannot contain a convergent subsequence either, proving that A is not precompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 5 Proofs of the main results 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 Closed and filled-in graphs In this subsection, we prove Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2, as well an analogue of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 that will later be used in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 We will show that each sequence (xn, tn) ∈ Gf(π) has a subsequence that converges to a limit in Gf(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since I(π) is closed, we can select a subsequence such that tn → t for some t ∈ I(π)∪{−∞, ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If t = ±∞, then Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18 tells us that (xn, tn) → (∗, ±∞) ∈ Gf(π) so we are done, so from now on we can assume that t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By going to a further subsequence, we can assume that we are in one of the following three cases: (i) tn < t for all n, (ii) tn > t for all n, and (iii) tn = t for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In case (i), we use the cadlag property of π and the fact that the betweenness is compatible with the topology to see, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14 (v), that xn ∈ ⟨π(tn−), π(tn+)⟩ −→ n→∞ {π(t−)} (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1) from which we conclude that (xn, tn) converges to � π(t−), t � ∈ Gf(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In case (ii), the same argument shows that (xn, tn) converges to � π(t+), t � ∈ Gf(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In case (iii), finally, using the compactness of ⟨π(t−), π(t+)⟩, we can select a further subsequence such that (xn, t) → (x, t) for 37 some x ∈ ⟨π(t−), π(t+)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since also in this case the limit (x, t) is an element of Gf(π), we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To see that the total order ⪯ on Gf(π) is compatible with the (induced) topology on Gf(π), it suffices to show that S := �� (x, s), (y, t) � ∈ Gf(π) : (x, s) ≺ (y, t) � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2) is an open subset of Gf(π)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If � (x, s), (y, t) � is an element of S, then either: (i) s < t, or: (ii) s = t ∈ R and x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In case (i), we can choose s < S < T < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then O := �� (x′, s′), (y′, t′) � ∈ Gf(π) : s′ < S, T < t′� (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3) is an open subset of Gf(π) such that � (x, s), (y, t) � ∈ O ⊂ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In case (ii), we recall that by definition (x, t) ⪯ (y, t) if x ≤π(t−),π(t+) z, where by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19 the total order ≤π(t−),π(t+) on ⟨π(t−), π(t+)⟩ is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It follows that we can choose ε > 0 small enough such that for z ∈ ⟨π(t−), π(t+)⟩, if d(z, x) < ε then (z, t) ≺ (y, t), while if d(z, y) < ε then (x, t) ≺ (z, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Next, we use the cadlag property of π to choose δ > 0 small enough such that d � π(s±), x � > ε for all t < s < t + δ and d � π(s±), y � > ε for all t − δ < s < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then O := �� (x′, s), (y′, u) � ∈ Gf(π) : |s − t| ∨ |u − t| < δ, d(x′, x) ∨ d(y′, y) < ε � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4) is an open subset of Gf(π) such that � (x, s), (y, t) � ∈ O ⊂ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Together, these observations prove that S is an open subset of Gf(π)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma is similar to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 (Characterisation of continuous graphs) Let X be a metrisable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' As- sume that G ∈ K+(R(X)) and (∗, ±∞) ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then G is the closed graph of a path π ∈ Πc(X) if and only if for each t ∈ R, the set {x ∈ X : (x, t) ∈ G} has at most one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof Clearly, if G is the closed graph of a path π ∈ Πc(X), then for each t ∈ R, the set {x ∈ X : (x, t) ∈ G} has at most one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Conversely, if G has this property and (∗, ±∞) ∈ G, then we define I(π) := � t ∈ R : ∃x ∈ X s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (x, t) ∈ G � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5) and we use the fact that G contains at each time at most one point to define π : I(π) → X by � π(t) � := {x ∈ X : (x, t) ∈ G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6) We observe that I(π) is the intersection of R with the image of G under the continuous map (x, t) �→ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since the continuous image of a compact set is compact, this proves that I(π) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To complete the proof, it suffices to show that π(tn) → π(t) for all tn, t ∈ I(π) such that tn → t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It suffices to show that {π(tn) : n ∈ N} is precompact and its only cluster point is π(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Equivalently, we may show that each subsequence of π(tn) contains a further subsequence that converges to π(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By the compactness of G, for any subsequence, we can select a further subsequence such that π(tn) → x for some x ∈ X with (x, t) ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' But then x = π(t) by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1, the filled-in graph of a path π ∈ Π(X) corresponds to an element of Ktot(R(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Properties (i) and (ii) now follow from the definition of the total order ⪯ on π and property (vi) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that conversely, (G, ⪯) ∈ Ktot(X) contains (∗, ±∞) and satisfies (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We claim that for each t ∈ R, there exist unique π(t−), π(t+) ∈ X such that π(t−) ⪯ π(t+) and St := � x ∈ X : (x, t) ∈ G � = ⟨π(t−), π(t+)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7) 38 Indeed, St is a compact metrisable set, so we can choose a countable dense set {xn : n ∈ N} ⊂ St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Set y0 := x0 and define yn as the maximum of xn and yn−1 in the total order ⪯ (n ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By the compactness of St, by going to a subsequence, we can assume that yn → π(t+) for some π(t+) ∈ St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then y′ ⪯ y for all y′ ∈ D and hence also for all y′ ∈ St since the order is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In the same way, we see that St has a (necessarily unique) minimal element π(t−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By (ii), we conclude that St = ⟨π(t−), π(t+)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We now define I(π) := � t ∈ R : ∃x ∈ X s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (x, t) ∈ G � and Is(π) := � t± : t ∈ I(π) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8) and we use the claim we have just proved to define π : Is(π) → X by ⟨π(t−), π(t+)⟩ := {x ∈ X : (x, t) ∈ G} with π(t−) ⪯ π(t+) � t ∈ I(π) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9) Since I(π) is the intersection of R with the image of G under the continuous map (x, t) �→ t, which is compact, we see that I(π) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To complete the proof, it suffices to show that π : Is(π) → X is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By symmetry, it suffices to show that if τn ∈ Is(π) and t ∈ I(π) satisfy τ n > t for all n and τ n → t as n → ∞, then π(τn) → π(t+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' As in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1, it suffices to show that each subsequence of π(τn) contains a further subsequence that converges to π(t+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By the compactness of G, for any subsequence, we can select a further subsequence such that π(τn) → x for some x ∈ X such that (x, t) ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By (iii), we have � π(t+), t � ⪯ � π(τn), τ n � for all n, so using the fact that the total order is compatible with the topology, we see that � π(t+), t � ⪯ (x, t), which using the fact that π(t+) is the maximal element of ⟨π(t−), π(t+)⟩ with respect to the order ⪯ identifies x as π(t+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 Polishness In this subsection, we prove Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let X be a metrisable space that is equipped with a betweenness that is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By a slight abuse of notation, for any G ∈ Ktot(R(X)), we set mT,δ(G) := sup � d(x1, x2) : (xi, ti) ∈ G, −T ≤ ti ≤ T ∀i = 1, 2, (x1, t1) ⪯ (x2, t2), t2 − t1 ≤ δ � mS T,δ(G) := sup � d � x2, ⟨x1, x3⟩ � : (xi, ti) ∈ G and − T ≤ ti ≤ T ∀i = 1, 2, 3, (x1, t1) ⪯ (x2, t2) ⪯ (x3, t3), t3 − t1 ≤ δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10) Then mT,δ(Gf(π)) = mT,δ(π) for each π ∈ Πc(X) and mS T,δ(Gf(π)) = mS T,δ(π) for each π ∈ Π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma is similar to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 (Upper semi-continuity) Let X be a metrisable space and assume that Gn, G ∈ Ktot(R(X)) satisfy Gn → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then, for each T < ∞ and δ > 0, mT,δ(G) ≥ lim sup n→∞ mT,δ(Gn) and mS T,δ(G) ≥ lim sup n→∞ mS T,δ(Gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11) Proof We only prove the statement for the Skorohod modulus of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The proof for the traditional modulus of continuity is basically the same, but a bit simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The proof will be very similar to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By the compactness of [0, ∞] we can select a subsequence for which limn→∞ mS T,δ(Gn) exists and is equal to the limit superior of the original sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let εn > 0 converge to zero and pick (xn i , tn i ) ∈ Gn with −T ≤ tn i ≤ T (i = 1, 2, 3), 39 (xn 1, tn 1) ⪯ (xn 2, tn 2) ⪯ (xn 3, tn 3), and tn 3 − tn 1 ≤ δ, such that d � xn 2, {xn 1, xn 3} � ≥ mS T,δ(Gn) − εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since Gn → G in the topology on Ktot(R(X)), by the first inequality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16) they also converge in the topology on K+(R(X)), so by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 there exists a compact C ⊂ R(X) such that Gn ⊂ C for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It follows that we can select a subsequence such that (xn i , tn i ) → (xi, ti) for some (xi, ti) ∈ G (i = 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Recall from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 that dtot = d⟨∞⟩ so by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5, convergence in Ktot(R(X)) implies that G⟨m⟩ n → G⟨m⟩ in the Hausdorff topology for any 1 ≤ m ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In particular, G⟨3⟩ n → G⟨3⟩, which by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 implies (x1, t1) ⪯ (x2, t2) ⪯ (x3, t3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Moreover −T ≤ ti ≤ T (i = 1, 2, 3) and t3 − t1 ≤ δ, so mS T,δ(G) ≥ d � x2, {x1, x3} � ≥ lim n→∞ � mS T,δ(Gn) − εn � = lim n→∞ mS T,δ(Gn), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12) where we have used that the betweenness is compatible with the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since we have chosen our subsequence such that the right-hand side is equal to the limit superior of the original sequence, this proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 We observe that if X is Polish, then, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19, so is R(X) and hence, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='11, also Ktot(R(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By identifying a path with its filled-in graph, we can identity Π(X) with a subset of Ktot(R(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The Skorohod topology on Π(X) is then the induced topology from Ktot(R(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Therefore, in view of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='17, it suffices to show that Π(X), viewed as a subset of Ktot(R(X)), is a Gδ-subset of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We start by showing that condition (i) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 can be replaced by (i)’ lim δ→0 mS T,δ(G) = 0 ∀T < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To see this, we argue as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If (i) does not hold, then for some t ∈ R, there exist (xi, ti) ∈ G (i = 1, 2, 3) with (x1, t1) ⪯ (x2, t2) ⪯ (x3, t3) and x2 ̸∈ ⟨x1, x3⟩, which implies that mS T,δ(G) ≥ d � x2, ⟨x1, x3⟩ � > 0 for all δ > 0 and T < ∞ such that −T ≤ t ≤ T, so (i)’ clearly does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Conversely, if (i)’ does not hold, then for some T < ∞ and ε > 0 we can choose δn > 0 tending to zero and (xn i , tn i ) ∈ G (i = 1, 2, 3) with (xn 1, tn 1) ⪯ (xn 2, tn 2) ⪯ (xn 3, tn 3) such that d � xn 2, ⟨xn 1, xn 3⟩ � ≥ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By the compactness of G, we can select a subsequence such that (xn i , tn i ) → (xi, ti) (i = 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then clearly t1 = t2 = t3 =: t for some −T ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since the order is compatible with the topology moreover (x1, t1) ⪯ (x2, t2) ⪯ (x3, t3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The fact that the betweenness is compatible with the topology allows us to conclude that d � x2, ⟨x1, x3⟩ � = limn→∞ d � xn 2, ⟨xn 1, xn 3⟩ � ≥ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This shows that x2 ̸∈ ⟨x1, x3⟩ and hence (i) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let H denote the set of all elements of Ktot(R(X)) that satisfy condition (ii) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By what we have just proved, Π(X) = � G ∈ H : lim δ→0 mS T,δ(G) = 0 ∀T < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13) It follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 that for each T < ∞ and ε, δ > 0, the set GT,ε,δ := � G ∈ Ktot(R(X)) : mS T,δ(L) ≥ ε � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14) is a closed subset of K+(X 2) and hence its complement Ac ε,δ is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' As a consequence, G := ∞ � N=1 ∞ � n=1 ∞ � m=1 Gc N,1/n,1/m (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='15) is a Gδ-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='13) says that Π(X) = G ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It is easy to see that H is a closed subset of Ktot(R(X)), and hence a Gδ-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since the intersection of two Gδ-sets is a Gδ-set, this yields the statement we wanted to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10, dS part and dS tot generate the same topology on Π(X), and by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16) convergence in any of these two metrics implies convergence in dH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Therefore, 40 to show that the conditions (i)–(iii) are equivalent, it suffices to show that if πn ∈ Π(X) and π ∈ Πc(X), then (iii) implies (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' More precisely, we will show that for any betweenness on X that is compatible with the topology, Gf(πn) → G(π) in the Hausdorff topology implies Gf(πn)⟨2⟩ −→ n→∞ G(π)⟨2⟩ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16) in the Hausdorff topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, convergence of Gf(πn) implies the existence of a compact set C ⊂ R(X) such that Gf(πn) ⊂ C for all n, which implies Gf(πn)⟨2⟩ ⊂ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To complete the proof, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, we need to prove the following two statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (i) For every � (x, s), (y, t) � ∈ G(π)⟨2⟩, there exist � (xn, sn), (yn, tn) � ∈ Gf(πn)⟨2⟩ such that � (xn, sn), (yn, tn) � → � (x, s), (y, t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (ii) If a sequence � (xn, sn), (yn, tn) � ∈ Gf(πn)⟨2⟩ has a cluster point � (x, s), (y, t) � ∈ R(X)2, then � (x, s), (y, t) � ∈ G(π)⟨2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove (i), we use the fact that Gf(πn) → G(π) to find (xn, sn), (yn, tn) ∈ Gf(πn) such that (xn, sn) → (x, s) and (yn, tn) → (y, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If s < t, then � (xn, sn), (yn, tn) � ∈ Gf(πn)⟨2⟩ for n large enough, so (i) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' On the other hand, if s = t, then � (xn, sn), (xn, sn) � ∈ Gf(πn)⟨2⟩ so (i) also holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove (ii), we use the fact that Gf(πn) → G(π) to conclude that any cluster point � (x, s), (y, t) � satisfies (x, s), (y, t) ∈ G(π) with s ≤ t, and hence by the continuity of π either s < t or (x, s) = (y, t), from which we conclude that � (x, s), (y, t) � ∈ G(π)⟨2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It remains to prove that Πc(X) is Polish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This is very similar to the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3, but simpler, so we only sketch the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1, we may identify Πc(X) with the subset of K+(R(X)) consisting of all G that contain (∗, ±∞) and have the property that for each t ∈ R, the set {x ∈ X : (x, t) ∈ G} has at most one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In this identification, the Hausdorff metric on K+(R(X)) induces the metric dH which generates the topology on Πc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We claim that the condition that for each t ∈ R, the set {x ∈ X : (x, t) ∈ G} has at most one element is equivalent to lim δ→0 mT,δ(G) = 0 ∀T < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='17) This follows by the same sort of argument as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3, where it was shown that condition (i)’ there is equivalent to condition (i) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Thus, identifying a path with its closed graph, we have Πc(X) = � G ∈ K+(X) : lim δ→0 mT,δ(G) = 0 ∀T < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='18) By the same argument as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3, it follows that Πc(X) is a Gδ-subset of K+(X) and hence, by Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='17, a Polish space if X is Polish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Our next aim is the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The proof of the final statement of that lemma needs a bit of preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that π ∈ Π(X) is not the trivial path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then we define G∗(π) := � (x, t) ∈ G(π) : t ∈ I(π) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='19) where I(π) denotes the closure of I(π) in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This is almost the same as the closed graph G(π), except that we include the points at infinity (∗, ±∞) only if their time coordinate lies in I(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since G∗(π) is a subset of G(π), it is naturally equipped with a total order that is compatible with the topology, so we can view it as an element of the space Ktot(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The reason why we usually work with G(π) instead of G∗(π) is that if we would use the latter throughout, we would 41 end up with a space of paths that contains three trivial paths, whose graphs would be {(∗, −∞)}, {(∗, −∞)}, and the union of these two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma says that as long as we restrict ourselves to nontrivial paths whose domain is an interval, it does not matter which definition of the closed graph we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 (Convergence of graphs) Assume that πn, π ∈ Π|(X) and that π is not the trivial path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then G(πn) → G(π) in the topology om K+(R(X)) if and only if G∗(πn) → G∗(π) in the topology om K+(R(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof Let π1, π2 ∈ Π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Each correspondence between G∗(π1) and G∗(π2) can be extended to a correspondence between G(π1) and G(π2) by adding the points � (∗, −∞), (∗, −∞) � and � (∗, +∞), (∗, +∞) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Adding these extra points does not change the supremum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8), so we see that dH � G(π1), G(π2) � ≤ dH � G∗(π1), G∗(π2) � � π1, π2 ∈ Π(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20) From this we see immediately that G∗(πn) → G∗(π) in the Hausdorff topology implies G(πn) → G(π) in the Hausdorff topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This part of the argument holds for general πn, π ∈ Π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To complete the proof, we must show that if πn, π ∈ Π|(X) and π is not the trivial path, then, conversely, G(πn) → G(π) in the Hausdorff topology implies G∗(πn) → G∗(π) in the Hausdorff topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, there exists a compact C ⊂ R(X) such that G(πn) ⊂ C for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since the G∗(πn) are subsets of G(πn), they are contained in C too, so by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 it suffices to show that: (i) For each (x, t) ∈ G∗(π), there exist (xn, tn) ∈ G∗(πn) such that (xn, tn) → (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (ii) If (x, t) is a cluster point of (xn, tn) ∈ G∗(πn), then (x, t) ∈ G∗(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove (i), we observe that since G∗(π) ⊂ G(π), for each (x, t) ∈ G∗(π), there exist (xn, tn) ∈ G(πn) such that (xn, tn) → (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If t ∈ R, then tn ∈ R for n large enough and hence (xn, tn) ∈ G∗(πn) and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If t = ∞, then by the fact that π ∈ Π|(X) and π is not the trivial path, we see that for each T < ∞, we must have I(πn) ∩ [T, ∞) ̸= ∅ for all n large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Using this, we see that there exist (xn, tn) ∈ G(πn) with tn < ∞ such that (xn, tn) → (x, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' But then (xn, tn) ∈ G∗(πn), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The proof when t = −∞ is the same, so the proof of (i) is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove (ii), we observe that since G∗(πn) ⊂ G(πn) → G(π), if (x, t) is a cluster point of (xn, tn) ∈ G∗(πn), then (x, t) ∈ G(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If t ∈ R, then clearly (x, t) ∈ G∗(π) and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If t = ∞, then by the assumption that π is not trivial we can choose (y, s) ∈ G(π) with s ∈ R, and by the assumption that G(πn) → G(π) we can choose (yn, sn) ∈ G(πn) such that (yn, sn) → (y, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since πn ∈ Π|(X) for all n, it follows that I(πn) contains (sn, tn) for each n and hence by the assumption that G(πn) → G(π), the domain I(π) contains (s, ∞), which implies that (∗, ∞) ∈ G∗(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The argument when t = −∞ is the same so we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The following lemma reveals a pleasant property of G∗(π) that G(π) does not have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 (Connected graphs) Assume that π ∈ Π(X) is not the trivial path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then π ∈ Π| c(X) if and only if G∗(π) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof If π ∈ Π| c(X), then G∗(π) is the image of the compact set I(π) under the continuous map from R to R(X) given by t �→ � π(t), t) (with ±∞ �→ (∗, ±∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since I(π) is connected and the continuous image of a connected set is connected, we conclude that G∗(π) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Conversely, if G∗(π) is connected, then I(π) must be connected and hence π ∈ Π|(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To see that π is moreover continuous, assume that conversely, π(t−) ̸= π(t+) for some t ∈ I(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then 42 we can define new paths π′, π′′ with domains I(π′) := (−∞, t] ∩ I(π) and I(π′′) := [0, ∞) ∩ I(π), by setting π′(s) := π(s) and π′′(s) := π(s) for s ̸= t, and π′(t±) := π(t−) and π′′(t±) := π(t+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1, G∗(π′) and G∗(π′′) are compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since G∗(π′) ∩ G∗(π′′) = ∅ and G∗(π′) ∪ G∗(π′′) = G∗(π), this proves that G∗(π) is not connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='5 By the first inequality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16), convergence πn → π in Π(X) implies convergence of Gf(πn) to Gf(π) in the Hausdorff topology, which by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 implies conver- gence of I(πn) ∪ {±∞} to I(π) ∪ {±∞} in K+(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, it is easy to see that if In are closed subintervals of R such that In ∪ {±∞} converges in K+(R) to a limit, then this limit must be of the form I ∪{±∞} for some (possibly empty) closed interval I ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This shows that Π|(X) is closed and in the same way we also see that Π↑(X) and Π↓(X) are closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Now assume that the betweenness is the trivial betweenness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that πn ∈ Π| c converge to π ∈ Π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We need to show that π ∈ Π| c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This is certainly true if π is the trivial path, so we assume from now on that π is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By the first inequality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='16), convergence of πn to π in the topology on Π(X) implies that G(πn) → G(π) in the Hausdorff topology, which by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 implies that also G∗(πn) → G∗(π) in the Hausdorff topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4, the graphs G∗(πn) are connected and hence Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9 implies that G∗(π) is connected, which by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 implies that π ∈ Π| c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 Compactness criteria In this subsection, we prove Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1 will play an essential role here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='1, we may identify Πc(X) with the subset of K+(R(X)) consisting of all G that contain (∗, ±∞) and have the property that for each t ∈ R, the set {x ∈ X : (x, t) ∈ G} has at most one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In this identification, the Hausdorff metric on K+(R(X)) induces the metric dH on Πc(X), which by the definition above Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 generates the topology on Πc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' For any G ∈ K+(R(X)), we define mT,δ(G) := sup � d(x1, x2) : (x1, t1), (x2, t2) ∈ G, −T ≤ t1 < t2 ≤ T, t2 − t1 ≤ δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='21) In the special case that G is (the closed graph of) a path in Πc(X), this coincides with the definition of the modulus of continuity in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let A ⊂ Πc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then A is precompact if and only if its closure A in K+(R(X)) is compact and A ⊂ Πc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20, A is a compact subset of K+(R(X)) if and only if A satisfies the compact containment condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Therefore, to complete the proof, it suffices to show that if A ⊂ Πc(X) satisfies the compact containment condition, then A ⊂ Πc(X) if and only if A is equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that A satisfies the compact containment condition and is equicontinuous, and that Gn ∈ A satisfy Gn → G for some G ∈ K+(R(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We need to show that G ∈ Πc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Clearly (∗, ±∞) ∈ G, so it suffices to show that for each t ∈ R, the set {x ∈ X : (x, t) ∈ G} has at most one element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that conversely, there exist (x1, t), (x2, t) ∈ G with t ∈ R and x1 ̸= x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, there exist (xn 1, tn 1), (xn 2, tn 2) ∈ Gn such that (xn i , tn i ) → (xi, t) as n → ∞ (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Choose T < ∞ such that −T < t < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then for each δ > 0 we have for n large enough that −T < tn 1, tn 2 < T, |tn 1 − tn 2| ≤ δ, and d(xn 1, xn 2) ≥ d(x1, x2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' This proves that sup π∈A mT,δ(π) ≥ d(x1, x2)/2 ∀δ > 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='22) contradicting the equicontinuity of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 43 Assume, on the other hand, that A satisfies the compact containment condition and is not equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let δn be positive constants tending to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since A is not equicontinuous, for some T < ∞ and ε > 0 we can find Gn ∈ A such that mT,δn(Gn) ≥ ε for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since A satisfies the compact containment condition, A is a compact subset of K+(R(X)), so by going to a subsequence we may assume that Gn → G for some G ∈ K+(R(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since mT,δn(Gn) ≥ ε we can find (xn 1, tn 1), (xn 2, tn 2) ∈ Gn such that −T ≤ tn 1 < tn 2 ≤ T, tn 2 −tn 1 ≤ δn, and d(xn 1, xn 2) ≥ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, going to a further subsequence if necessary, we can assume that (xn i , tn i ) → (xi, t) as n → ∞ (i = 1, 2) for some (x1, t), (x2, t) ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then −T ≤ t ≤ T and d(x1, x2) ≥ ε, which shows that {x ∈ X : (x, t) ∈ G} has more than one element and hence A is not contained in Πc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7 By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2, identifying a path with its filled-in graph, we may identify Π(X) with the subset of Ktot(R(X)) consisting of all (G, ⪯) that contain (∗, ±∞) and satisfy conditions (i) and (ii) of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In this identification, the metrics dpart and dtot on Ktot(R(X)) induce metrics dS part and dS tot on Π(X) that both generate the Skorohod topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let A ⊂ Π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then A is precompact in the Skorohod topology if and only if its closure A in Ktot(R(X)) is compact and A ⊂ Π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='12 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20, A is a compact subset of Ktot(R(X)) if and only if A satisfies the compact containment condition and lim ε→0 sup G∈A mε(G) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='23) where mε(G) denotes the mismatch modulus of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To complete the proof we will prove the following three statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' I Assume that A is satisfies the compact containment condition and is Skorohod-equicon- tinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then A satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' II Assume that A is Skorohod-equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then A ⊂ Π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' III Assume that A is a compact subset of Ktot(R(X)) and that A ⊂ Π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then A is Skorohod-equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Now if A satisfies the compact containment condition and is Skorohod-equicontinuous, then by our earlier remarks I implies that A is a compact subset of Ktot(R(X)) and II implies that A ⊂ Π(X), so A is precompact in the Skorohod topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Conversely, if A is precompact in the Skorohod topology, then A is a compact subset of Ktot(R(X)) and hence by our earlier remarks A satisfies the compact containment condition, and moreover A ⊂ Π(X) which by III implies that A is Skorohod-equicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We start by proving I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since A satisfies the compact containment condition, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20, we see that there exists a compact C ⊂ R(X) such that G ⊂ C for all G ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since supG∈A mε(G) is nondecreasing as a function of ε, the limit in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='23) always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let εn be positive constants, tending to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If the limit in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='23) is positive, then there exists a δ > 0 such that for each n we can find a G ∈ A and (xn i , sn i ), (yn i , tn i ) ∈ G (i = 1, 2) with (xn 1, sn 1) ⪯ (yn 1 , tn 1), (yn 2 , tn 2) ⪯ (xn 2, sn 2) such that dsqz � (xn 1, sn 1), (xn 2, sn 2) � ∨ dsqz � (yn 1 , tn 1), (yn 2 , tn 2) � ≤ εn, dsqz � (xn i , sn i ), (yn i , tn i ) � ≥ δ (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since G ⊂ C for all G ∈ A, by going to a subsequence, we may assume that (xn i , sn i ) → (x, s) and (yn i , tn i ) → (y, t) (i = 1, 2) for some (x, s), (y, t) ∈ R(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then dsqz � (x, s), (y, t) � ≥ δ and hence (x, s) ̸= (y, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since (xn 1, sn 1) ⪯ (yn 1 , tn 1) and (yn 2 , tn 2) ⪯ (xn 2, sn 2) we have sn 1 ≤ tn 1 and tn 2 ≤ sn 2 for all n which implies s = t and hence x ̸= y, since (x, t) ̸= (y, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By the structure of R(X), this 44 implies t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let (xn −, sn −) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (xn +, sn +)) be the smallest (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' largest) of the points (xn i , sn i ) (i = 1, 2) with respect to the order ⪯, and define (yn ±, tn ±) similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since G is totally ordered, by going to a subsequence, we can assume that we are in one of the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (xn −, sn −) ⪯ (yn −, tn −) for all n, or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' (yn −, tn −) ⪯ (xn −, sn −) for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let us assume that we are in case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then (xn −, sn −) ⪯ (yn −, tn −) ⪯ (xn +, sn +) for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since the betweenness is compatible with the topology, d � yn −, ⟨xn −, xn +⟩ � −→ n→∞ d(y, x) > 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='24) which contradicts the Skorohod-equicontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Case 2 is completely the same, exchanging the roles of x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We next prove II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that πn ∈ A converge in Ktot(R(X)) to a limit G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Recall from Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3 that dpart = d⟨2⟩ ≤ d⟨m⟩ ≤ d⟨∞⟩ = dtot for all m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' In particular, πn → G in Ktot(X) implies that π⟨m⟩ n → G⟨m⟩ in the Hausdorff topology for all m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It suffices to check that G satisfies conditions (i) and (ii) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Condition (ii) easily follows from the fact that π⟨2⟩ n → G⟨2⟩ in the Hausdorff topology, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It remains to prove that G satisfies condition (i) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that conversely, for some t ∈ R, there exist (x1, t), (x2, t), (x3, t) ∈ G with (x1, t) ⪯ (x2, t) ⪯ (x3, t) such that x2 ̸∈ ⟨x1, x3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since π⟨3⟩ n → G⟨3⟩ in the Hausdorff topology, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 there exist τ n i ∈ Is(πn) ∪ {±∞} (i = 1, 2, 3) with τ n 1 ≤ τ n 2 ≤ τ n 3 such that τ n i → t and πn(τ n i ) → xi (i = 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Using the fact that the betweenness is compatible with the topology, we see that d � πn(τ n 2 ), ⟨πn(τ n 1 ), πn(τ n 3 )⟩ � −→ n→∞ d � x2, ⟨x1, x3⟩ � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='25) which is easily seen to contradict Skorohod-equicontinuity, completing the proof of II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To prove III, finallly, we will show that if A is a compact subset of Ktot(R(X)) and A is not Skorohod-equicontinuous, then A is not contained in Π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let δn be positive constants, tending to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since A is not Skorohod-equicontinuous, there exists a ε > 0, T < ∞, πn ∈ A, and τ n i ∈ Is(πn) (i = 1, 2, 3) such that τ1 ≤ τ2 ≤ τ3, −T ≤ τ n 1, τ n 3 ≤ T, τ n 3 − τ n 1 ≤ δn, and d � π(τ n 2 ), ⟨π(τ n 1 ), π(τ n 3 )⟩ � ≥ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since A is a compact subset of Ktot(R(X)), we can select a subsequence such that πn → G for some G ∈ Ktot(R(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then π⟨3⟩ n → G⟨3⟩ in the Hausdorff topology, so by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 there exist C ⊂ R(X)3 such that π⟨3⟩ n ⊂ C for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It follows that by going to a further subsequence we can assume that � πn(τ n i ), τ n i � → (xi, ti) as n → ∞ for some (xi, t) ∈ G (i = 1, 2, 3) with (x1, t1) ⪯ (x2, t2) ⪯ (x3, t3) and −T ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Using the fact that the betweenness is compatible with the topology, we see that d(x2, ⟨x1, x3⟩) ≥ ε, which shows that G is not the filled-in graph of a path π ∈ Π(X) and hence A is not contained in Π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='4 Paths on fixed domains In this subsection, we prove Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 We start by proving the statement for dH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let T := {t > 0 : f(t−) = f(t)}, which is dense in [0, ∞) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let G := Gf(f), Gn := Gf(fn), Gt := Gf(f �� [0,t]), and Gt n := Gf(fn �� [0,t]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We first prove the implication ⇒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since Gn → G, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, there exists a compact set C such that Gn ⊂ C for all n, and hence also Gt n ⊂ C for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8, it follows that {Gt n : n ∈ N} is compact, so to prove that Gt n → Gt, it suffices to show that Gt is the only subsequential limit of the Gt n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let G∗ be such a subsequential limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since Gt n ⊂ Gn it is clear from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 that G∗ ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let ψ denote the projection ψ(x, s) := s and let ψ(G∗) denote the image of G∗ under ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='3, ψ(G∗) = [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' It follows that G∗ ⊂ Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To 45 prove the opposite inclusion, assume that (y, s) ∈ Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If s < t, then we use that by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6, there exist (xn, sn) ∈ Gn such that (xn, sn) → (x, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since s < t, we have (xn, sn) ∈ Gt n for n large enough and hence (x, s) ∈ G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' If s = t, then we use that ψ(G∗) = [0, t] to conclude that there must be at least one y′ ∈ X such that (y′, t) ∈ G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since G∗ ⊂ Gt we must have y, y′ ∈ ⟨f(t−), f(t)⟩ = {f(t)}, where we have used that t ∈ T, so we conclude that y′ = y, concluding the proof that G∗ = Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We next prove the implication ⇐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since Gt n → Gt for each t ∈ T, using Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='20, we see that there exists a compact set C such that Gn ⊂ C for all n, so by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='8 it suffices to show that if G∗ is a subsequential limit of the Gn, then G∗ = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since Gt n ⊂ Gn for each n it is clear from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 that Gt ⊂ G∗ for each t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We claim that conversely, for each (x, s) ∈ G∗ and s < t ∈ T, we have (x, s) ∈ Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Indeed, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='6 for some subsequence there exist (xn, sn) ∈ Gn such that (xn, sn) → (x, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since sn < t for n large enough, it follows that (x, s) ∈ Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' These arguments show that {(x, t) ∈ G∗ : t < ∞} = {(x, t) ∈ G : t < ∞}, which is enough to conclude G∗ = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We next prove the statement for dS tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let π := f, πn := fn, πt := f �� [0,t], and πt n := fn �� [0,t], which we view as elements of the path space Π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We first prove the implication ⇒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7, dS tot(πn, π) → 0 implies that {πn : n ∈ N} is Skorohod-equicontinuous and satisfies the compact containment condition, which implies the same is true for {πt n : n ∈ N} for any t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Therefore, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7, it suffices to show that all subsequential limits of the πt n are equal to πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Since by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='10, convergence in dS tot implies convergence in dH tot, we can use what we have already proved for dH tot to draw the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The implication ⇐ follows in the same way, where now we use that if {πt n : n ∈ N} is Skorohod-equicontinuous and satisfies the compact containment condition for any t ∈ T, then the same is true for {πn : n ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='9 We view DI(X) as a subset of Π(X) as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then F is compact as a subset of DI(X) if and only if its closure F in the larger space Π(X) is compact and satisfies F ⊂ DI(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='7, F is a compact subset of Π(X) if an only if conditions (i) and (ii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To complete the proof, we will show that, assuming (i) and (ii), one has F ⊂ DI(X) if and only if (iii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' We first show that (iii) implies that F ⊂ DI(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume that (iii) holds and let fn ∈ DI(X) and π ∈ Π(X) satisfy fn → π in the Skorohod topology associated with the given betweenness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then clearly I(π) = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' To show that π ∈ DI(X) assume that conversely π(t−) ̸= π(t+) for some t ∈ ∂I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then, by the fact that dS part(fn, π) → 0, there exist sn, tn ∈ I with sn < tn such that fn(sn) → π(t−) and fn(tn) → π(t+), which is easily seen to contradict (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Assume, on the other hand, that (iii) does not hold for some t ∈ ∂I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Let δn be positive constants, tending to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then there exists an ε > 0 such that for each n, we can find fn ∈ F and sn ∈ I with |sn − t| ≤ δn and d � fn(sn), fn(t) � ≥ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By (i) and (ii), F is compact in Π(X) so by going to a subsequence we can assume that fn → π for some π ∈ Π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' By (i), we can moreover assume that fn(sn) → x and fn(t) → y for some x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Then d(x, y) ≥ ε and (x, t), (y, t) ∈ Gf(π), which by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='14 (v) shows that π(t−) ̸= π(t+) and hence F is not contained in DI(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Acknowledgments We thank Jan Seidler for useful discussions and for his help in understanding [AU29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 46 References [AU29] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Alexandroff and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Urysohn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' M´emoire sur les espaces topologiques compacts, d´edi´e `a Monsieur D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Egoroff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Verhandelingen Amsterdam 14(1) (1929).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' [Bil99] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Billingsley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Convergence of Probability Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' John Wiley & Sons, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' [Bou58] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Bourbaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' ´El´ements de Math´ematique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 1: Les Structures Fondamen- tales de l’Analyse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Livre III: Topologie G´en´erale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Chap.' metadata={'source': 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[SSS17] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Schertzer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Sun and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Swart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' The Brownian web, the Brownian net, and their universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Pages 270–368 in: P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Contucci and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Giardin`a (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=') Advances in Disordered Systems, Random Processes and Some Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Cambridge University Press, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' [Whi02] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Whitt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Stochastic-Process Limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' An introduction to stochastic-process limits and their application to queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' Springer, New York, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} +page_content=' 47' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E5T4oBgHgl3EQfhA8F/content/2301.05637v1.pdf'} diff --git a/etE3T4oBgHgl3EQffAo5/content/tmp_files/2301.04548v1.pdf.txt b/etE3T4oBgHgl3EQffAo5/content/tmp_files/2301.04548v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2e8290beae05a67158e55b63d629d615be96269 --- /dev/null +++ b/etE3T4oBgHgl3EQffAo5/content/tmp_files/2301.04548v1.pdf.txt @@ -0,0 +1,3423 @@ +Orbital design of Berry curvature: pinch points and giant dipoles induced by crystal fields +Maria Teresa Mercaldo,1 Canio Noce,1, 2 Andrea D. Caviglia,3 Mario Cuoco,2, 1 and Carmine Ortix1 +1Dipartimento di Fisica “E. R. Caianiello”, Universit`a di Salerno, IT-84084 Fisciano (SA), Italy +2SPIN-CNR, IT-84084 Fisciano (SA), Italy +3Department of Quantum Matter Physics, University of Geneva, +24 Quai Ernest Ansermet, CH-1211 Geneva, Switzerland +The Berry curvature (BC) – a quantity encoding the geometric properties of the electronic wavefunctions in a +solid – is at the heart of different Hall-like transport phenomena, including the anomalous Hall and the non- +linear Hall and Nernst effects. In non-magnetic quantum materials with acentric crystalline arrangements, local +concentrations of BC are generally linked to single-particle wavefunctions that are a quantum superposition +of electron and hole excitations. BC-mediated effects are consequently observed in two-dimensional systems +with pairs of massive Dirac cones and three-dimensional bulk crystals with quartets of Weyl cones. Here, we +demonstrate that in materials equipped with orbital degrees of freedom local BC concentrations can arise even +in the complete absence of hole excitations. In these solids, the crystals fields appearing in very low-symmetric +structures trigger BCs characterized by hot-spots and singular pinch points. These characteristics naturally yield +giant BC dipoles and large non-linear transport responses in time-reversal symmetric conditions. +I. +INTRODUCTION +Quantum materials can be generally defined as those solid- +state structures hosting physical phenomena which, even at +the macroscopic scale, cannot be captured by a purely classi- +cal description [1]. Among such quantum phenomena, those +related to the geometric properties of the electronic wavefunc- +tions play undoubtedly a primary role. In an N-band crys- +talline system, the cell-periodic part of the electronic Bloch +waves defines a mapping from the Brillouin zone (BZ) to +a complex space naturally equipped with a geometric struc- +ture – its tangent space defines a Fubini-Study metric [2] that +measures the infinitesimal distance between Bloch states at +different points of the BZ. The imaginary part of this quan- +tum geometric tensor [3, 4] corresponds to the well-known +Berry curvature (BC), which, when integrated over the full +BZ, gives the Chern number cataloguing two-dimensional in- +sulators [5]. In metallic systems with partially filled bands, +the BC summed over all occupied states can result in a non- +vanishing Berry phase if the system breaks time-reversal sym- +metry. This Berry phase regulates the intrinsic part of the +anomalous Hall conductivity of magnetic metals [6–9]. +Materials with an acentric crystal structure can possess non- +vanishing concentrations of BC even if magnetic order is ab- +sent. Probing the BC of these non-centrosymmetric and non- +magnetic materials via charge transport measurements usually +requires externally applied magnetic fields. For instance, in +time-reversal invariant Weyl semimetals, such as TaAs [10], +the strong BC arising from the Weyl nodes can be revealed +using the planar Hall effect [11] – a physical consequence of +the negative longitudinal magnetoresistance associated with +the chiral anomaly of Weyl fermions [12]. Recently, it has +been also shown that the planar Hall effect can display an +anomalous antisymmetric response [13, 14], which, at least +in two-dimensional materials, is entirely due to an unbalance +in the BC distribution triggered by the Zeeman-induced spin +splitting of the electronic bands. +In the absence of external magnetic fields, a BC charge +transport diagnostic for non-magnetic materials requires to go +beyond the linear response regime [15–19]. Hall-like currents +appearing as a non-linear (quadratic) response to a driving +electric field can have an intrinsic contribution governed by +the Berry curvature dipole (BCD), which is essentially the first +moment of the Berry curvature in momentum space. In three- +dimensional systems, non-vanishing BCDs have been linked +to the presence of tilted Weyl cones, and have been shown to +exist both in type-I and in type-II Weyl semimetals [20] such +as MoTe2 [21] and the ternary compound TaIrTe4 [22, 23]. +Furthermore, the Rashba semicondutor BiTeI has been pre- +dicted to host a BCD that is strongly enhanced across its +pressure-induced topological phase transitions [24]. +In two-dimensional materials, the appearance of BCDs is +subject to stringent symmetry constraints: the largest sym- +metry group is Cs, which is composed by the identity and a +single vertical mirror line [25]. The concomitant presence +of spin-orbit coupled massive Dirac cones with substantial +BC and such unusually low-symmetry crystalline environ- +ments have suggested the surface states of SnTe [26] in the +low-temperature ferroelectric phase [27], monolayer transi- +tion metal dichalcogenides in the so-called 1Td phase [28– +30], and bilayer WTe2 as material structures hosting sizable +BCDs [31, 32]. Spin-orbit free two-dimensional materials, in- +cluding monolayer and bilayer graphene, have been also put +forward as materials with relatively large BCDs [33]. In these +systems, it is the interplay between the trigonal warping of the +Fermi surface and the presence of massive Dirac cones due to +inversion symmetry breaking that triggers dipolar concentra- +tion of Berry curvatures [34]. +Finite concentrations of BC and BCDs are symmetry al- +lowed also in systems that do not feature quartets of Weyl +cones and pairs of massive Dirac cones. +The anomalous +massless Dirac cones at the surface of three-dimensional +strong topological insulators [35] as well as conventional +two-dimensional electron gases (2DEG) with Rashba spin- +orbit coupling [36] are generally characterized by finite lo- +cal BC concentrations when subject to trigonal crystal fields. +The existence of BC in 2DEGs, which has been experimen- +tally probed through “anomalous” planar Hall effect measure- +ments [36], provides a new avenue for investigations. It shows +arXiv:2301.04548v1 [cond-mat.mes-hall] 11 Jan 2023 + +2 +in fact that Berry curvature-mediated effects can be generated +entirely from conduction electrons. This overcomes the re- +quirement of materials with narrow gaps in which the elec- +tronic wavefunctions at the Fermi level are a quantum su- +perposition of electron and hole excitations, and extends the +palette of non-magnetic materials displaying BC effects to, +for instance, doped semiconductors with gaps in the eV range. +It also proves that it is possible to trigger BC effects in con- +ventional electron liquids with competing instabilities towards +other many-body quantum phases. +In a spin-orbit coupled 2DEG, the BC is however triggered +by crystalline anisotropy terms, which are cubic in momen- +tum and linked to the out-of-plane component of the spin tex- +tures [37, 38]. Consequently, the BC does not possess the +characteristic “hot-spots” appearing in close proximity to near +degeneracy between two bands where the Bloch wavefunc- +tions are rapidly changing. The absence of such BC hot-spots +forbids, in turn, large enhancements of the BCD, which is a +central quest for material design. This motivates the funda- +mental question on whether and how an electron system can +develop strong local BC concentrations in time-reversal sym- +metric conditions even in the complete asbence of hole exci- +tations. Here, we provide a positive answer to this question +by showing that spin-orbit free metallic systems with an ef- +fective pseudo-spin one orbital degree of freedom can display +BC hot-spots and characteristic BC singular pinch points that +yield dipoles order of magnitudes larger than those triggered +by spin-orbit coupling in a 2DEG. +II. +RESULTS +A. +Model Hamiltonian from symmetry principles +Let us first consider a generic single-valley two-level sys- +tem in two dimensions with spin degree of freedom only. The +corresponding energy spectrum is assumed to accurately rep- +resent the electronic bands close to the Fermi level of the +metal in question. As long as we consider materials with- +out long-range magnetic order, the two Fermi surfaces must +originate from one of the four time-reversal invariant point +of the Brillouin zone (BZ) (n1b1 +n2b2)/2 with b1,2 the two +primitive reciprocal lattice vectors of the BZ and n1,2 = 0,1 +[39]. Time-reversal symmetry guarantees that the two bands +will be Kramers’ degenerate at the time-reversal invariant mo- +menta (TRIM). The effective Hamiltonian in the vicinity of +the TRIM can be captured using a conventional k · p theory +that keeps track of the point group symmetries of the crystal. +To make things concrete, let us assume that the low-energy +conduction bands are centered around the Γ point of the BZ +and we are dealing with an acentric crystal with C3v point +group symmetry. This is the largest acentric symmetry group +without C2T symmetry, C2 indicating a twofold rotation sym- +metry with out-of-plane axis and T time-reversal, and thus al- +lows for local BC concentrations [13]. The generators of C3v +are the threefold rotation symmetry C3 and a vertical mirror +symmetry, which, without loss of generality, we take as Mx +sending x → −x. The threefold rotation symmetry can be rep- +resented as e−iπσz/3 while the mirror symmetry as iσx [40]. +Momentum and spin transform under C3 and Mx as follows +C3 : k± → e±2πi/3k±; σ± → e±2πi/3σ± +σz → σz +Mx : k± → −k±; σy,z → −σy,z +σx → σx +(1) +where k± = kx ± iky and σ± = σx ± iσy. Furthermore, the +Hamiltonian must satisfy the time-reversal symmetry con- +straint H(k) = TH(−k)T−1, with the time-reversal operator +that, as usual, can be represented as T = iσyK and K the com- +plex conjugation. When expanded up to linear order in k, the +form of the Hamiltonian reads as H(k) = αR (kxσy −kyσx). +The Dirac cone energy spectrum predicted by this Hamilto- +nian violates the fermion doubling theorem [41] and hence +can occur only on the isolated surfaces of three-dimensional +strong topological insulators [42]. And indeed H(k) coin- +cides with the effective Hamiltonian for the surface states of +the topological insulators in the Bi2Se3 material class [40, 43, +44]. In a genuine two-dimensional system such anomalous +states cannot be present, and an even number of Kramers’ re- +lated pair of bands must exist at each Fermi energy. Conse- +quently, the effective Hamiltonian must be equipped with an +additional term that is quadratic in momentum and such that +it doubles the number of states at each energy. Time-reversal +symmetry implies that terms quadratic in momentum are cou- +pled to the identity matrix. Therefore, we arrive at the well- +known Hamiltonian of a two-dimensional electron gas with +Rashba-like spin-orbit coupling that reads +H(k) = ¯h2k2 +2m σ0 +αR (kxσy −kyσx). +(2) +The corresponding energy spectrum consisting of two shifted +parabolas is schematically shown in Fig. 1(a). Although the +crystalline symmetry requirements are fulfilled, the Hamilto- +nian above does not predict any finite BC local concentration. +This is because the d vector associated to the Hamiltonian +d = +� +−αRky,αRkx,0 +� +is confined to a two-dimensional plane +at all momenta. +There are two different ways to lift the d vector out-of-plane +and thus trigger a non-vanishing BC [45]. The first one con- +sists in introducing a constant mass ∆σz. This term removes +the Kramers’ degeneracy at the TRIM [see Fig. 1(b)] and +therefore breaks time-reversal invariance. It can be realized +by externally applying an out-of-plane magnetic field or by +inducing long-range magnetic order. The BC then generally +displays an hot-spot located at the TRIM and a circular sym- +metric distribution [see Fig. 1(c)]. Moreover, time-reversal +symmetry breaking implies that the Berry phase accumulated +by electrons on the Fermi surface is non-vanishing [6]. The +second route explicitly takes into account trigonal warping +terms which are cubic in momentum and couple to the Pauli +matrix σz. Such terms preserve time-reversal invariance, and +thus create a BC distribution with an angular dependence such +that the Berry phase accumulated over any symmetry-allowed +Fermi line cancels out [35]. Perhaps more importantly, the BC +triggered by crystalline anisotropy terms [36] does not display +a hot-spot, thus suggesting that in systems with conventional + +3 +FIG. 1. (a) Schematic band structure of a two-dimensional electron gas with Rashba spin-orbit coupling. (b) An out-of-plane magnetic field +breaks the Kramers’ degeneracy at k = 0 and triggers a finite BC. (c) The local BC has a circular profile with an hot spot at the Γ point of +the BZ. (d) Schematic band structure of a two-dimensional electron system characterized by an L = 1 orbital multiplet in a trigonal crystalline +environment. (e) An additional crystalline symmetry lowering splits completely the energy levels at the Γ point of the BZ even if time-reversal +symmetry is preserved. The presence of mirror symmetry protects crossing at finite momenta. (f) A characteristic time-reversal symmetric +BC profile with the presence of hot-spots and singular pinch points. The BC has been obtained using the model Hamiltonian Eq. (5) with +∆ = −0.2E0, ∆m = 0.12E0, αR = 1.0E0/k0 +F and αm = 0.5E0/k0 +F with E0 = ¯h2(k0 +F)2/(2m) and k0 +F a characteristic Fermi wavevector. +quasiparticles and a single internal degree of freedom time- +reversal symmetry breaking is a prerequisite for large local +BC enhancements. +We now refute this assertion by showing that in systems +with orbital degrees of freedom the formation of BC hot- +spots is entirely allowed even in time-reversal symmetric con- +ditions. +Consider for instance a system of p orbitals. +In +a generic centrosymmetric crystal, interorbital hybridization +away from the TRIM can only occur with terms that are +quadratic in momentum. However, and this is key, in an acen- +tric crystal interorbital mixing terms linear in momentum are +symmetry allowed. These mixing terms, often referred to as +orbital Rashba coupling [46–48], are able to induce BC hot +spots with time-reversal symmetry, as we now show. We as- +sume as before an acentric crystal with C3v point group, and +electrons that are effectively spinless due to SU(2) spin sym- +metry conservation: we are thus removing spin-orbit coupling +all together. In the pz, py, px orbital basis, the generators of +the point group are represented by +Mx = +� +� +1 0 +0 +0 1 +0 +0 0 −1 +� +�; +C3 = +� +� +1 +0 +0 +0 +cos 2π +3 +sin 2π +3 +0 −sin 2π +3 +cos 2π +3 +� +�, +The two px,y orbitals form a two-dimensional irreducible rep- +resentation (IRREP) whereas the pz orbital represents a one- +dimensional IRREP. The form of the effective Hamiltonian +away from the TRIM can be captured using symmetry con- +straints. Specifically, any generic 3 × 3 Hamiltonian can be +expanded in terms of the nine Gell-Mann matrices [49] Λi [see +Methods] as +H(k) = +8 +∑ +i=0 +bi(k)Λi. +(3) +The invariance of the Hamiltonian requires that the compo- +nents of the Hamiltonian vector b(k) should have the same +behavior as the corresponding Gell-Mann matrices Λi. This +means that they should belong to the same representation of +the crystal point group [50]. From the representation of the +Λi’s [see Methods and Table 1] and those of the polynomials +of k [see Table 1], we find that the effective Hamiltonian up +to linear order in momentum reads as +H(k) = ∆ +� +Λ3 + 1 +√ +3Λ8 +� +−αR [kxΛ5 +kyΛ2]. +(4) +Here the parameter ∆ quantifies the energetic splitting be- +tween the px,y doublet and the pz singlet. The second term in +the Hamiltonian corresponds instead to the pseudo-spin one + +Energy Spectra +Berry Curvature +b +2+(k)/2max +a +C +1.0 +0.8 +Spin +0.6 ++ Magnetic Field +0.4 +0.2 +22(k)/2max +Orbital +1.0 +0.5 +X +0 ++ Crystal Field +-0.5 +-1.04 +massless Dirac Hamiltonian [51, 52] predicted to occur for +instance in the kagome lattice with a staggered magnetic π +flux [51]. Pseudo-spin one Dirac fermions are not subject to +any fermion multiplication theorem [53]. Therefore, a dou- +bling of the number of states at each energy is not strictly +required. However, since we are interested in systems without +the concomitant presence of electrons and holes, we will in- +troduce a term ¯h2k2Λ0/(2m) with an equal effective mass for +all three bands. The ensuing Hamiltonian can be then seen +as a generalization of the Rashba 2DEG to an SU(3) sys- +tem with the effect of the trigonal crystal field that leads to +a partial splitting of the energy levels at the TRIM, entirely +allowed by the absence of Kramers’ theorem. Despite the +spectral properties [c.f. Fig. 1(d)] have a strong resemblance +to those obtained in a time-reversal broken 2DEG, a direct +computation [see Methods] shows that the BC associated to +the Hamiltonian above is vanishing for all momenta. Break- +ing time-reversal symmetry introducing a constant mass term +∝ Λ7 or considering crystalline anisotropy terms that are cubic +in momentum represent two possible routes to trigger a finite +Berry curvature. The crux of the story is that in the present +SU(3) system at hand, another possibility exists. It only relies +on the crystal field effects that are generated by lowering the +crystalline point group to Cs. From the representations of the +Gell-Mann matrices and the polynomials of k in this group, +we find that the effective Hamiltonian reads +H(k) = ¯h2k2 +2m Λ0 +∆ +� +Λ3 + 1 +√ +3Λ8 +� ++∆m +� +1 +2Λ3 − +√ +3 +2 Λ8 +� +−αR [kxΛ5 +kyΛ2]−αmkxΛ7. +(5) +Nothing prevents to have the interorbital mixing terms ∝ Λ2,5 +with different amplitudes. Without loss of generality, in the +remainder we will consider a single parameter αR. +In the +Hamiltonian above, we have also neglected a constant term +∝ Λ1. For materials with an high-temperature trigonal struc- +ture, its amplitude ∆1 is expected to be of the same order of +magnitude as ∆m. In this regime [see the Supplemental Mate- +rial], a term ∝ Λ1 has a very weak effect on the energy spec- +trum and BC properties, and can be thus disregarded [54]. The +energy spectrum reported in Fig. 1(e) shows that the effect of +the crystal symmetry lowering is twofold. First, there is an +additional energy splitting between the px,y implying that all +levels at the Γ point of the BZ are singly degenerate. Second, +the two px,y orbitals have band degeneracies along the mir- +ror symmetric kx = 0 line of the BZ. Such mirror-symmetry +protected crossings give rise to BC singular pinch points [see +Fig. 1(f) and the Supplemental Material]. It is the presence +of these pinch points that represents the hallmark of the non- +trivial geometry of the electronic wavefunctions associated to +the p-orbital manifold. Note that the BC also displays hot- +spots [see Fig. 1(f)] with BC sources and sinks averaging to +zero on any mirror symmetric Fermi surface as mandated by +time-reversal invariance. +C3v E 2 C3 2σv polynomials of k +Gell-Mann matrices +A1 1 +1 +1 +1, k2x +k2y +Λ3 +Λ8/ +√ +3,Λ0 +A2 1 +1 +−1 +– +Λ7 +E +2 +−1 +0 +� +kx,ky +� +{Λ1,Λ4}, {Λ2,Λ5} +� +Λ6,Λ3/2− +√ +3Λ8/2 +� +Cs E 2σv polynomials of k Gell-Mann matrices +A′ 1 +1 +1, ky, k2x, k2y +Λ1, Λ2, Λ3, Λ8 +A′′ 1 −1 +kx +Λ4, Λ5, Λ6, Λ7 +TABLE I. Character table for the point groups C3v and Cs. We also +indicate the representation of the Gell-Mann matrices and the poly- +nomials of momentum k. The model Hamiltonians reported in the +main text can be obtained by additionally using the time-reversal +symmetry constraint H⋆(−kx,−ky) = H(kx,ky). +B. +Material realizations +Before analyzing the origin and physical consequence of +the BC and its characteristic pinch points, we now intro- +duce a material platform naturally equipped with orbital de- +grees of freedom and the required low crystalline symme- +try: [111] interfaces of transition metal oxides hosting two- +dimensional d electron systems of t2g orbital character such as +SrTiO3 [55, 56], KTaO3 [57], and SrVO3-based heterostruc- +tures. When compared to conventional semiconductor het- +erostructures, complex oxide interfaces consist of d elec- +trons with different symmetries, a key element in determin- +ing their many-body ground states that include, notably, un- +conventional superconductivity [58]. In the high-temperature +cubic phases of these materials, the octahedral crystal field +pins the low-energy physics to a degenerate t2g manifold, +which spans an effective angular momentum one subspace, +precisely as the p orbitals discussed above [59]. +The re- +duced symmetry at interfaces lift their energetic degeneracy +and modify their orbital character. At the [111] interface the +transition metal atoms form a stacked triangular lattice with +three interlaced layers [see Fig. 2(a),(b)]. This results in a +triangular planar crystal field that hybridizes the |xy⟩, |xz⟩ +and |yz⟩ orbitals to form an |a1g⟩ = (|xy⟩+|xz⟩+|yz⟩)/ +√ +3 +one-dimensional IRREP whereas the two states |e′ +g±⟩ = +� +|xy⟩+ω±1 |xz⟩+ω±2 |yz⟩ +� +/ +√ +3, with ω = e2πi/3, form the +two-dimensional IRREP . +The energetic ordering of the levels depends on the +microscopic details of the interface. +For example, at +the (111)LaAlO3/SrTiO3 interface, x-ray absorption spec- +troscopy [60] sets the |a1g⟩ state at lower energy [see +Fig. 2(e)]. +By further considering the structural inversion +symmetry inherently present at the heterointerface, we thus +formally reach the situation we discussed for the set of p +orbitals, be it for the trigonal symmetry that excludes any +local concentrations of BC. However, and this is key, low- +temperature phase transitions in oxides lower the crystal sym- +metry, often realising a tetragonal or orthorhombic phase with +oxygen octahedra rotations and (anti)polar cation displace- +ments. Let’s consider the paradigmatic case of SrTiO3. A + +5 +FIG. 2. (a) Schematic representation of an ABO3 pervoskite cubic +unit cell displaying the three interlaced transition metal [111] planes. +(b) Corresponding top view along the [111] crystallographic direc- +tion. We only show the B transition metal atoms. (c) and (d) show +the effect of a tetragonal distortion with the [001] direction being the +tetragonal axis. The distortion breaks the threefold rotation symme- +try around the [111] axis but leaves a residual mirror symmetry. (e) +Evolution of the orbital states at the Γ point of the BZ with quenched +angular momentum. +structural transition occurring at around 105 K, from the cubic +phase to a tetragonal structure [61] [see Fig. 2(c)], breaks the +threefold rotational symmetry leaving a single residual mir- +ror line. Assuming the tetragonal axis to be along the [001] +direction, the surviving mirror symmetry at the [111] inter- +face corresponds to M[¯110] [see Fig. 2(d)]. +This structural +distortion lifts the degeneracy of the e′ +g doublet. The bond- +ing and antibonding states |e′ +g+⟩±|e′ +g−⟩ have opposite mirror +M[¯110] eigenvalues and realize two distinct one-dimensional +IRREP [see Fig. 2(e)]. +SrTiO3-based heterointerfaces un- +dergo additional tetragonal to locally triclinic structural dis- +tortions at temperatures below ≃ 70 K which involves small +displacements of the Sr atoms along the [111] directions con- +voluted with TiO6 oxygen-octahedron antiferrodistortive rota- +tions [62]. In addition, below about 50 K, SrTiO3 and KTaO3 +approach a ferroelectric instability that is accompanied by +strong polar quantum fluctuations. This regime is character- +ized by a soft transverse phonon mode that involves off-center +displacement of the Ti ions with respect to the surrounding oc- +tahedron of oxygen ions [63], which, in the static limit, would +correspond to a ferroelectric order parameter. This can poten- +tially enhance the interorbital hybridization terms allowed in +acentric crystalline environments, and thus boost the appear- +ance of large BC concentrations. +C. +Berry curvature dipole +Having identified (111)-oriented oxide heterointerfaces as +ideal material platforms, we next analyze the specific prop- +erties of the BC and its first moment. We first notice that +in the case of a two-level spin system the local Berry cur- +vature of the spin-split bands, if non-vanishing, is opposite. +Due to the concomitant presence of both spin-bands at each +Fermi energy, the spin split bands cancel their respective lo- +cal BC except for those momenta which are occupied by one +spin band. In the SU(3) system at hand, there is a similar +sum rule stating that at each momentum k the BC of the three +bands [c.f. Fig. 3(a)] sum to zero. However, and as mentioned +above, the orbital bands are not subject to fermion multiplica- +tion theorems. In certain energy ranges a single orbital band +is occupied [c.f. Fig. 3(a,b)] and BC cancellations are not +at work. There is also another essential difference between +the BC associated to spin and orbital degrees of freedom. In +general, the commutation and anticommutation relations of +the SU(N) Lie algebra define symmetric and antisymmetric +structure constants, which, in turn, define the star and cross +products of generic SU(N) vectors [64]. Differently from an +SU(3) system spanning an angular momentum one subspace, +in SU(2) spin systems the symmetric structure constant van- +ishes identically. The ensuing absence of star products bk ⋆bk +precludes the appearance of BC with time-reversal symmetry +as long as crystalline anisotropies are not taken into account +[see Methods]. On the other hand, for SU(3) the presence +of all three purely imaginary Gell-Mann matrices Λ2,5,7, to- +gether with the “mass” terms Λ3,8, is a sufficient condition to +obtain time-reversal symmetric BC concentrations even when +accounting only for terms that are linear in momentum [see +Methods]. This, however, strictly requires that all rotation +symmetries must be broken. +Next, we analyze the properties of the band resolved local +BC starting from the lowest energy band, which corresponds +to the (|xy⟩+|xz⟩+|yz⟩)/ +√ +3 state at (111) LAO/STO het- +erointerfaces. Fig. 3(c) shows a characteristic BC profile. It +displays two opposite poles centered on the ky = 0 line. These +sources and sinks of BC are equidistant from the mirror sym- +metric kx = 0 line since the BC, as any genuine pseudoscalar, +must be odd under vertical mirror symmetry operations, i.e. +Ω(kx,ky) = −Ω(−kx,ky). Note that the combination of time- +reversal symmetry and vertical mirror implies that the BC +will be even sending ky → −ky, thus guaranteeing that, taken +by themselves, the BC hot-spots will be centered around the +ky = 0 line. Their finite kx values coincide with the points +where the (direct) energy gap between the n = 1 and the n = 2 +bands is minimized [see Fig. 3(a,b) and Supplemental Mate- +rial], and thus the interorbital mixing is maximal. The prop- +erties of the BC are obviously reflected in the BCD local den- +sity ∂kxΩ(kx,ky): it possesses [see Fig. 1(f)] a positive area +strongly localized at the center of the BZ that is neutralized + +[0,0,1] +a +C +[0,1,0] +[1,0,0] +M[1,1,0] +[1,1,1] +b +d +e' +t2 g +a1 g +(111) Interface +Bulk +(111) Interface ++ tetragonal distorsion +Cs symmetry +Cs symmetry +e6 +FIG. 3. (a) Ordering of the crystal field split t2g (p) orbitals with their associated band index. (b) Energy spectrum of the model Eq. (5) obtained +using the parameter set ∆ = −0.2E0, ∆m = 0.01E0, αR = αm = 1.0E0/k0 +F. (c),(d),(e) show the ensuing band-resolved Berry curvature. (f), +(g),(h) are the corresponding BC dipole densities ∂kxΩ. Note that the presence of mirror symmetry guarantees that the orthogonal dipole +density ∂kyΩ averages to zero. +by two mirror symmetric negative regions present at finite kx. +Let us next consider the Berry curvature profile arising from +the two degenerate e′ +g states that are split by the threefold rota- +tion symmetry breaking. Fig. 3(d) shows the BC profile of the +lowest energy band: it is entirely dominated by the BC pinch +points induced by the mirror symmetry protected degeneracies +on the kx = 0 line. The BC also displays a nodal ring around +the pinch point, and thus possesses a characteristic d-wave +character around the singular point. This can be understood +by constructing a k·p theory around each of the two time- +reversal related degeneracies. To do so, we first recall that the +two bands deriving from the e′ +g states have opposite Mx mirror +eigenvalue along the full mirror line kx ≡ 0 of the BZ. Close +to the degeneracies, Mx can be therefore represented as σz. +Under Mx, kx → −kx whereas ky → ky. Moreover, the Pauli +matrices σx,y → −σx,y. An effective two-band model close +to the degeneracies must then have the following form at the +leading order: +Hef f = vxkx σx +βkx δky σy +vyδky σz, +(6) +where δky is the momentum measured relatively to the mir- +ror symmetry-protected degeneracy and we have neglected +the quadratic term coupling to the identity k2σ0 that does not +affect the BC. Using the usual formulation of the BC for a +two-band model [see the Methods section], it is possible to +show that the Hamiltonian above is characterized by a zero- +momentum pinch-point with two nodal lines [see the Supple- +mental Material] and d-wave character. It is interesting to note +that this also implies that the “effective” time-reversal sym- +metry inverting the sign of k around the pinch point is bro- +ken [65]. Perhaps even more importantly, the d-wave charac- +ter implies a very large BCD density in the immediate neigh- +borhood of the pinch point [see Fig. 3(g)]. Similar properties +are encountered when considering the highest energy band , +with the difference that the pinch-point has an opposite an- +gular dependence [see Fig. 3(e)] and consequently the BCD +density has opposite sign [c.f. Fig. 3(h)]. +Having the band-resolved BC and BCD density profiles +in our hands, we finally discuss their characteristic finger- +prints in the BCD defined by Dx = +� +k ∂kxΩ(k) f0, with +� +k = +� d2k/(2π)2 and f0 being the equilibrium Fermi-Dirac distri- +bution function. By continuously sweeping the Fermi energy, +we find that the BCD shows cusps and inflection points [see +Fig. 4(a)] , which, as we now discuss, are a direct consequence +of Lifshitz transitions and their associated van Hove singular- +ities [see Supplemental Material]. Starting from the bottom +of the first band, the magnitude of the BCD continuously in- +creases until it reaches a maximum where the dipole is larger +than the inverse of the Fermi momentum of a 2DEG 1/k0 +F +and thus gets an enhancement of three order of magnitudes +with respect to a Rashba 2DEG [36]. In this region, there +are two distinct Fermi lines encircling electronic pockets at fi- +nite values of k [c.f. Fig. 4(b)], which subsequently merge on +two disconnected regions in momentum space [c.f. Fig. 4(c)]. +Since the states in the immediate vicinity of the center of the +BZ are not occupied, the BCD is entirely dominated by the +two mirror symmetric negative hot-spots of Fig. 3(f). By fur- +ther increasing the chemical potential, the internal Fermi line +collapses at the Γ point and therefore a first Lifshitz transition +occurs [c.f. Fig. 4(d)]. In this regime, the BCD has exponen- + +23 +21 +■>100 +>100 +0.2 +0.2 +7.5 +n=3 +T100 +T100 +4E +0.5 +n=2 +5.0 +0.1 +0.1 +50 + 50 +a +2.5 +0.0 +0.0 +0.0 +n=l1 +ky +ky +0 +K +0 +0 +T +-0.1 +-0.1 +-2.5 +-50 +-50 +-0.5 +-0.2 +-0.2 +-5.0 +100 +100 +-0.5 +0.0 +0.5 +0.2 +-0.2-0.1 +-0.1 +0.0 +0.1 +0.2 +0.0 +¥0.1 +-0.2 +-7.5 +■<-100 +■<-100 +d +c +e +kx/k? +kx/ kg +k/kg +/0.5 +1 E/80 +0k22 +0kxQ23 +0kx21 +■>5·103 +■>5·103 +0.4 +0.4 +4·103 + 4.103 +10.0 +0.5 +2·103 + 2.103 +75 +0.2 +0.2 +200 +_ 200 +50 +0.0 +0.0 +0.0 +k +-0.5 +0 +kylk C +25 +-0.2 +-0.2 +-200 +-200 +-0.5 +kxlk +-0.4 +-0.4 +-2·103 +-2·103 +0 +b +-4·103 +0.0 +0.5 +-0.5 +-0.4 -0.2 +-0.4 -0.2 +0.0 +0.2 +0.4 +0.0 +0.2 +0.4 +<-5·103 +■<-5·103 +k/ kg +-25 +h +f +kx/k? +6 +kx/ k?7 +tially small values due to the fact that the strong positive BCD +density area around the center of the BZ counteracts the mir- +ror symmetric negative hot-spots. By further increasing the +chemical potential, a second Lifshitz transition signals the oc- +cupation of the first eg band with two pockets centered around +the ky = 0 line [see Fig. 4(e)]. This Lifshitz transition coin- +cides with a rapid increase of the BCD due to the contribution +coming from the local BCD density regions external to the +BC nodal ring of Fig. 3(g). The subsequent sharp negative +peak originates from a third Lifshitz transition in which the +two electronic pockets of the second band merge, and almost +concomitantly a tiny pocket of the third band centered around +Γ arises [see Fig. 4(f)]. By computing the band resolved BCD +[see the Supplemental Material] one finds that it is this small +pocket the cause of the negative sharp peak. For large enough +chemical potentials, the BCD develops an additional peak cor- +responding to the fermiology of Fig. 4(g). This peak, which is +again larger than 1/k0 +F, can be understood by noticing that due +to the BC local sum rule the momenta close to the center of the +BZ do not contribute to the BCD. On the other hand, the re- +gions external to the BC nodal ring are unoccupied by the third +band and consequently have a net positive BCD local density. +Thermal smearing can affect the strongly localized peaks at +lower chemical potential but will not alter the presence of this +broader peak. Note that the BCD gets amplified by increas- +ing the interorbital mixing parameter αR but retains similar +properties [see Fig. 4 and the Supplemental Material]. The +strength of BC-mediated effects depends indeed on the ratio +between the characteristic orbital Rashba energy 2mα2 +R(m)/¯h2 +and the crystal field splittings ∆(m). The BCD properties and +values comparable to the Fermi wavelength are hence com- +pletely generic. +Let us finally discuss the role of spin-orbit coupling. It +can be included in our model Hamiltonian Eq. 5 as Hso = +λso (Lx ⊗τx +Ly ⊗τy +Lz ⊗τz), where λso is the spin-orbit +coupling strength, the L = 1 angular momentum matrices cor- +respond to the Gell-Mann matrices Λ2,Λ5,Λ7, and the Pauli +matrices τx,y,z act in spin space. Its effect can be analyzed +using conventional (degenerate) perturbation theory. At the +center of the Brillouin zone, Hso is completely inactive – the +eigenstates of the Hamiltonian Eq. 5 are orbital eigenstates +and the off-diagonal terms in orbital space Λ2,5,7 cannot give +any correction at first order in λso. The situation is different +at finite values of momentum. The two spin-orbit free degen- +erate eigenstates are a superposition of the different orbitals +(due to the orbital Rashba coupling). Therefore, the spin-orbit +coupling term will lift their degeneracy resulting in a Rashba- +like splitting of the bands. +In order to explore the consequence of this spin splitting on +the Berry curvature, let us denote with |ψ↑ +0(k)⟩ and |ψ↓ +0(k)⟩ +the two spin-orbit free degenerate eigenstates at each value of +the momentum. Note that |ψ0⟩ is a three-component spinor +for the orbital degrees of freedom. When accounting pertur- +batively for spin-orbit coupling the eigenstates will be a super- +position of the spin degenerate eigenstates and will generally +read +|ψ+(k)⟩ = cosθ(k)eiφ(k) |ψ↑ +0(k)⟩+sinθ(k)|ψ↓ +0(k)⟩ +|ψ−(k)⟩ = −sinθ(k)eiφ(k) |ψ↑ +0(k)⟩+cosθ(k)|ψ↓ +0(k)⟩ +Here, the momentum dependence of the phase φ and the angle +θ is a “by-product” of the orbital Rashba coupling: the effect +of spin-orbit coupling, which is off-diagonal in orbital space, +is modulated by the momentum-dependent orbital content of +the eigenstates |ψ↑,↓ +0 (k)⟩. The abelian Berry connection of +the two spin-split states A+,− +kx,ky = ⟨ψ+,−(k)|i∂kx,kyψ+,−(k)⟩ +will therefore contain two terms: the first one is the spin- +independent Berry connection A0 +kx,ky = ⟨ψ0(k)|i∂kx,kyψ0(k)⟩; +the second term is instead related to the derivatives of the +phase φ and angle θ. This Berry connection is opposite for the ++,− states and coincides with the Berry connection of a two- +level spin system [45]. This also implies that the Berry curva- +ture of a Kramers’ pair of bands Ω+,−(k) = Ω(k) ± Ωso(k). +The contribution of the Berry curvature Ωso is opposite for +the time-reversed partners and the net effect only comes from +the difference between the Fermi lines of two partner bands. +However, the purely orbital Berry curvature Ω(k), which can +be calculated directly from Eq. 5, sums up. The values of the +BCD presented in Fig. 4 are thus simply doubled in the pres- +ence of a weak but finite spin-orbit coupling. +III. +CONCLUSIONS +In this study, we have shown an intrinsic pathway to de- +sign large concentrations of Berry curvature in time-reversal +symmetric conditions making use only of the orbital angular +momentum electrons acquire when bound to atomic nuclei. +Such mechanism is different in nature with respect to that ex- +ploited in topological semimetals and narrow-gap semicon- +ductors where the geometric properties of the electronic wave- +functions originate from the coupling between electron and +hole excitations. The orbital design of Berry curvature is also +inherently different from the time-reversal symmetric spin- +orbit mechanism [35, 36] , which strongly relies on crystalline +anisotropy terms. We have shown in fact that the Berry curva- +ture triggered by orbital degrees of freedom features both hot- +spots and singular pinch-points. Furthermore, due to the crys- +talline symmetry constraints the Berry curvature is naturally +equipped with a non-vanishing Berry curvature dipole. These +characteristics yield a boost of three orders of magnitude in +the quantum non-linear Hall effect. In (111) LaAlO3-SrTiO3 +heterointerfaces where the characteristic Fermi wavevector +k0 +F ≃ 1 nm−1, the Berry curvature dipole Dx ≃ 1nm. +The +corresponding non-linear Hall voltage can be evaluated us- +ing the relation [31, 33] Vyxx = e3 τ Dx |Ix|2/(2¯h2σ2 +xxW), with +the characteristic relaxation time τ ≃ 1 pS and the longitudi- +nal conductance σxx ≃ 5 mS. In a typical Hall bar of width +W ≃ 10µ m sourced with a current Ix ≃ 100 µA, the non lin- +ear Hall voltage Vyxx ≃ 2 µV, which is compatible with the +strong non-linear Hall signal experimentally detected [36]. +The findings of our study carry a dramatic impact on the +developing area of condensed matter physics dubbed orbitron- + +8 +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +◆ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +○ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +□ +-2 +-1 +0 +1 +2 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +μ/ℰ0 +DxkF +0 +αR +● 1.0 +■ 1.5 +◆ 2.0 +▲ 2.5 +▼ 3.0 +○ 3.5 +□ 4.0 +a +-2 -1 +0 +1 +2 +-2 +-1 +0 +1 +2 +kx/kF +0 +ky/kF +0 +-2 -1 +0 +1 +2 +-2 +-1 +0 +1 +2 +kx/kF +0 +ky/kF +0 +-2 -1 +0 +1 +2 +-2 +-1 +0 +1 +2 +kx/kF +0 +ky/kF +0 +-2 -1 +0 +1 +2 +-2 +-1 +0 +1 +2 +kx/kF +0 +ky/kF +0 +-2 -1 +0 +1 +2 +-2 +-1 +0 +1 +2 +kx/kF +0 +ky/kF +0 +-2 -1 +0 +1 +2 +-2 +-1 +0 +1 +2 +kx/kF +0 +ky/kF +0 +b +c +d +e +f +g +μ/ℰ0=-0.70 +μ/ℰ0=-0.60 +μ/ℰ0=-0.20 +μ/ℰ0=0.05 +μ/ℰ0=0.25 +μ/ℰ0=0.60 +FIG. 4. (a) Behavior of the Berry curvature dipole obtained by changing the chemical potential µ measured in units of E0. The different curves +correspond to the different values of αR measured in units of E0/k0 +F. The other model parameters have been instead fixed as ∆ = −0.2E0, +∆m = −0.01E0, αm = αR. (b),(c),(d),(e),(f) display the Fermi lines and the band-resolved occupied regions in momentum space for αR = αm = +1.5E0/k0 +F. +ics [66]. Electrons in solids can carry information by exploit- +ing either their intrinsic spin or their orbital angular momen- +tum. Generation, detection and manipulation of information +using the electron spin is at the basis of spintronics. The Berry +curvature distribution we have unveiled in our study is ex- +pected to trigger also an orbital Hall effect, whose origin is +rooted in the geometric properties of the electronic wavefunc- +tions, and can be manipulated using the orbital degrees of free- +dom. This opens a number of possibilities for orbitronic de- +vices. This is even more relevant considering that our findings +can be applied to a wide class of materials whose electronic +properties can be described with an effective L = 1 orbital +multiplet. These include other complex oxide heterointerfaces +as well as spin-orbit free semiconductors where p-orbitals can +be exploited. Since Dirac quasiparticles are not required in +the orbital design of Berry curvature, it is possible to reach +carrier densities large enough to potentially exploit electron- +electron and electron-phonon interactions effects in the con- +trol of Berry curvature-mediated effects. For instance, orbital +selective metal-insulator transitions can be used to switch on +and off the electronic transport channels responsible for the +Berry curvature and its dipole. We envision that this capabil- +ity can be used to design orbitronic and electronic transistors +relying on the geometry of the quantum wavefunctions. +METHODS +Representation of the Gell-Mann matrices in the symmetry +groups +Apart from the identity matrix Λ0, the eight Gell-Mann matri- +ces can be defined as +Λ1 = +� +� +0 1 0 +1 0 0 +0 0 0 +� +�, +Λ2 = +� +� +0 −i 0 +i +0 +0 +0 +0 +0 +� +�, +Λ3 = +� +� +1 +0 +0 +0 −1 0 +0 +0 +0 +� +�, +Λ4 = +� +� +0 0 1 +0 0 0 +1 0 0 +� +�, +Λ5 = +� +� +0 0 −i +0 0 +0 +i 0 +0 +� +�, +Λ6 = +� +� +0 0 0 +0 0 1 +0 1 0 +� +�, +Λ7 = +� +� +0 0 +0 +0 0 −i +0 i +0 +� +�, +Λ8 = +� +� +� +1 +√ +3 +0 +0 +0 +1 +√ +3 +0 +0 +0 +−2 +√ +3 +� +� +�. +Let us now check the properties of these eight Gell-Mann ma- +trices under time-reversal symmetry. Since we are consider- +ing electrons that are effectively spinless due to the SU(2) +spin symmetry, the time-reversal operator can be represented +as K. Hence, the three Gell-Mann matrices Λ2,Λ5,Λ7 are +odd under time-reversal, i.e. T−1Λ2,5,7T = −Λ2,5,7, whereas +the remaining matrices are even under time-reversal. Sim- +ilarly, Λ1,2,3,8 are even under the vertical mirror symmetry +whereas Λ4,5,6,7 are odd. Let us finally talk about the three- +fold rotational symmetry. Since the rotation symmetry oper- +ator C3 = exp[2πiΛ7/3], the transformation properties of the +Gell-Mann matrices are determined by the commutation rela- + +9 +tions [Λ7,Λi]. The commutation relations are listed as follows: +[Λ7,Λ1] = iΛ4 +[Λ7,Λ2] = iΛ5 +[Λ7,Λ4] = −iΛ1 +[Λ7,Λ5] = −iΛ2 +[Λ7,Λ6] = 2i +� +Λ3 +2 − +√ +3 +2 Λ8 +� +� +Λ7, Λ3 +2 − +√ +3 +2 Λ8 +� += −2iΛ6 +� +Λ7,Λ3 + Λ8 +√ +3 +� += 0 +The results above indicate that the three pairs of operators +{Λ1,Λ4}, {Λ2,Λ5}, and +� +Λ6, Λ3 +2 − +√ +3 +2 Λ8 +� +behave as vector +under the threefold rotation symmetry and therefore form two- +dimensional IRREPS. +Berry curvature of SU(2) and SU(3) systems +For SU(2) systems, a generic Hamiltonian can be written +in terms of Pauli matrices σi as H(k) = d0(k)σ0 + d(k) · σσσ, +where σ0 is the 2×2 identity matrix and the Pauli matrix vec- +tor σσσ = (σx,σyσz). The Berry curvature can be expressed in +terms of d vector +Ω±(k) = ∓ +1 +2|d(k)|3 d(k)·[∂kxd(k)×∂kyd(k)] . +(7) +For SU(3) system, we can proceed analogously using the Gell- +Mann matrices introduced above. The Hamiltonian of a sys- +tem described by three electronic degrees of freedom in a 3×3 +manifold can be written as H(k) = b0(k)Λ0 +b(k)·ΛΛΛ, where +b0(k) is a scalar and b(k) is an eight dimensional vector. The +Gell-Mann matrices satisfy an algebra which is a generaliza- +tion of the SU(2). In particular, we have that +ΛaΛb = 2 +3δab +(dabc +ifabc)Λc, +(8) +where repeated indices are summed over. +In the equation +above, we have introduced the antisymmetric and symmetric +structure factors of SU(3) that are defined respectively as +fabc = − i +4Tr([Λa,Λb]Λc) , +dabc = 1 +4Tr({Λa,Λb}Λc). +From these one defines three bilinear operations of SU(3) vec- +tors: the dot (scalar) product v · w = vawa, the cross product +(v×w)a = fabcvbwc, and the star product (v⋆w)a = dabcvbwc. +The star product is a symmetric vector product which does not +play any role for SU(2) since dabc = 0. Moreover, the band- +resolved Berry curvature is given by [49, 64]: +Ωn(k) = −4(γk,nbk +bk ⋆bk) +(3γ2 +k,n −|bk|2)3 ·{[γk,n∂kxbk+ +(9) ++ ∂kx(bk ⋆bk)]×[γk,n∂kybk +∂ky(bk ⋆bk)] +� +where we introduced γk,n = +2 +√ +3|bk|cos +� +θk + 2π +3 n +� +, θk = +1 +3 arccos +� √ +3bk·(bk⋆bk) +|bk|3 +� +, and bk is a shorthand for b(k). Gener- +ally speaking, the Berry curvature in Eq. (7) can be split in two +contributions as Ωn(k) = Ω(0) +n (k) + Ω(⋆) +n (k), with Ω(0) +n (k) = +−4 +(γk,n)3 +(3γ2 +k,n−|bk|2)3 bk ·[∂kxbk ×∂kybk] that strongly resembles the +BC expression for SU(2) systems of Eq. (6). In trigonal sys- +tems described by the effective Hamiltonian in Eq. (4) we +have that for any momentum k and for any value of the pa- +rameters, the b vector associated with the Hamiltonian is +such that (γk,nbk +bk ⋆bk) is always orthogonal to the vector +in the curly braces in the expression of the Berry curvature +Eq. (9). Hence Ωn(k) = 0. On the contrary, assuming a Cs +point-group symmetry the effective Hamiltonian of Eq. (5) de- +fines bk = +� +0,αRky,∆+ 1 +2∆m,0,αRkx,0,αmkx, ∆ +√ +3 − +√ +3 +2 ∆m +� +, +for which we get that Ω(0) +n (k) = 0, and the BC is substantially +given by Ω(⋆) +n (k). In other words, the terms obtained by do- +ing the star product, bk ⋆ bk are those that yield the non-zero +BC. We point out that the BC is proportional to the combi- +nation of parameters α2 +R αm (2∆+∆m). A non-vanishing BC +can be thus obtained even in the absence of the Gell-Mann +matrix Λ8. This, on the other hand, would correspond to val- +ues of the crystal field splitting ∆m = 2∆/3 implying a very +strong distortion of the crystal from the trigonal arrangement. +The presence of the constant term ∝ Λ8 is thus essential to de- +scribe systems with a parent high-temperature trigonal crystal +structure. +Calculation of the Berry curvature dipole +The first moment of the Berry curvature, the Berry cur- +vature dipole, for each energy band n is given by Dx,n = +� +d2k +(2π)2 ∂kxΩn(k)f0(k), where f0(k) is the equilibrium Fermi- +Dirac distribution function. At zero temperature, this expres- +sion can be rewritten as a line integral over the Fermi line +Dx,n = +� +d2k +(2π)2 Ωn(k)∂En +∂kx +δ(En − µ) , +(10) +where En = En(k) (n = 1,2,3) are the energy bands and µ +is the chemical potential. 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B 104, 085114 (2021). +[65] The effective time-reversal symmetry would imply that the BC +should be an odd function of momentum. +[66] Dongwook Go, Daegeun Jo, Hyun-Woo Lee, Mathias Kl¨aui, +and Yuriy Mokrousov, “Orbitronics: Orbital currents in solids,” +EPL (Europhysics Letters) 135, 37001 (2021). + +Supplemental Material for: +“Orbital design of Berry curvature: pinch points and giant dipoles induced by crystal fields” +Maria Teresa Mercaldo,1 Canio Noce,1, 2 Andrea D. Caviglia,3 Mario Cuoco,2, 1 and Carmine Ortix1 +1Dipartimento di Fisica “E. R. Caianiello”, Universit`a di Salerno, IT-84084 Fisciano (SA), Italy +2SPIN-CNR, IT-84084 Fisciano (SA), Italy +3Department of Quantum Matter Physics, University of Geneva, +24 Quai Ernest Ansermet, CH-1211 Geneva, Switzerland +I. +BERRY CURVATURE PROPERTIES +In Fig. 1 we show the connection between the presence of mirror-symmetry protected crossings and the Berry curvature pinch- +points. This is further highlighted in Fig. 2 where we display the Berry curvature distribution associated with the effective k·p +theory model close to a mirror symmetry protected degeneracy and reading (see the main text) +He f f = vxkxσx +βkxδkyσy +vyδkyσz, +(1) +where δky is the momentum measured relatively to the mirror symmetry-protected degeneracy. +Fig. 3 illustrates the relation between the occurrence of the Berry curvature hot-spots and the energetic distance between +the |a1g⟩ and the crystal field split |e′ +g⟩ state. Finally, in Fig. 4 we show for comparison the Berry curvature of a Rashba two- +dimensional electron gas with crystalline anisotropy terms that are cubic in momentum. The corresponding Hamiltonian reads: +H2DEG = ¯h2k2 +2m σ0 +αSR (kxσy −kyσx)+ λ +2 +� +k3 ++ +k3 +− +� +σz +(2) +where k± = kx ±iky. +FIG. 1. a Energy dispersion of the orbital model along the mirror symmetric line of the BZ kx = 0. At finite ky there are two mirror symmetry- +protected crossings b Contour plot and c tridimensional plot of the band-resolved Berry curvature corresponding to the second band Ω2(k) in +the region close to the mirror-symmetry protected crossing where the BC pinch point is formed. We have used the same set of parameters of +Fig. 3 in the main article, i.e. ∆ = −0.2E0, ∆m = −0.01E0, and αR = αm = 1.0E0/k0 +F. +arXiv:2301.04548v1 [cond-mat.mes-hall] 11 Jan 2023 + +0.5 +Q22 +0.14 +Q22 +>200 +0.12 +-200 +100 +QF 0.10 +100 +K +0.0 +0 +-100 +0.06 +-200 +/0.04 +kx=0 +-100 +0.04 +<-200 +-0.5 +kx/ k? +0.14 +1.0 +0.0 +1.0 +-0.04 +0 +0.04 +0.10 +0.04 +kx/ k? +b +a +c2 +FIG. 2. Density plot of the Berry curvature corresponding to the effective two-band model of Eq. 1 with a pinch point at zero momentum +assuming vx = vy = β. +n=3 +n=2 +n=1 +1.0 +0.0 +1.0 +-0.5 +0.0 +0.5 +kx/kF +0 +En/ℰ0 +ky=0 +αR=1.0 +αR +1. +2. +3. +-0.4 -0.2 +0.0 +0.2 +0.4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +kx/kF +0 +(E2-E1)/ℰ0 +ky=0 +αR +1. +2. +3. +-0.4 -0.2 +0.0 +0.2 +0.4 +-50 +0 +50 +kx/kF +0 +Ω1 +ky=0 +● +● +● +● +● +● +● +● +● +● +● +● +● +● +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +● kx where (E2-E1) is minimum +■ kx where Ω1 is maximum +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +αR +k x/kF +0 +a +b +c +d +FIG. 3. (a) Energy dispersion along the ky = 0 line for ∆ = −0.2E0, ∆m = −0.01E0, αR = αm = 1.0E0/k0 +F. (b) The corresponding behavior +of the energy difference between the first two bands E2 − E1. We show also other values of the interorbital mixing parameter αR.(c) The +local behavior of the Berry curvature of the first band Ω1 for different values of αR shows that large enhancements of the Berry curvature, i.e. +the hot-spots, are strongly related to the E2 − E1 minima. This is further shown in (d) where we compare the location in k space of the BC +hot-spots and the E2 −E1 minima. +FIG. 4. Density plot of the Berry curvature for a spin-orbit coupled electron gas with warping terms cubic in momentum with effective +Hamiltonian Eq. 2. Here we have chosen αSR = 1 and λ = 0.2. +II. +RELATION BETWEEN VAN HOVE SINGULARITIES AND PROPERTIES OF THE BERRY CURVATURE DIPOLE +As mentioned in the main text, the cusps and inflection points in the Berry curvature dipole are direct consequence of Lifshitz +transitions and their associated van Hove singularities. We recall that the generic sequence of Lifshitz transition in our model is +as follows (see also Fig.4 of the main text). First, there are two “small” electron pockets centered at finite momentum that then +merge to create a single electron pocket (centered at Γ) delimited by two “concentric” Fermi lines. Note that this Fermiology is +strongly reminiscent of a conventional isotropic Rashba two-dimensional electron gas in the low-density regime. The internal +Fermi line then collapses at zero momentum and a single Fermi line survives. Such intraband Lifshitz transitions are then + +1.0 +.4 +4 +0.5 +2 +0.0 +0 +-0.5 +-2 +-1.0 +0.5 +1.0 +-4 +1.0 +-0.5 +0.0 +<-4 +Kx()+U +2 +0.4 +0.2 +y/y +0 +0 +1 +-0.2 +2 +2 +0 +2 +1 +1 +kx/kg +-0.43 +followed by the interband Lifshitz transition due to the appearance of two pockets of the second band that then merge with the +almost concomitant appearance of a tiny pocket of the third (highest-in-energy) band. As shown in Fig. 5 all these Lifshitz +transitions are characterized by van Hove singularities. Indeed, at all Lifshitz transitions there is a divergence in the slope of +the density of states. We point out that at the first intraband Lifshitz transition, the singularities due to the saddle points of the +energy levels are not integrable and a divergence in the density of states occurs. This is because the saddle points are distributed +over lines of the Brillouin zone – similarly to the zero-energy van Hove divergence of a Rashba two-dimensional electron gas. +Importantly, while the cusps and inflection points of the Berry curvature dipole can be all related to Lifshitz transitions (and +hence van Hove singularities), the opposite does not hold. The direct comparison between Berry curvature dipole and density +of states [see Fig. 5] indeed shows that the divergence of the density of states is accompanied by a featureless Berry curvature +dipole. Note that the features of the Berry curvature dipole are preserved by changing the interorbital mixing parameter αR as +shown in Fig. 6. +●●●●●●●●●●●●●●●● +● +● +● +● +● +● +● +● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ +◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ +● +n=1 +■ +n=2 +◆ +n=3 +0.0 +0.2 +0.4 +-2 +-1 +0 +1 +2 +gn(E) +E/ℰ0 +●●●●●●●●●●●●●●●●●●●●●●●●● +● +● +● +● +●● +● +● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +■ +■ +■ +■ +■ +■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ +◆ +◆ +◆ +◆◆◆ +◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ +● n=1 +■ n=2 +◆ n=3 +-2 +-1 +0 +1 +2 +-0.4 +-0.2 +0.0 +0.2 +0.4 +μ/ℰ0 +Dx,n +a +b +FIG. 5. (a) Band-resolved density of states gn(E) and (b) Berry curvature dipole for ∆ = −0.2E0, ∆m = 0.01E0, αOR = 2.0E0/k0 +F, αm = αOR. +In panel (a) the vertical lines indicate the various Lifshitz transitions. +● +● +● +● +● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +■ +■ +■ +■ +■ +■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ +◆ +◆ +◆ +◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ +● n=1 +■ n=2 +◆ n=3 +0 +1 +2 +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +μ/ℰ0 +Dx,n kF +0 +αR=1.0 +●●●●●●●●●●●●●●●●●●●●●● +● +● +● +● +● +● +● +● +● +● +● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +■ +■ +■ +■ +■ +■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ +◆ +◆ +◆ +◆◆ +◆ +◆ +◆ +◆ +◆ +◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ +● n=1 +■ n=2 +◆ n=3 +-2 +-1 +0 +1 +2 +-0.4 +-0.2 +0.0 +0.2 +0.4 +μ/ℰ0 +Dx,n kF +0 +αR=2.0 +●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +● +● +● +● +● +● +● +●● +● +● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +■ +■ +■ +■ +■ +■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ +◆ +◆ +◆ +◆◆ +◆ +◆ +◆ +◆ +◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ +● n=1 +■ n=2 +◆ n=3 +-2 +-1 +0 +1 +2 +-1.0 +-0.5 +0.0 +0.5 +1.0 +μ/ℰ0 +Dx,n kF +0 +αR=3.0 +●●●●●●●●●●●●●●●●●●●●●●●●●●●● +● +● +● +● +● +● +● +● +●● +● +● +●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +■ +■ +■ +■ +■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ +◆ +◆ +◆ +◆◆ +◆ +◆ +◆ +◆ +◆ +◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ +● n=1 +■ n=2 +◆ n=3 +-2 +-1 +0 +1 +2 +-1.5 +-1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +μ/ℰ0 +Dx,n kF +0 +αR=4.0 +a +b +c +d +FIG. 6. The band-resolved Berry curvature dipole for different values of the interorbital mixing parameter αR. The other parameters are the +same as in the main text. + +4 +a g +eg +eg +0.00 +0.05 +0.10 +0.15 +-0.3 +-0.2 +-0.1 +0.0 +0.1 +0.2 +Δ1/ℰ0 +En(Γ)/ℰ0 +Δm +Δ1,c +' +' +1 +FIG. 7. Behavior of the energy levels of the a1g and e′g doublet (split by the crystal field ∆m) as a function of the ∆1 mixing term. The critical +value ∆1,c where a level crossing is obtained is much larger than ∆m = 0.01E0. The trigonal crystal field splitting ∆ = −0.2E0 as in the main +text. +III. +EFFECT OF A CONSTANT Λ1 TERM +As mentioned in the main text, reducing the point-group symmetry from C3v to Cs makes a constant term in Λ1 symmetry +allowed. This term is not present in our model Hamiltonian. It is important to note that such term yields a mixing between two +states with equal Mx mirror eigenvalue (for p orbitals these correspond to the |pz⟩ and |py⟩ states). Consequently, a constant +Λ1 term lifts the energy degeneracy at the center of the BZ of the e′ +g (|px⟩, |py⟩) states precisely as the constant term in Λ3/2− +√ +3Λ8/2 does. However, there is an important difference. The constant Λ3/2 − +√ +3Λ8/2 term, appearing in Eq.5 of the main +text, corresponds to a direct splitting between the two states of the doublet. On the contrary, the splitting due to the constant Λ1 +term is suppressed by the energy difference – caused by the trigonal crystal field – between the a1g (|pz⟩) state and the e′ +g doublet. +Since the strengths ∆1,∆m of the constant terms in Λ1 and Λ3/2 − +√ +3Λ8/2 are expected to have the same order of magnitude +(both terms are proportional to the degree of lattice distortion from C3v) one finds that the constant term ∝ Λ1 has a much weaker +effect on the electronic properties: the band structure and the behavior of the Berry curvature are essentially determined by ∆m. +In particular, neither the existence of Berry curvature hot-spots nor the pinch points are affected by ∆1, in this regime. +To illustrate this point, we show in Fig. 7 the behavior of the energies (at the Γ point) of the three a1g and e′ +g states by increasing +the strength of the constant Λ1 term at fixed value of the crystal field splittings ∆,∆m. One finds that a non-zero ∆1 is able to +change the level ordering – and thus have a dramatic effect on the band structure – only for ∆1 ≫ ∆m. +Even though this regime is not realistic, we have explored it for completeness of our study. In Fig.8 we display the behavior of +the energy bands on the ky ≡ 0 line and the mirror symmetric line kx = 0. On the latter, one clearly sees that at all momenta the +bands are non-degenerate, and the mirror symmetry-protected crossings discussed in the main text are absent. This is because +the constant term ∝ Λ1 moves the crossings towards the center of the BZ until (for ∆1 = ∆1,c of Fig.7) they merge and annihilate +each other. +-1.0 +0.0 +1.0 +-0.5 +0.0 +0.5 +kx/kF +0 +En/ℰ0 +ky=0 +-1.0 +0.0 +1.0 +-0.5 +0.0 +0.5 +ky/kF +0 +En/ℰ0 +kx=0 +a +b +FIG. 8. Energy bands of the model Hamiltonian Eq.5 of the main text including a (large) constant term ∝ Λ1. Here we have used a trigonal +crystal field ∆ = −0.2E0, the additional crystal field splitting due to rotational symmetry breaking ∆m = 0.01E0, and the strengths of the orbital +Rashba coupling αOR = 1.0E0/k0 +F, αm = αOR. For the constant term ∝ Λ1 we considered a strength ∆1 = 0.15E0. + +5 +This qualitative difference in the electronic bands is reflected in different characteristics of the Berry curvature whose behavior, +in the ∆1 ≫ ∆m regime, is shown in Fig. 9. The Berry curvature is characterized by hot-spots with a dipolar profile, as mandated +by time-reversal symmetry. However, due to the absence of mirror symmetry-protected crossings, the Berry curvature pinch +points cannot exist. We note that the presence of hot-spots guarantees values of the Berry curvature dipole [see Fig. 10] of the +order of inverse of the Fermi wavevector k0 +F, and thus of the same order of magnitude of the ∆1 ≡ 0 case. This proves that the +size of the Berry curvature dipole is a robust feature of our symmetry-based model Hamiltonian. +FIG. 9. Density plot of the Berry curvature for the model Hamiltonian Eq.5 of the main text supplemented by a (large) constant term ∝ Λ1. +The annihilation of the mirror symmetry-protected crossings results in the absence of pinch points in the Berry curvature. We have used the +same parameter set of Fig. 8. +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +Δ1/ℰ0 +● 0 +■ 0.15 +-1 +0 +1 +2 +-0.4 +-0.2 +0.0 +0.2 +0.4 +μ/ℰ0 +DxkF +0 +FIG. 10. Behavior of the Berry curvature dipole as a function of the chemical potential for the model Eq.5 of the main text and a generalized +model which includes the constant term ∝ Λ1 with a strength ∆1 ≫ ∆m and that of the orbital Rashba coupling αR = αm = 2.0E0/k0 +F. All other +values of the model parameters are same as in Fig. 8. +IV. +ADDITIONAL FIGURES +Fig. 11 shows a comparison of the behavior of the Berry curvature dipole as a function of the chemical potential by changing +the relative strength between the three-fold rotational symmetric interorbital mixing ∝ αR and the term ∝ Λ7 only allowed in the +absence of rotational symmetries parametrized by αm. Finally Fig. 12 shows the local Berry curvature and the Berry curvature +dipole density for a set of parameters different from the one reported in the main text. + +21(k) +22(k) +23(k) +0.4F +0.4F +0.4F +15 +10 +15 +10 +0.2 +0.2 +10 +0.2 +ky/kg +5 +5 +0.0 +0.0 +0.0 +0 +0 +-0.2 +-0.2 +-0.2 +-5 +-5 +-0.4 +-0.4 +-0.4 +-5 +-10 +-10 +-0.4 -0.2 +0.4 +0.0 +0.2 +-0.4 -0.2 +¥0.0 +0.2 +0.4 +-0.4 -0.2 +0.0 +0.2 +0.4 +-10 +-15 +-15 +kx/ kg +kx/k? +kxlkg +b +a +c6 +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +αm +● 0.3 +■ 2.0 +-2 +-1 +0 +1 +2 +-0.4 +-0.2 +0.0 +0.2 +0.4 +μ/ℰ0 +DxkF +0 +αR=2.0 +a +-2 -1 +0 +1 +2 +-2 +-1 +0 +1 +2 +kx/kF +0 +ky/kF +0 +-2 -1 +0 +1 +2 +-2 +-1 +0 +1 +2 +kx/kF +0 +ky/kF +0 +-2 -1 +0 +1 +2 +-2 +-1 +0 +1 +2 +kx/kF +0 +ky/kF +0 +-2 -1 +0 +1 +2 +-2 +-1 +0 +1 +2 +kx/kF +0 +ky/kF +0 +-0.2 +0.0 +0.2 +-0.2 +0.0 +0.2 +kx/kF +0 +ky/kF +0 +-2 -1 +0 +1 +2 +-2 +-1 +0 +1 +2 +kx/kF +0 +ky/kF +0 +b +c +d +e +f +g +μ/ℰ0=-1.00 +μ/ℰ0=-0.50 +μ/ℰ0=0.05 +μ/ℰ0=0.15 +μ/ℰ0=0.60 +FIG. 11. a Berry curvature dipole as a function of µ for two values of αm (namely αm = 0.3E0/k0 +F and αm = 2.0E0/k0 +F) and for αR = 2.0E0/k0 +F, +∆ = −0.2E0 and ∆m = 0.01E0. b,c,d,e,f display the Fermi lines and the band-resolved occupied regions in momentum space for αR = 2.0E0/k0 +F +and αm = 0.3E0/k0 +F. Note that the absence of the Berry curvature dipole peak at negative values of the chemical potential for αm = 0.3E0/k0 +F +is consistent with the fact that the Fermi lines retain an almost symmetric profile in this region. +FIG. 12. Density plots of Berry curvature and Berry curvature dipole for ∆ = −0.2E0;∆M = 0.12E0,αR = 1.0E0/k0 +F,αM = 0.5E0/k0 +F. + +22(k) +21(k) +23(k) +0.4 +0.4 +0.4 +4 +4 +4 +0.2 +0.2 +0.2 +2 +2 +2 +0.0 +0.0 +0.0 +ky +ky +0 +0 +0 +-0.2 +-0.2 +-0.2 +-2 +-0.4 +-0.4 +-2 +-0.4 +-2 +-0.4 -0.2 0.0 +0.4 +-0.4 -0.2 0.0 0.2 +0.4 +-0.4 -0.2 0.00.2 +0.4 +-4 +-4 +b +a +c +kx/k? +kx/ k? +k/kg +0k21 +0k22 +0kxQ23 +100 +60 +100 +0.4 +0.4 +0.4 +0.2 +0.2 +0.2 +40 +50 +50 +ky/kg +0.0 +0.0 +0.0 +ky +ky +20 +0 +0 +-0.2 +-0.2 +-0.2 +-0.4 +-0.4 +-0.4 +-50 +-50 +0 +-0.4 -0.2 0.00.2 +0.4 +-0.4 -0.2 0.0 +¥0.2 +0.4 +-0.4 -0.2 0.0 0.2 +0.4 +d +kx/ kg +f +e +kx/kg +kx/k? +-20 +-100 +-100 \ No newline at end of file diff --git a/etE3T4oBgHgl3EQffAo5/content/tmp_files/load_file.txt b/etE3T4oBgHgl3EQffAo5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e32ad0d9097d11853dca8c35769f3637f53c2f47 --- /dev/null +++ b/etE3T4oBgHgl3EQffAo5/content/tmp_files/load_file.txt @@ -0,0 +1,1808 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf,len=1807 +page_content='Orbital design of Berry curvature: pinch points and giant dipoles induced by crystal fields Maria Teresa Mercaldo,1 Canio Noce,1, 2 Andrea D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Caviglia,3 Mario Cuoco,2, 1 and Carmine Ortix1 1Dipartimento di Fisica “E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Caianiello”, Universit`a di Salerno, IT-84084 Fisciano (SA), Italy 2SPIN-CNR, IT-84084 Fisciano (SA), Italy 3Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest Ansermet, CH-1211 Geneva, Switzerland The Berry curvature (BC) – a quantity encoding the geometric properties of the electronic wavefunctions in a solid – is at the heart of different Hall-like transport phenomena, including the anomalous Hall and the non- linear Hall and Nernst effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In non-magnetic quantum materials with acentric crystalline arrangements, local concentrations of BC are generally linked to single-particle wavefunctions that are a quantum superposition of electron and hole excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' BC-mediated effects are consequently observed in two-dimensional systems with pairs of massive Dirac cones and three-dimensional bulk crystals with quartets of Weyl cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Here, we demonstrate that in materials equipped with orbital degrees of freedom local BC concentrations can arise even in the complete absence of hole excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In these solids, the crystals fields appearing in very low-symmetric structures trigger BCs characterized by hot-spots and singular pinch points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' These characteristics naturally yield giant BC dipoles and large non-linear transport responses in time-reversal symmetric conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' INTRODUCTION Quantum materials can be generally defined as those solid- state structures hosting physical phenomena which, even at the macroscopic scale, cannot be captured by a purely classi- cal description [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Among such quantum phenomena, those related to the geometric properties of the electronic wavefunc- tions play undoubtedly a primary role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In an N-band crys- talline system, the cell-periodic part of the electronic Bloch waves defines a mapping from the Brillouin zone (BZ) to a complex space naturally equipped with a geometric struc- ture – its tangent space defines a Fubini-Study metric [2] that measures the infinitesimal distance between Bloch states at different points of the BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The imaginary part of this quan- tum geometric tensor [3, 4] corresponds to the well-known Berry curvature (BC), which, when integrated over the full BZ, gives the Chern number cataloguing two-dimensional in- sulators [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In metallic systems with partially filled bands, the BC summed over all occupied states can result in a non- vanishing Berry phase if the system breaks time-reversal sym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This Berry phase regulates the intrinsic part of the anomalous Hall conductivity of magnetic metals [6–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Materials with an acentric crystal structure can possess non- vanishing concentrations of BC even if magnetic order is ab- sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Probing the BC of these non-centrosymmetric and non- magnetic materials via charge transport measurements usually requires externally applied magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' For instance, in time-reversal invariant Weyl semimetals, such as TaAs [10], the strong BC arising from the Weyl nodes can be revealed using the planar Hall effect [11] – a physical consequence of the negative longitudinal magnetoresistance associated with the chiral anomaly of Weyl fermions [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Recently, it has been also shown that the planar Hall effect can display an anomalous antisymmetric response [13, 14], which, at least in two-dimensional materials, is entirely due to an unbalance in the BC distribution triggered by the Zeeman-induced spin splitting of the electronic bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In the absence of external magnetic fields, a BC charge transport diagnostic for non-magnetic materials requires to go beyond the linear response regime [15–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Hall-like currents appearing as a non-linear (quadratic) response to a driving electric field can have an intrinsic contribution governed by the Berry curvature dipole (BCD), which is essentially the first moment of the Berry curvature in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In three- dimensional systems, non-vanishing BCDs have been linked to the presence of tilted Weyl cones, and have been shown to exist both in type-I and in type-II Weyl semimetals [20] such as MoTe2 [21] and the ternary compound TaIrTe4 [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Furthermore, the Rashba semicondutor BiTeI has been pre- dicted to host a BCD that is strongly enhanced across its pressure-induced topological phase transitions [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In two-dimensional materials, the appearance of BCDs is subject to stringent symmetry constraints: the largest sym- metry group is Cs, which is composed by the identity and a single vertical mirror line [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The concomitant presence of spin-orbit coupled massive Dirac cones with substantial BC and such unusually low-symmetry crystalline environ- ments have suggested the surface states of SnTe [26] in the low-temperature ferroelectric phase [27], monolayer transi- tion metal dichalcogenides in the so-called 1Td phase [28– 30], and bilayer WTe2 as material structures hosting sizable BCDs [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Spin-orbit free two-dimensional materials, in- cluding monolayer and bilayer graphene, have been also put forward as materials with relatively large BCDs [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In these systems, it is the interplay between the trigonal warping of the Fermi surface and the presence of massive Dirac cones due to inversion symmetry breaking that triggers dipolar concentra- tion of Berry curvatures [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Finite concentrations of BC and BCDs are symmetry al- lowed also in systems that do not feature quartets of Weyl cones and pairs of massive Dirac cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The anomalous massless Dirac cones at the surface of three-dimensional strong topological insulators [35] as well as conventional two-dimensional electron gases (2DEG) with Rashba spin- orbit coupling [36] are generally characterized by finite lo- cal BC concentrations when subject to trigonal crystal fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The existence of BC in 2DEGs, which has been experimen- tally probed through “anomalous” planar Hall effect measure- ments [36], provides a new avenue for investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' It shows arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='04548v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='mes-hall] 11 Jan 2023 2 in fact that Berry curvature-mediated effects can be generated entirely from conduction electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This overcomes the re- quirement of materials with narrow gaps in which the elec- tronic wavefunctions at the Fermi level are a quantum su- perposition of electron and hole excitations, and extends the palette of non-magnetic materials displaying BC effects to, for instance, doped semiconductors with gaps in the eV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' It also proves that it is possible to trigger BC effects in con- ventional electron liquids with competing instabilities towards other many-body quantum phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In a spin-orbit coupled 2DEG, the BC is however triggered by crystalline anisotropy terms, which are cubic in momen- tum and linked to the out-of-plane component of the spin tex- tures [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Consequently, the BC does not possess the characteristic “hot-spots” appearing in close proximity to near degeneracy between two bands where the Bloch wavefunc- tions are rapidly changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The absence of such BC hot-spots forbids, in turn, large enhancements of the BCD, which is a central quest for material design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This motivates the funda- mental question on whether and how an electron system can develop strong local BC concentrations in time-reversal sym- metric conditions even in the complete asbence of hole exci- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Here, we provide a positive answer to this question by showing that spin-orbit free metallic systems with an ef- fective pseudo-spin one orbital degree of freedom can display BC hot-spots and characteristic BC singular pinch points that yield dipoles order of magnitudes larger than those triggered by spin-orbit coupling in a 2DEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Model Hamiltonian from symmetry principles Let us first consider a generic single-valley two-level sys- tem in two dimensions with spin degree of freedom only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The corresponding energy spectrum is assumed to accurately rep- resent the electronic bands close to the Fermi level of the metal in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' As long as we consider materials with- out long-range magnetic order, the two Fermi surfaces must originate from one of the four time-reversal invariant point of the Brillouin zone (BZ) (n1b1 +n2b2)/2 with b1,2 the two primitive reciprocal lattice vectors of the BZ and n1,2 = 0,1 [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Time-reversal symmetry guarantees that the two bands will be Kramers’ degenerate at the time-reversal invariant mo- menta (TRIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The effective Hamiltonian in the vicinity of the TRIM can be captured using a conventional k · p theory that keeps track of the point group symmetries of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' To make things concrete, let us assume that the low-energy conduction bands are centered around the Γ point of the BZ and we are dealing with an acentric crystal with C3v point group symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This is the largest acentric symmetry group without C2T symmetry, C2 indicating a twofold rotation sym- metry with out-of-plane axis and T time-reversal, and thus al- lows for local BC concentrations [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The generators of C3v are the threefold rotation symmetry C3 and a vertical mirror symmetry, which, without loss of generality, we take as Mx sending x → −x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The threefold rotation symmetry can be rep- resented as e−iπσz/3 while the mirror symmetry as iσx [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Momentum and spin transform under C3 and Mx as follows C3 : k± → e±2πi/3k±;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' σ± → e±2πi/3σ± σz → σz Mx : k± → −k±;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' σy,z → −σy,z σx → σx (1) where k± = kx ± iky and σ± = σx ± iσy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Furthermore, the Hamiltonian must satisfy the time-reversal symmetry con- straint H(k) = TH(−k)T−1, with the time-reversal operator that, as usual, can be represented as T = iσyK and K the com- plex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' When expanded up to linear order in k, the form of the Hamiltonian reads as H(k) = αR (kxσy −kyσx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The Dirac cone energy spectrum predicted by this Hamilto- nian violates the fermion doubling theorem [41] and hence can occur only on the isolated surfaces of three-dimensional strong topological insulators [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' And indeed H(k) coin- cides with the effective Hamiltonian for the surface states of the topological insulators in the Bi2Se3 material class [40, 43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In a genuine two-dimensional system such anomalous states cannot be present, and an even number of Kramers’ re- lated pair of bands must exist at each Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Conse- quently, the effective Hamiltonian must be equipped with an additional term that is quadratic in momentum and such that it doubles the number of states at each energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Time-reversal symmetry implies that terms quadratic in momentum are cou- pled to the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Therefore, we arrive at the well- known Hamiltonian of a two-dimensional electron gas with Rashba-like spin-orbit coupling that reads H(k) = ¯h2k2 2m σ0 +αR (kxσy −kyσx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (2) The corresponding energy spectrum consisting of two shifted parabolas is schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Although the crystalline symmetry requirements are fulfilled, the Hamilto- nian above does not predict any finite BC local concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This is because the d vector associated to the Hamiltonian d = � −αRky,αRkx,0 � is confined to a two-dimensional plane at all momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' There are two different ways to lift the d vector out-of-plane and thus trigger a non-vanishing BC [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The first one con- sists in introducing a constant mass ∆σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This term removes the Kramers’ degeneracy at the TRIM [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 1(b)] and therefore breaks time-reversal invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' It can be realized by externally applying an out-of-plane magnetic field or by inducing long-range magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The BC then generally displays an hot-spot located at the TRIM and a circular sym- metric distribution [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 1(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Moreover, time-reversal symmetry breaking implies that the Berry phase accumulated by electrons on the Fermi surface is non-vanishing [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The second route explicitly takes into account trigonal warping terms which are cubic in momentum and couple to the Pauli matrix σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Such terms preserve time-reversal invariance, and thus create a BC distribution with an angular dependence such that the Berry phase accumulated over any symmetry-allowed Fermi line cancels out [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Perhaps more importantly, the BC triggered by crystalline anisotropy terms [36] does not display a hot-spot, thus suggesting that in systems with conventional 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (a) Schematic band structure of a two-dimensional electron gas with Rashba spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (b) An out-of-plane magnetic field breaks the Kramers’ degeneracy at k = 0 and triggers a finite BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (c) The local BC has a circular profile with an hot spot at the Γ point of the BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (d) Schematic band structure of a two-dimensional electron system characterized by an L = 1 orbital multiplet in a trigonal crystalline environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (e) An additional crystalline symmetry lowering splits completely the energy levels at the Γ point of the BZ even if time-reversal symmetry is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The presence of mirror symmetry protects crossing at finite momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (f) A characteristic time-reversal symmetric BC profile with the presence of hot-spots and singular pinch points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The BC has been obtained using the model Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (5) with ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2E0, ∆m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='12E0, αR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0E0/k0 F and αm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5E0/k0 F with E0 = ¯h2(k0 F)2/(2m) and k0 F a characteristic Fermi wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' quasiparticles and a single internal degree of freedom time- reversal symmetry breaking is a prerequisite for large local BC enhancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We now refute this assertion by showing that in systems with orbital degrees of freedom the formation of BC hot- spots is entirely allowed even in time-reversal symmetric con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Consider for instance a system of p orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In a generic centrosymmetric crystal, interorbital hybridization away from the TRIM can only occur with terms that are quadratic in momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' However, and this is key, in an acen- tric crystal interorbital mixing terms linear in momentum are symmetry allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' These mixing terms, often referred to as orbital Rashba coupling [46–48], are able to induce BC hot spots with time-reversal symmetry, as we now show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We as- sume as before an acentric crystal with C3v point group, and electrons that are effectively spinless due to SU(2) spin sym- metry conservation: we are thus removing spin-orbit coupling all together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In the pz, py, px orbital basis, the generators of the point group are represented by Mx = � � 1 0 0 0 1 0 0 0 −1 � �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' C3 = � � 1 0 0 0 cos 2π 3 sin 2π 3 0 −sin 2π 3 cos 2π 3 � �, The two px,y orbitals form a two-dimensional irreducible rep- resentation (IRREP) whereas the pz orbital represents a one- dimensional IRREP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The form of the effective Hamiltonian away from the TRIM can be captured using symmetry con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Specifically, any generic 3 × 3 Hamiltonian can be expanded in terms of the nine Gell-Mann matrices [49] Λi [see Methods] as H(k) = 8 ∑ i=0 bi(k)Λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (3) The invariance of the Hamiltonian requires that the compo- nents of the Hamiltonian vector b(k) should have the same behavior as the corresponding Gell-Mann matrices Λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This means that they should belong to the same representation of the crystal point group [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' From the representation of the Λi’s [see Methods and Table 1] and those of the polynomials of k [see Table 1], we find that the effective Hamiltonian up to linear order in momentum reads as H(k) = ∆ � Λ3 + 1 √ 3Λ8 � −αR [kxΛ5 +kyΛ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (4) Here the parameter ∆ quantifies the energetic splitting be- tween the px,y doublet and the pz singlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The second term in the Hamiltonian corresponds instead to the pseudo-spin one Energy Spectra Berry Curvature b 2+(k)/2max a C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='8 Spin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='6 + Magnetic Field 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 22(k)/2max Orbital 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 X 0 + Crystal Field 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='04 massless Dirac Hamiltonian [51, 52] predicted to occur for instance in the kagome lattice with a staggered magnetic π flux [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Pseudo-spin one Dirac fermions are not subject to any fermion multiplication theorem [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Therefore, a dou- bling of the number of states at each energy is not strictly required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' However, since we are interested in systems without the concomitant presence of electrons and holes, we will in- troduce a term ¯h2k2Λ0/(2m) with an equal effective mass for all three bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The ensuing Hamiltonian can be then seen as a generalization of the Rashba 2DEG to an SU(3) sys- tem with the effect of the trigonal crystal field that leads to a partial splitting of the energy levels at the TRIM, entirely allowed by the absence of Kramers’ theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Despite the spectral properties [c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 1(d)] have a strong resemblance to those obtained in a time-reversal broken 2DEG, a direct computation [see Methods] shows that the BC associated to the Hamiltonian above is vanishing for all momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Break- ing time-reversal symmetry introducing a constant mass term ∝ Λ7 or considering crystalline anisotropy terms that are cubic in momentum represent two possible routes to trigger a finite Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The crux of the story is that in the present SU(3) system at hand, another possibility exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' It only relies on the crystal field effects that are generated by lowering the crystalline point group to Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' From the representations of the Gell-Mann matrices and the polynomials of k in this group, we find that the effective Hamiltonian reads H(k) = ¯h2k2 2m Λ0 +∆ � Λ3 + 1 √ 3Λ8 � +∆m � 1 2Λ3 − √ 3 2 Λ8 � −αR [kxΛ5 +kyΛ2]−αmkxΛ7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (5) Nothing prevents to have the interorbital mixing terms ∝ Λ2,5 with different amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Without loss of generality, in the remainder we will consider a single parameter αR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In the Hamiltonian above, we have also neglected a constant term ∝ Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' For materials with an high-temperature trigonal struc- ture, its amplitude ∆1 is expected to be of the same order of magnitude as ∆m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In this regime [see the Supplemental Mate- rial], a term ∝ Λ1 has a very weak effect on the energy spec- trum and BC properties, and can be thus disregarded [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The energy spectrum reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 1(e) shows that the effect of the crystal symmetry lowering is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' First, there is an additional energy splitting between the px,y implying that all levels at the Γ point of the BZ are singly degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Second, the two px,y orbitals have band degeneracies along the mir- ror symmetric kx = 0 line of the BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Such mirror-symmetry protected crossings give rise to BC singular pinch points [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 1(f) and the Supplemental Material].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' It is the presence of these pinch points that represents the hallmark of the non- trivial geometry of the electronic wavefunctions associated to the p-orbital manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Note that the BC also displays hot- spots [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 1(f)] with BC sources and sinks averaging to zero on any mirror symmetric Fermi surface as mandated by time-reversal invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' C3v E 2 C3 2σv polynomials of k Gell-Mann matrices A1 1 1 1 1, k2x +k2y Λ3 +Λ8/ √ 3,Λ0 A2 1 1 −1 – Λ7 E 2 −1 0 � kx,ky � {Λ1,Λ4}, {Λ2,Λ5} � Λ6,Λ3/2− √ 3Λ8/2 � Cs E 2σv polynomials of k Gell-Mann matrices A′ 1 1 1, ky, k2x, k2y Λ1, Λ2, Λ3, Λ8 A′′ 1 −1 kx Λ4, Λ5, Λ6, Λ7 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Character table for the point groups C3v and Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We also indicate the representation of the Gell-Mann matrices and the poly- nomials of momentum k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The model Hamiltonians reported in the main text can be obtained by additionally using the time-reversal symmetry constraint H⋆(−kx,−ky) = H(kx,ky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Material realizations Before analyzing the origin and physical consequence of the BC and its characteristic pinch points, we now intro- duce a material platform naturally equipped with orbital de- grees of freedom and the required low crystalline symme- try: [111] interfaces of transition metal oxides hosting two- dimensional d electron systems of t2g orbital character such as SrTiO3 [55, 56], KTaO3 [57], and SrVO3-based heterostruc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' When compared to conventional semiconductor het- erostructures, complex oxide interfaces consist of d elec- trons with different symmetries, a key element in determin- ing their many-body ground states that include, notably, un- conventional superconductivity [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In the high-temperature cubic phases of these materials, the octahedral crystal field pins the low-energy physics to a degenerate t2g manifold, which spans an effective angular momentum one subspace, precisely as the p orbitals discussed above [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The re- duced symmetry at interfaces lift their energetic degeneracy and modify their orbital character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' At the [111] interface the transition metal atoms form a stacked triangular lattice with three interlaced layers [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 2(a),(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This results in a triangular planar crystal field that hybridizes the |xy⟩, |xz⟩ and |yz⟩ orbitals to form an |a1g⟩ = (|xy⟩+|xz⟩+|yz⟩)/ √ 3 one-dimensional IRREP whereas the two states |e′ g±⟩ = � |xy⟩+ω±1 |xz⟩+ω±2 |yz⟩ � / √ 3, with ω = e2πi/3, form the two-dimensional IRREP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The energetic ordering of the levels depends on the microscopic details of the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' For example, at the (111)LaAlO3/SrTiO3 interface, x-ray absorption spec- troscopy [60] sets the |a1g⟩ state at lower energy [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 2(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' By further considering the structural inversion symmetry inherently present at the heterointerface, we thus formally reach the situation we discussed for the set of p orbitals, be it for the trigonal symmetry that excludes any local concentrations of BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' However, and this is key, low- temperature phase transitions in oxides lower the crystal sym- metry, often realising a tetragonal or orthorhombic phase with oxygen octahedra rotations and (anti)polar cation displace- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Let’s consider the paradigmatic case of SrTiO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' A 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (a) Schematic representation of an ABO3 pervoskite cubic unit cell displaying the three interlaced transition metal [111] planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (b) Corresponding top view along the [111] crystallographic direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We only show the B transition metal atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (c) and (d) show the effect of a tetragonal distortion with the [001] direction being the tetragonal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The distortion breaks the threefold rotation symme- try around the [111] axis but leaves a residual mirror symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (e) Evolution of the orbital states at the Γ point of the BZ with quenched angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' structural transition occurring at around 105 K, from the cubic phase to a tetragonal structure [61] [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 2(c)], breaks the threefold rotational symmetry leaving a single residual mir- ror line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Assuming the tetragonal axis to be along the [001] direction, the surviving mirror symmetry at the [111] inter- face corresponds to M[¯110] [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 2(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This structural distortion lifts the degeneracy of the e′ g doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The bond- ing and antibonding states |e′ g+⟩±|e′ g−⟩ have opposite mirror M[¯110] eigenvalues and realize two distinct one-dimensional IRREP [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 2(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' SrTiO3-based heterointerfaces un- dergo additional tetragonal to locally triclinic structural dis- tortions at temperatures below ≃ 70 K which involves small displacements of the Sr atoms along the [111] directions con- voluted with TiO6 oxygen-octahedron antiferrodistortive rota- tions [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In addition, below about 50 K, SrTiO3 and KTaO3 approach a ferroelectric instability that is accompanied by strong polar quantum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This regime is character- ized by a soft transverse phonon mode that involves off-center displacement of the Ti ions with respect to the surrounding oc- tahedron of oxygen ions [63], which, in the static limit, would correspond to a ferroelectric order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This can poten- tially enhance the interorbital hybridization terms allowed in acentric crystalline environments, and thus boost the appear- ance of large BC concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Berry curvature dipole Having identified (111)-oriented oxide heterointerfaces as ideal material platforms, we next analyze the specific prop- erties of the BC and its first moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We first notice that in the case of a two-level spin system the local Berry cur- vature of the spin-split bands, if non-vanishing, is opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Due to the concomitant presence of both spin-bands at each Fermi energy, the spin split bands cancel their respective lo- cal BC except for those momenta which are occupied by one spin band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In the SU(3) system at hand, there is a similar sum rule stating that at each momentum k the BC of the three bands [c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3(a)] sum to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' However, and as mentioned above, the orbital bands are not subject to fermion multiplica- tion theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In certain energy ranges a single orbital band is occupied [c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3(a,b)] and BC cancellations are not at work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' There is also another essential difference between the BC associated to spin and orbital degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In general, the commutation and anticommutation relations of the SU(N) Lie algebra define symmetric and antisymmetric structure constants, which, in turn, define the star and cross products of generic SU(N) vectors [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Differently from an SU(3) system spanning an angular momentum one subspace, in SU(2) spin systems the symmetric structure constant van- ishes identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The ensuing absence of star products bk ⋆bk precludes the appearance of BC with time-reversal symmetry as long as crystalline anisotropies are not taken into account [see Methods].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' On the other hand, for SU(3) the presence of all three purely imaginary Gell-Mann matrices Λ2,5,7, to- gether with the “mass” terms Λ3,8, is a sufficient condition to obtain time-reversal symmetric BC concentrations even when accounting only for terms that are linear in momentum [see Methods].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This, however, strictly requires that all rotation symmetries must be broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Next, we analyze the properties of the band resolved local BC starting from the lowest energy band, which corresponds to the (|xy⟩+|xz⟩+|yz⟩)/ √ 3 state at (111) LAO/STO het- erointerfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3(c) shows a characteristic BC profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' It displays two opposite poles centered on the ky = 0 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' These sources and sinks of BC are equidistant from the mirror sym- metric kx = 0 line since the BC, as any genuine pseudoscalar, must be odd under vertical mirror symmetry operations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Ω(kx,ky) = −Ω(−kx,ky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Note that the combination of time- reversal symmetry and vertical mirror implies that the BC will be even sending ky → −ky, thus guaranteeing that, taken by themselves, the BC hot-spots will be centered around the ky = 0 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Their finite kx values coincide with the points where the (direct) energy gap between the n = 1 and the n = 2 bands is minimized [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3(a,b) and Supplemental Mate- rial], and thus the interorbital mixing is maximal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The prop- erties of the BC are obviously reflected in the BCD local den- sity ∂kxΩ(kx,ky): it possesses [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=" 1(f)] a positive area strongly localized at the center of the BZ that is neutralized [0,0,1] a C [0,1,0] [1,0,0] M[1,1,0] [1,1,1] b d e' t2 g a1 g (111) Interface Bulk (111) Interface + tetragonal distorsion Cs symmetry Cs symmetry e6 FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (a) Ordering of the crystal field split t2g (p) orbitals with their associated band index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (b) Energy spectrum of the model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (5) obtained using the parameter set ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2E0, ∆m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='01E0, αR = αm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0E0/k0 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (c),(d),(e) show the ensuing band-resolved Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (f), (g),(h) are the corresponding BC dipole densities ∂kxΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Note that the presence of mirror symmetry guarantees that the orthogonal dipole density ∂kyΩ averages to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' by two mirror symmetric negative regions present at finite kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Let us next consider the Berry curvature profile arising from the two degenerate e′ g states that are split by the threefold rota- tion symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3(d) shows the BC profile of the lowest energy band: it is entirely dominated by the BC pinch points induced by the mirror symmetry protected degeneracies on the kx = 0 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The BC also displays a nodal ring around the pinch point, and thus possesses a characteristic d-wave character around the singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This can be understood by constructing a k·p theory around each of the two time- reversal related degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' To do so, we first recall that the two bands deriving from the e′ g states have opposite Mx mirror eigenvalue along the full mirror line kx ≡ 0 of the BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Close to the degeneracies, Mx can be therefore represented as σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Under Mx, kx → −kx whereas ky → ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Moreover, the Pauli matrices σx,y → −σx,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' An effective two-band model close to the degeneracies must then have the following form at the leading order: Hef f = vxkx σx +βkx δky σy +vyδky σz, (6) where δky is the momentum measured relatively to the mir- ror symmetry-protected degeneracy and we have neglected the quadratic term coupling to the identity k2σ0 that does not affect the BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Using the usual formulation of the BC for a two-band model [see the Methods section], it is possible to show that the Hamiltonian above is characterized by a zero- momentum pinch-point with two nodal lines [see the Supple- mental Material] and d-wave character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' It is interesting to note that this also implies that the “effective” time-reversal sym- metry inverting the sign of k around the pinch point is bro- ken [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Perhaps even more importantly, the d-wave charac- ter implies a very large BCD density in the immediate neigh- borhood of the pinch point [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3(g)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Similar properties are encountered when considering the highest energy band , with the difference that the pinch-point has an opposite an- gular dependence [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3(e)] and consequently the BCD density has opposite sign [c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3(h)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Having the band-resolved BC and BCD density profiles in our hands, we finally discuss their characteristic finger- prints in the BCD defined by Dx = � k ∂kxΩ(k) f0, with � k = � d2k/(2π)2 and f0 being the equilibrium Fermi-Dirac distri- bution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' By continuously sweeping the Fermi energy, we find that the BCD shows cusps and inflection points [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 4(a)] , which, as we now discuss, are a direct consequence of Lifshitz transitions and their associated van Hove singular- ities [see Supplemental Material].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Starting from the bottom of the first band, the magnitude of the BCD continuously in- creases until it reaches a maximum where the dipole is larger than the inverse of the Fermi momentum of a 2DEG 1/k0 F and thus gets an enhancement of three order of magnitudes with respect to a Rashba 2DEG [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In this region, there are two distinct Fermi lines encircling electronic pockets at fi- nite values of k [c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 4(b)], which subsequently merge on two disconnected regions in momentum space [c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 4(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Since the states in the immediate vicinity of the center of the BZ are not occupied, the BCD is entirely dominated by the two mirror symmetric negative hot-spots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' By fur- ther increasing the chemical potential, the internal Fermi line collapses at the Γ point and therefore a first Lifshitz transition occurs [c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 4(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In this regime, the BCD has exponen- 23 21 ■>100 >100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 n=3 T100 T100 4E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 n=2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='1 50 50 a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 n=l1 ky ky 0 K 0 0 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 50 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 ¥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 ■<-100 ■<-100 d c e kx/k?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' kx/ kg k/kg /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 1 E/80 0k22 0kxQ23 0kx21 ■>5·103 ■>5·103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 4·103 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='103 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 2·103 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='103 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 200 _ 200 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 0 kylk C 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 200 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 kxlk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 2·103 2·103 0 b 4·103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 <-5·103 ■<-5·103 k/ kg 25 h f kx/k?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 6 kx/ k?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='7 tially small values due to the fact that the strong positive BCD density area around the center of the BZ counteracts the mir- ror symmetric negative hot-spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' By further increasing the chemical potential, a second Lifshitz transition signals the oc- cupation of the first eg band with two pockets centered around the ky = 0 line [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 4(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This Lifshitz transition coin- cides with a rapid increase of the BCD due to the contribution coming from the local BCD density regions external to the BC nodal ring of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The subsequent sharp negative peak originates from a third Lifshitz transition in which the two electronic pockets of the second band merge, and almost concomitantly a tiny pocket of the third band centered around Γ arises [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 4(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' By computing the band resolved BCD [see the Supplemental Material] one finds that it is this small pocket the cause of the negative sharp peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' For large enough chemical potentials, the BCD develops an additional peak cor- responding to the fermiology of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 4(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This peak, which is again larger than 1/k0 F, can be understood by noticing that due to the BC local sum rule the momenta close to the center of the BZ do not contribute to the BCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' On the other hand, the re- gions external to the BC nodal ring are unoccupied by the third band and consequently have a net positive BCD local density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Thermal smearing can affect the strongly localized peaks at lower chemical potential but will not alter the presence of this broader peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Note that the BCD gets amplified by increas- ing the interorbital mixing parameter αR but retains similar properties [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 4 and the Supplemental Material].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The strength of BC-mediated effects depends indeed on the ratio between the characteristic orbital Rashba energy 2mα2 R(m)/¯h2 and the crystal field splittings ∆(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The BCD properties and values comparable to the Fermi wavelength are hence com- pletely generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Let us finally discuss the role of spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' It can be included in our model Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 5 as Hso = λso (Lx ⊗τx +Ly ⊗τy +Lz ⊗τz), where λso is the spin-orbit coupling strength, the L = 1 angular momentum matrices cor- respond to the Gell-Mann matrices Λ2,Λ5,Λ7, and the Pauli matrices τx,y,z act in spin space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Its effect can be analyzed using conventional (degenerate) perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' At the center of the Brillouin zone, Hso is completely inactive – the eigenstates of the Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 5 are orbital eigenstates and the off-diagonal terms in orbital space Λ2,5,7 cannot give any correction at first order in λso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The situation is different at finite values of momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The two spin-orbit free degen- erate eigenstates are a superposition of the different orbitals (due to the orbital Rashba coupling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Therefore, the spin-orbit coupling term will lift their degeneracy resulting in a Rashba- like splitting of the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In order to explore the consequence of this spin splitting on the Berry curvature, let us denote with |ψ↑ 0(k)⟩ and |ψ↓ 0(k)⟩ the two spin-orbit free degenerate eigenstates at each value of the momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Note that |ψ0⟩ is a three-component spinor for the orbital degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' When accounting pertur- batively for spin-orbit coupling the eigenstates will be a super- position of the spin degenerate eigenstates and will generally read |ψ+(k)⟩ = cosθ(k)eiφ(k) |ψ↑ 0(k)⟩+sinθ(k)|ψ↓ 0(k)⟩ |ψ−(k)⟩ = −sinθ(k)eiφ(k) |ψ↑ 0(k)⟩+cosθ(k)|ψ↓ 0(k)⟩ Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' the momentum dependence of the phase φ and the angle θ is a “by-product” of the orbital Rashba coupling: the effect of spin-orbit coupling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' which is off-diagonal in orbital space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' is modulated by the momentum-dependent orbital content of the eigenstates |ψ↑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='↓ 0 (k)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The abelian Berry connection of the two spin-split states A+,− kx,ky = ⟨ψ+,−(k)|i∂kx,kyψ+,−(k)⟩ will therefore contain two terms: the first one is the spin- independent Berry connection A0 kx,ky = ⟨ψ0(k)|i∂kx,kyψ0(k)⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' the second term is instead related to the derivatives of the phase φ and angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This Berry connection is opposite for the +,− states and coincides with the Berry connection of a two- level spin system [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This also implies that the Berry curva- ture of a Kramers’ pair of bands Ω+,−(k) = Ω(k) ± Ωso(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The contribution of the Berry curvature Ωso is opposite for the time-reversed partners and the net effect only comes from the difference between the Fermi lines of two partner bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' However, the purely orbital Berry curvature Ω(k), which can be calculated directly from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 5, sums up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The values of the BCD presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 4 are thus simply doubled in the pres- ence of a weak but finite spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' CONCLUSIONS In this study, we have shown an intrinsic pathway to de- sign large concentrations of Berry curvature in time-reversal symmetric conditions making use only of the orbital angular momentum electrons acquire when bound to atomic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Such mechanism is different in nature with respect to that ex- ploited in topological semimetals and narrow-gap semicon- ductors where the geometric properties of the electronic wave- functions originate from the coupling between electron and hole excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The orbital design of Berry curvature is also inherently different from the time-reversal symmetric spin- orbit mechanism [35, 36] , which strongly relies on crystalline anisotropy terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We have shown in fact that the Berry curva- ture triggered by orbital degrees of freedom features both hot- spots and singular pinch-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Furthermore, due to the crys- talline symmetry constraints the Berry curvature is naturally equipped with a non-vanishing Berry curvature dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' These characteristics yield a boost of three orders of magnitude in the quantum non-linear Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In (111) LaAlO3-SrTiO3 heterointerfaces where the characteristic Fermi wavevector k0 F ≃ 1 nm−1, the Berry curvature dipole Dx ≃ 1nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The corresponding non-linear Hall voltage can be evaluated us- ing the relation [31, 33] Vyxx = e3 τ Dx |Ix|2/(2¯h2σ2 xxW), with the characteristic relaxation time τ ≃ 1 pS and the longitudi- nal conductance σxx ≃ 5 mS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In a typical Hall bar of width W ≃ 10µ m sourced with a current Ix ≃ 100 µA, the non lin- ear Hall voltage Vyxx ≃ 2 µV, which is compatible with the strong non-linear Hall signal experimentally detected [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='The findings of our study carry a dramatic impact on the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='developing area of condensed matter physics dubbed orbitron- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='8 ■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='■ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='■ ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 μ/ℰ0 DxkF 0 αR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 ■ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 ◆ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 ▲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 ▼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 a 2 -1 0 1 2 2 1 0 1 2 kx/kF 0 ky/kF 0 2 -1 0 1 2 2 1 0 1 2 kx/kF 0 ky/kF 0 2 -1 0 1 2 2 1 0 1 2 kx/kF 0 ky/kF 0 2 -1 0 1 2 2 1 0 1 2 kx/kF 0 ky/kF 0 2 -1 0 1 2 2 1 0 1 2 kx/kF 0 ky/kF 0 2 -1 0 1 2 2 1 0 1 2 kx/kF 0 ky/kF 0 b c d e f g μ/ℰ0=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='70 μ/ℰ0=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='60 μ/ℰ0=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='20 μ/ℰ0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='05 μ/ℰ0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='25 μ/ℰ0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='60 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (a) Behavior of the Berry curvature dipole obtained by changing the chemical potential µ measured in units of E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The different curves correspond to the different values of αR measured in units of E0/k0 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The other model parameters have been instead fixed as ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2E0, ∆m = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='01E0, αm = αR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (b),(c),(d),(e),(f) display the Fermi lines and the band-resolved occupied regions in momentum space for αR = αm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5E0/k0 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' ics [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Electrons in solids can carry information by exploit- ing either their intrinsic spin or their orbital angular momen- tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Generation, detection and manipulation of information using the electron spin is at the basis of spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The Berry curvature distribution we have unveiled in our study is ex- pected to trigger also an orbital Hall effect, whose origin is rooted in the geometric properties of the electronic wavefunc- tions, and can be manipulated using the orbital degrees of free- dom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This opens a number of possibilities for orbitronic de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This is even more relevant considering that our findings can be applied to a wide class of materials whose electronic properties can be described with an effective L = 1 orbital multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' These include other complex oxide heterointerfaces as well as spin-orbit free semiconductors where p-orbitals can be exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Since Dirac quasiparticles are not required in the orbital design of Berry curvature, it is possible to reach carrier densities large enough to potentially exploit electron- electron and electron-phonon interactions effects in the con- trol of Berry curvature-mediated effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' For instance, orbital selective metal-insulator transitions can be used to switch on and off the electronic transport channels responsible for the Berry curvature and its dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We envision that this capabil- ity can be used to design orbitronic and electronic transistors relying on the geometry of the quantum wavefunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' METHODS Representation of the Gell-Mann matrices in the symmetry groups Apart from the identity matrix Λ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' the eight Gell-Mann matri- ces can be defined as Λ1 = � � 0 1 0 1 0 0 0 0 0 � �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Λ2 = � � 0 −i 0 i 0 0 0 0 0 � �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Λ3 = � � 1 0 0 0 −1 0 0 0 0 � �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Λ4 = � � 0 0 1 0 0 0 1 0 0 � �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Λ5 = � � 0 0 −i 0 0 0 i 0 0 � �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Λ6 = � � 0 0 0 0 0 1 0 1 0 � �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Λ7 = � � 0 0 0 0 0 −i 0 i 0 � �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Λ8 = � � � 1 √ 3 0 0 0 1 √ 3 0 0 0 −2 √ 3 � � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Let us now check the properties of these eight Gell-Mann ma- trices under time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Since we are consider- ing electrons that are effectively spinless due to the SU(2) spin symmetry, the time-reversal operator can be represented as K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Hence, the three Gell-Mann matrices Λ2,Λ5,Λ7 are odd under time-reversal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' T−1Λ2,5,7T = −Λ2,5,7, whereas the remaining matrices are even under time-reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Sim- ilarly, Λ1,2,3,8 are even under the vertical mirror symmetry whereas Λ4,5,6,7 are odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Let us finally talk about the three- fold rotational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Since the rotation symmetry oper- ator C3 = exp[2πiΛ7/3], the transformation properties of the Gell-Mann matrices are determined by the commutation rela- 9 tions [Λ7,Λi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The commutation relations are listed as follows: [Λ7,Λ1] = iΛ4 [Λ7,Λ2] = iΛ5 [Λ7,Λ4] = −iΛ1 [Λ7,Λ5] = −iΛ2 [Λ7,Λ6] = 2i � Λ3 2 − √ 3 2 Λ8 � � Λ7, Λ3 2 − √ 3 2 Λ8 � = −2iΛ6 � Λ7,Λ3 + Λ8 √ 3 � = 0 The results above indicate that the three pairs of operators {Λ1,Λ4}, {Λ2,Λ5}, and � Λ6, Λ3 2 − √ 3 2 Λ8 � behave as vector under the threefold rotation symmetry and therefore form two- dimensional IRREPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Berry curvature of SU(2) and SU(3) systems For SU(2) systems, a generic Hamiltonian can be written in terms of Pauli matrices σi as H(k) = d0(k)σ0 + d(k) · σσσ, where σ0 is the 2×2 identity matrix and the Pauli matrix vec- tor σσσ = (σx,σyσz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The Berry curvature can be expressed in terms of d vector Ω±(k) = ∓ 1 2|d(k)|3 d(k)·[∂kxd(k)×∂kyd(k)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (7) For SU(3) system, we can proceed analogously using the Gell- Mann matrices introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The Hamiltonian of a sys- tem described by three electronic degrees of freedom in a 3×3 manifold can be written as H(k) = b0(k)Λ0 +b(k)·ΛΛΛ, where b0(k) is a scalar and b(k) is an eight dimensional vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The Gell-Mann matrices satisfy an algebra which is a generaliza- tion of the SU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In particular, we have that ΛaΛb = 2 3δab +(dabc +ifabc)Λc, (8) where repeated indices are summed over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In the equation above, we have introduced the antisymmetric and symmetric structure factors of SU(3) that are defined respectively as fabc = − i 4Tr([Λa,Λb]Λc) , dabc = 1 4Tr({Λa,Λb}Λc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' From these one defines three bilinear operations of SU(3) vec- tors: the dot (scalar) product v · w = vawa, the cross product (v×w)a = fabcvbwc, and the star product (v⋆w)a = dabcvbwc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The star product is a symmetric vector product which does not play any role for SU(2) since dabc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Moreover, the band- resolved Berry curvature is given by [49, 64]: Ωn(k) = −4(γk,nbk +bk ⋆bk) (3γ2 k,n −|bk|2)3 ·{[γk,n∂kxbk+ (9) + ∂kx(bk ⋆bk)]×[γk,n∂kybk +∂ky(bk ⋆bk)] � where we introduced γk,n = 2 √ 3|bk|cos � θk + 2π 3 n � , θk = 1 3 arccos � √ 3bk·(bk⋆bk) |bk|3 � , and bk is a shorthand for b(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Gener- ally speaking, the Berry curvature in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (7) can be split in two contributions as Ωn(k) = Ω(0) n (k) + Ω(⋆) n (k), with Ω(0) n (k) = −4 (γk,n)3 (3γ2 k,n−|bk|2)3 bk ·[∂kxbk ×∂kybk] that strongly resembles the BC expression for SU(2) systems of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In trigonal sys- tems described by the effective Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (4) we have that for any momentum k and for any value of the pa- rameters, the b vector associated with the Hamiltonian is such that (γk,nbk +bk ⋆bk) is always orthogonal to the vector in the curly braces in the expression of the Berry curvature Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Hence Ωn(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' On the contrary, assuming a Cs point-group symmetry the effective Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (5) de- fines bk = � 0,αRky,∆+ 1 2∆m,0,αRkx,0,αmkx, ∆ √ 3 − √ 3 2 ∆m � , for which we get that Ω(0) n (k) = 0, and the BC is substantially given by Ω(⋆) n (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In other words, the terms obtained by do- ing the star product, bk ⋆ bk are those that yield the non-zero BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We point out that the BC is proportional to the combi- nation of parameters α2 R αm (2∆+∆m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' A non-vanishing BC can be thus obtained even in the absence of the Gell-Mann matrix Λ8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This, on the other hand, would correspond to val- ues of the crystal field splitting ∆m = 2∆/3 implying a very strong distortion of the crystal from the trigonal arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The presence of the constant term ∝ Λ8 is thus essential to de- scribe systems with a parent high-temperature trigonal crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Calculation of the Berry curvature dipole The first moment of the Berry curvature, the Berry cur- vature dipole, for each energy band n is given by Dx,n = � d2k (2π)2 ∂kxΩn(k)f0(k), where f0(k) is the equilibrium Fermi- Dirac distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' At zero temperature, this expres- sion can be rewritten as a line integral over the Fermi line Dx,n = � d2k (2π)2 Ωn(k)∂En ∂kx δ(En − µ) , (10) where En = En(k) (n = 1,2,3) are the energy bands and µ is the chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We have used the latter expression (10) to evaluate the BCD, where Dx = ∑3 n=1 Dx,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Competing interests: The authors declare no competing fi- nancial or non-financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Author contributions: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' conceived and supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' M.' metadata={'source': 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Keimer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Shukla, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Strempfer, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Bernhard, “Electric-field-induced polar order and localiza- tion of the confined electrons in LaAlO3/SrTiO3 heterostruc- tures,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 110, 136805 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' [64] Ansgar Graf and Fr´ed´eric Pi´echon, “Berry curvature and quan- tum metric in N-band systems: An eigenprojector approach,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' B 104, 085114 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' [65] The effective time-reversal symmetry would imply that the BC should be an odd function of momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' [66] Dongwook Go, Daegeun Jo, Hyun-Woo Lee, Mathias Kl¨aui, and Yuriy Mokrousov, “Orbitronics: Orbital currents in solids,” EPL (Europhysics Letters) 135, 37001 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Supplemental Material for: “Orbital design of Berry curvature: pinch points and giant dipoles induced by crystal fields” Maria Teresa Mercaldo,1 Canio Noce,1, 2 Andrea D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Caviglia,3 Mario Cuoco,2, 1 and Carmine Ortix1 1Dipartimento di Fisica “E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Caianiello”, Universit`a di Salerno, IT-84084 Fisciano (SA), Italy 2SPIN-CNR, IT-84084 Fisciano (SA), Italy 3Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest Ansermet, CH-1211 Geneva, Switzerland I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' BERRY CURVATURE PROPERTIES In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 1 we show the connection between the presence of mirror-symmetry protected crossings and the Berry curvature pinch- points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This is further highlighted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 2 where we display the Berry curvature distribution associated with the effective k·p theory model close to a mirror symmetry protected degeneracy and reading (see the main text) He f f = vxkxσx +βkxδkyσy +vyδkyσz, (1) where δky is the momentum measured relatively to the mirror symmetry-protected degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3 illustrates the relation between the occurrence of the Berry curvature hot-spots and the energetic distance between the |a1g⟩ and the crystal field split |e′ g⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 4 we show for comparison the Berry curvature of a Rashba two- dimensional electron gas with crystalline anisotropy terms that are cubic in momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The corresponding Hamiltonian reads: H2DEG = ¯h2k2 2m σ0 +αSR (kxσy −kyσx)+ λ 2 � k3 + +k3 − � σz (2) where k± = kx ±iky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' a Energy dispersion of the orbital model along the mirror symmetric line of the BZ kx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' At finite ky there are two mirror symmetry- protected crossings b Contour plot and c tridimensional plot of the band-resolved Berry curvature corresponding to the second band Ω2(k) in the region close to the mirror-symmetry protected crossing where the BC pinch point is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We have used the same set of parameters of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3 in the main article, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2E0, ∆m = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='01E0, and αR = αm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0E0/k0 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='04548v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='mes-hall] 11 Jan 2023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 Q22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='14 Q22 >200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='12 200 100 QF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='10 100 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='06 200 /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='04 kx=0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='04 <-200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 kx/ k?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='04 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='04 kx/ k?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' b a c2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Density plot of the Berry curvature corresponding to the effective two-band model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 1 with a pinch point at zero momentum assuming vx = vy = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' n=3 n=2 n=1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 kx/kF 0 En/ℰ0 ky=0 αR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 αR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 kx/kF 0 (E2-E1)/ℰ0 ky=0 αR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 50 0 50 kx/kF 0 Ω1 ky=0 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ kx where (E2-E1) is minimum ■ kx where Ω1 is maximum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='35 αR k x/kF 0 a b c d FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (a) Energy dispersion along the ky = 0 line for ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2E0, ∆m = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='01E0, αR = αm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0E0/k0 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (b) The corresponding behavior of the energy difference between the first two bands E2 − E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We show also other values of the interorbital mixing parameter αR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (c) The local behavior of the Berry curvature of the first band Ω1 for different values of αR shows that large enhancements of the Berry curvature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' the hot-spots, are strongly related to the E2 − E1 minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This is further shown in (d) where we compare the location in k space of the BC hot-spots and the E2 −E1 minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Density plot of the Berry curvature for a spin-orbit coupled electron gas with warping terms cubic in momentum with effective Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Here we have chosen αSR = 1 and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' RELATION BETWEEN VAN HOVE SINGULARITIES AND PROPERTIES OF THE BERRY CURVATURE DIPOLE As mentioned in the main text, the cusps and inflection points in the Berry curvature dipole are direct consequence of Lifshitz transitions and their associated van Hove singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We recall that the generic sequence of Lifshitz transition in our model is as follows (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 of the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' First, there are two “small” electron pockets centered at finite momentum that then merge to create a single electron pocket (centered at Γ) delimited by two “concentric” Fermi lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Note that this Fermiology is strongly reminiscent of a conventional isotropic Rashba two-dimensional electron gas in the low-density regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The internal Fermi line then collapses at zero momentum and a single Fermi line survives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Such intraband Lifshitz transitions are then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 <-4 Kx()+U 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 y/y 0 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 2 2 0 2 1 1 kx/kg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='43 followed by the interband Lifshitz transition due to the appearance of two pockets of the second band that then merge with the almost concomitant appearance of a tiny pocket of the third (highest-in-energy) band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 5 all these Lifshitz transitions are characterized by van Hove singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Indeed, at all Lifshitz transitions there is a divergence in the slope of the density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We point out that at the first intraband Lifshitz transition, the singularities due to the saddle points of the energy levels are not integrable and a divergence in the density of states occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This is because the saddle points are distributed over lines of the Brillouin zone – similarly to the zero-energy van Hove divergence of a Rashba two-dimensional electron gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Importantly, while the cusps and inflection points of the Berry curvature dipole can be all related to Lifshitz transitions (and hence van Hove singularities), the opposite does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The direct comparison between Berry curvature dipole and density of states [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 5] indeed shows that the divergence of the density of states is accompanied by a featureless Berry curvature dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Note that the features of the Berry curvature dipole are preserved by changing the interorbital mixing parameter αR as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ n=1 ■ n=2 ◆ n=3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 2 1 0 1 2 gn(E) E/ℰ0 ●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ■ ■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆ ◆ ◆ ◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ n=1 ■ n=2 ◆ n=3 2 1 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 μ/ℰ0 Dx,n a b FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' (a) Band-resolved density of states gn(E) and (b) Berry curvature dipole for ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2E0, ∆m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='01E0, αOR = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0E0/k0 F, αm = αOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In panel (a) the vertical lines indicate the various Lifshitz transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ■ ■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆ ◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ n=1 ■ n=2 ◆ n=3 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='10 μ/ℰ0 Dx,n kF 0 αR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ■ ■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ n=1 ■ n=2 ◆ n=3 2 1 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 μ/ℰ0 Dx,n kF 0 αR=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ■ ■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ n=1 ■ n=2 ◆ n=3 2 1 0 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 μ/ℰ0 Dx,n kF 0 αR=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ■ ■ ■ ■ ■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆ ◆ ◆ ◆◆ ◆ ◆ ◆ ◆ ◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ n=1 ■ n=2 ◆ n=3 2 1 0 1 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 μ/ℰ0 Dx,n kF 0 αR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 a b c d FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The band-resolved Berry curvature dipole for different values of the interorbital mixing parameter αR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The other parameters are the same as in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 4 a g eg eg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content="2 Δ1/ℰ0 En(Γ)/ℰ0 Δm Δ1,c ' ' 1 FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Behavior of the energy levels of the a1g and e′g doublet (split by the crystal field ∆m) as a function of the ∆1 mixing term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The critical value ∆1,c where a level crossing is obtained is much larger than ∆m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='01E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The trigonal crystal field splitting ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2E0 as in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' EFFECT OF A CONSTANT Λ1 TERM As mentioned in the main text, reducing the point-group symmetry from C3v to Cs makes a constant term in Λ1 symmetry allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This term is not present in our model Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' It is important to note that such term yields a mixing between two states with equal Mx mirror eigenvalue (for p orbitals these correspond to the |pz⟩ and |py⟩ states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Consequently, a constant Λ1 term lifts the energy degeneracy at the center of the BZ of the e′ g (|px⟩, |py⟩) states precisely as the constant term in Λ3/2− √ 3Λ8/2 does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' However, there is an important difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The constant Λ3/2 − √ 3Λ8/2 term, appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 of the main text, corresponds to a direct splitting between the two states of the doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' On the contrary, the splitting due to the constant Λ1 term is suppressed by the energy difference – caused by the trigonal crystal field – between the a1g (|pz⟩) state and the e′ g doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Since the strengths ∆1,∆m of the constant terms in Λ1 and Λ3/2 − √ 3Λ8/2 are expected to have the same order of magnitude (both terms are proportional to the degree of lattice distortion from C3v) one finds that the constant term ∝ Λ1 has a much weaker effect on the electronic properties: the band structure and the behavior of the Berry curvature are essentially determined by ∆m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In particular, neither the existence of Berry curvature hot-spots nor the pinch points are affected by ∆1, in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' To illustrate this point, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 7 the behavior of the energies (at the Γ point) of the three a1g and e′ g states by increasing the strength of the constant Λ1 term at fixed value of the crystal field splittings ∆,∆m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' One finds that a non-zero ∆1 is able to change the level ordering – and thus have a dramatic effect on the band structure – only for ∆1 ≫ ∆m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Even though this regime is not realistic, we have explored it for completeness of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='8 we display the behavior of the energy bands on the ky ≡ 0 line and the mirror symmetric line kx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' On the latter, one clearly sees that at all momenta the bands are non-degenerate, and the mirror symmetry-protected crossings discussed in the main text are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This is because the constant term ∝ Λ1 moves the crossings towards the center of the BZ until (for ∆1 = ∆1,c of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='7) they merge and annihilate each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 kx/kF 0 En/ℰ0 ky=0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 ky/kF 0 En/ℰ0 kx=0 a b FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Energy bands of the model Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 of the main text including a (large) constant term ∝ Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Here we have used a trigonal crystal field ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2E0, the additional crystal field splitting due to rotational symmetry breaking ∆m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='01E0, and the strengths of the orbital Rashba coupling αOR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0E0/k0 F, αm = αOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' For the constant term ∝ Λ1 we considered a strength ∆1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='15E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 5 This qualitative difference in the electronic bands is reflected in different characteristics of the Berry curvature whose behavior, in the ∆1 ≫ ∆m regime, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The Berry curvature is characterized by hot-spots with a dipolar profile, as mandated by time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' However, due to the absence of mirror symmetry-protected crossings, the Berry curvature pinch points cannot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We note that the presence of hot-spots guarantees values of the Berry curvature dipole [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 10] of the order of inverse of the Fermi wavevector k0 F, and thus of the same order of magnitude of the ∆1 ≡ 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' This proves that the size of the Berry curvature dipole is a robust feature of our symmetry-based model Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Density plot of the Berry curvature for the model Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 of the main text supplemented by a (large) constant term ∝ Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' The annihilation of the mirror symmetry-protected crossings results in the absence of pinch points in the Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' We have used the same parameter set of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Δ1/ℰ0 0 ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='15 1 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 μ/ℰ0 DxkF 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Behavior of the Berry curvature dipole as a function of the chemical potential for the model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='5 of the main text and a generalized model which includes the constant term ∝ Λ1 with a strength ∆1 ≫ ∆m and that of the orbital Rashba coupling αR = αm = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0E0/k0 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' All other values of the model parameters are same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' ADDITIONAL FIGURES Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 11 shows a comparison of the behavior of the Berry curvature dipole as a function of the chemical potential by changing the relative strength between the three-fold rotational symmetric interorbital mixing ∝ αR and the term ∝ Λ7 only allowed in the absence of rotational symmetries parametrized by αm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Finally Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 12 shows the local Berry curvature and the Berry curvature dipole density for a set of parameters different from the one reported in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 21(k) 22(k) 23(k) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4F 15 10 15 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 ky/kg 5 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 5 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 5 10 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 ¥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 10 15 15 kx/ kg kx/k?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' kxlkg b a c6 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ αm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='3 ■ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 2 1 0 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='4 μ/ℰ0 DxkF 0 αR=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 a 2 -1 0 1 2 2 1 0 1 2 kx/kF 0 ky/kF 0 2 -1 0 1 2 2 1 0 1 2 kx/kF 0 ky/kF 0 2 -1 0 1 2 2 1 0 1 2 kx/kF 0 ky/kF 0 2 -1 0 1 2 2 1 0 1 2 kx/kF 0 ky/kF 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2 kx/kF 0 ky/kF 0 2 -1 0 1 2 2 1 0 1 2 kx/kF 0 ky/kF 0 b c d e f g μ/ℰ0=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='00 μ/ℰ0=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='50 μ/ℰ0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='05 μ/ℰ0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='15 μ/ℰ0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='60 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' a Berry curvature dipole as a function of µ for two values of αm (namely αm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='3E0/k0 F and αm = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0E0/k0 F) and for αR = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0E0/k0 F, ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2E0 and ∆m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='01E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' b,c,d,e,f display the Fermi lines and the band-resolved occupied regions in momentum space for αR = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='0E0/k0 F and αm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='3E0/k0 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Note that the absence of the Berry curvature dipole peak at negative values of the chemical potential for αm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='3E0/k0 F is consistent with the fact that the Fermi lines retain an almost symmetric profile in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content=' Density plots of Berry curvature and Berry curvature dipole for ∆ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='2E0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='∆M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etE3T4oBgHgl3EQffAo5/content/2301.04548v1.pdf'} +page_content='12E0,αR = 1.' metadata={'source': 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Email: morlokan@uni-muenster.de, +Tel. +49-251-83-39069 +Stephan Klemme, Institut für Mineralogie, Corrensstraße 24, 48149 Münster, +Germany. Email: stephan.klemme@uni-muenster.de +Iris Weber, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 Münster, +Germany. Email: sonderm@uni-muenster.de +Aleksandra Stojic, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 +Münster, Germany. Email: a.stojic@uni-muenster.de; +Martin Sohn, Hochschule Emden/Leer, Constantiaplatz 4, 26723 Emden, +Germany, Email: martin.sohn@hs-emden-leer.de +Harald Hiesinger, Institut für Planetologie, Wilhelm-Klemm-Str. 10, 48149 +Münster, Germany. Email: hiesinger@uni-muenster.de + +© 2017 This manuscript version is made available under the CC-BY-NC-ND 4.0 + + + + + + + + + + +2 + +Abstract +In a study to provide ground-truth data for mid-infrared observations of the surface of Mercury with +the MERTIS (Mercury Radiometer and Thermal Infrared Spectrometer) instrument onboard the +ESA/JAXA BepiColombo mission, we have studied 17 synthetic glasses. These samples have the +chemical compositions of characteristic Hermean surface areas based on MESSENGER data. +The samples have been characterized using optical microscopy, EMPA and Raman spectroscopy. Mid- +infrared spectra have been obtained from polished thin sections using Micro-FTIR, and of powdered +size fractions of bulk material (0-25, 25-63, 93-125 and 125-250 μm) in the 2.5-18 µm range. +The synthetic glasses display mostly spectra typical for amorphous materials with a dominating, +single Reststrahlen Band (RB) at 9.5 µm - 10.7 µm. RB Features of crystalline forsterite are found in +some cases at 9.5-10.2 µm, 10.4-11.2 µm, and at 11.9 µm. Dendritic crystallization starts at a MgO +content higher than 23 wt.% MgO. +The Reststrahlen Bands, Christiansen Features (CF), and Transparency Features (TF) shift depending +on the SiO2 and MgO contents. Also a shift of the Christiansen Feature of the glasses compared with +the SCFM (SiO2/(SiO2+CaO+FeO+MgO)) index is observed. This shift could potentially help distinguish +crystalline and amorphous material in remote sensing data. A comparison between the degree of +polymerization of the glass and the width of the characteristic strong silicate feature shows a weak +positive correlation. +A comparison with a high-quality mid-IR spectrum of Mercury shows some moderate similarity to the +results of this study, but does not explain all features. + + + + + + +3 + +1. Introduction +Infrared spectroscopy allows determining the mineralogical composition of planetary surfaces via +remote sensing. The Mercury Radiometer and Thermal Infrared Spectrometer (MERTIS) +spectrometer onboard the future ESA/JAXA BepiColombo mission to Mercury will allow such remote +sensing observations by mapping spectral features of the Hermean surface in the 7-14 µm range, +with a spatial resolution of ~500 m (Benkhoff et al., 2010; Hiesinger et al., 2010). In order to correctly +interpret remote sensing data, laboratory spectra of suitable analog material are of vital importance +(Helbert et al., 2007; Maturilli et al., 2008). The IRIS (InfraRed and Raman for Interplanetary +Spectroscopy) laboratory in Münster therefore generates spectra from analog material similar to +those materials expected to occur on the surface of Mercury. + +Surface regolith and exposed rocks of terrestrial planets and their moons are modified by impact +events throughout their lifetimes (Hörz and Cintala, 1997). The investigation of how these related +processes affect the spectral properties of the rocks is important for the correct interpretation of +infrared data from planetary bodies. Higher impact shock, for example, results in amorphous phases +produced in solid state transformation (such as maskelynite), or melt glass (e.g., Stöffler, 1966; +Wünnemann et al., 2008; Osinski and Pierrazo, 2012; Jaret et al., 2015a). Under shock metamorphic +conditions, minerals transform from crystalline to a solid amorphous state including diaplectic +glasses like maskelynite at pressures of ~25 - ~40 GPa. Melting of feldspar starts at ~35 to ~45 GPa. +Over 60 GPa rocks melt completely, which may result in quenched melt glass (e.g., Stöffler, 1966, +1971, 1984; Chao, 1967; von Engelhardt and Stöffler, 1968; Stöffler and Langenhorst, 1994; French, +1998; Johnson, 2012). Impact glass lacks a far-range order of its atomic constituents and represents +the amorphous building block of a material, typically generated in events involving high shock +pressure and temperatures (French, 1998; Speck et al., 2011). +In our study, we present the first mid-infrared reflectance data for synthetic glasses as analogs for +melt glass based on the respective chemical compositions derived from remote sensing and model + +4 + +data for the surface of Mercury. Using synthetic materials allows us to produce more realistic analogs +for Mercury surface rocks. To date, there are no Hermean meteorites we know of (Weber et al., +2016; Goodrich et al., 2017), chemical remote sensing data based on X-ray Spectrometer (XRS) and +the Gamma-Ray and Neutron Spectrometer (GNRS) is the best information source available so far to +deduce the surface composition of Mercury (Weider et al., 2015; vander Kaaden et al., 2016). We +expect the surface rocks and regolith not only to consist of glassy material, a mixture of components +of various shock stages seems more likely. Every respective shock stage will have different spectral +characteristics (e.g. Morlok et al., 2016a, 2016b, 2017). Therefore, the spectra of the synthetic +glasses produced in this study will serve as the endmember for studies of glass mixtures where +amorphous and crystalline components are mixed to varying degrees. Also, areas that underwent +more recent volcanism could be less affected by impact alteration. Such areas comprise large areas +of the Mercurian surface (e.g. They could provide crystalline minerals (e.g. Deneva et al., 2013; +Goudge et al., 2014; vander Kaaden et al., 2016). +To produce the synthetic glasses, we use the average chemical composition of surface regions +identified in the MESSENGER data Compositions G1 and G2 (Charlier et al., 2013), which are the +average compositions for larger areas in the equatorial region and the southern hemisphere of +Mercury. They are distinguished by their variation in the Ca and Al contents (Tab.1) and cover both +high-reflectance volcanic plains and low-reflectance rocks. Stockstill-Cahill et al. (2013) and Weider et +al. (2012) present average compositions for the Mg poor, alkali-rich northern volcanic plains (NVPa) +area, and the Mg-rich intercrater plains and heavily cratered terrain (IcP-HCTa) (Tab.1). Peplowski et +al. (2015) present compositions of the high-Al and low-Mg Interior plains (CBC), i.e., the area inside +the young Caloris impact crater and the low-Al and Mg-Northern Terrane (NC), i.e., northernmost +part of Mercury above 60° northern latitude. Further areas are High-Mg Terranes (HMC) and an +Intermediate composition (IC) (Tab.1). +Compositional data presented in vander Kaaden et al. (2015) and Weider et al. (2015) are the high- +Mg (and low Al) region (HMR) and a sub region of the HMR with Ca and S enrichments (HMR-CaS), +the low-Mg plains of the Caloris basin (CB), Mg-rich and poor parts of the northern volcanic plains + +5 + +(NP-HMg; NP-LMg, respectively), the Rachmaninoff basin (RaB), an area with high-Mg near the high- +Al northern plains (HAl), a large pyroclastic deposit (PD), and the average of inter crater plains and +various, cratered terrains (IT) (Weider et al., 2015). +We analyze four size fractions (0-25, 25-63, 63-125, 125-250 µm), motivated to better account for +the high porosity and large grain size variations of surface regolith. Variation in grain size causes +changes in the intensity of the characteristic Reststrahlen Bands (RB), fundamental mode absorption +features in the 7-14 µm region, resulting in a loss of spectral contrast with decreasing grain size. An +earlier study in the visible and near-infrared range (Sprague et al., 2007) indicated a high abundance +of grains smaller 30 µm in size comprising the surface regolith on Mercury. Therefore, the +corresponding RBs are expected to be weak in the remote sensing data of Mercury (Salisbury and +Eastes, 1985; Salisbury and Wald, 1992; Mustard and Hayes, 1997). In addition, the transparency +feature (TF), a characteristic additional spectral feature for small grain sizes, appears around 11-13 +µm in the smallest grain size fractions below 50 µm (e.g., Salisbury, 1993). Potential TF features have +been observed in ground based infrared observation of Mercury, indicating a high abundance of such +fine-grained material in the regolith. This motivates the need for spectral data especially of the fine- +grained size fractions (e.g., Cooper et al., 2001; Sprague et al., 2007). +Earlier reflectance and emission studies in the mid-infrared of synthetic glasses as analogues +for impact melt glass were made by Byrnes et al. (2007) and Lee et al. (2010). They analyzed +synthetic quartzofeldspathic glasses and found correlations between the band positions of +characteristic dominant features and SiO2 contents or Si/O ratios. Comparable results for synthetic +glass with basaltic (low SiO2) to intermediate (high SiO2) composition were obtained by DuFresne et +al. (2009), Minitti et al. (2002), and Minitti and Hamilton (2010). McMillan and Piriou (1982), Speck et +al. (2011) and King et al (2004) provide additional overview of the infrared properties of silicate glass. +Glasses from laser pulse experiments with a Martian soil analog JSC Mars-1 were analyzed by +Basilevsky et al. (2000), Moroz et al. (2009), and Morris et al. (2000), resulting in spectra dominated +by a strong single band in the 9.2-10.5 µm wavelength range. Earlier reflectance and emission studies +of natural impact melt glass formed during impacts in the mid-infrared were performed by Thomson + +6 + +and Schultz (2002), Gucsik et al. (2004), Faulques et al. (2001), Fröhlich et al. (2013) and Morlok et al. +(2016a and b). Spectra of these samples are dominated by a broad RB in the 8.9-10.3 µm range, with +only few other features in the mid-infrared. A complementary study of silicate glasses with +Mercurian and other planetary compositions was made by Cannon et al. (2016). + +2. Samples and Techniques +2.1 Sample Compositions and Preparation of Glasses +The respective chemical composition used for the synthetic glass analogs of surface areas on +Mercury are based on Charlier et al., 2013 (Char), Stockstill-Cahill et al., 2013 (Stock), Peplowski et +al., 2015 (Pep), and vander Kaaden et al., 2015 (VdK) (comparable to those in vander Kaaden et al., +2016). The various studies and models did not always present the same range of oxide components, +we therefore limited the composition used in this study to SiO2, TiO2, Al2O3, Fe2O3, MgO, CaCO3, +Na2O, and K2O for better comparability. Components below 0.5 wt.% were omitted for individual +mixtures for simplification. Starting material compositions are given in Table 1. +The oxide and carbonate starting mixtures were finely ground to a powder in an agate mortar under +acetone and then dried. The resulting mixtures were placed in medium sized Pt crucibles in which +they were slowly heated to 1000°C to de-carbonate. Subsequently, the mixtures were heated and +melted in a conventional box furnace at 1450°C for 2h. They were quenched immediately after +complete liquefaction, the crucibles were swiftly taken out of the furnace and submerged in +water. The samples were vitrified within 10 secs. +The samples were melted in a box furnace in air. Oxygen fugacity was not controlled. This may affect +phase equilibria slightly, but only when high amounts of Fe, the only redox sensitive major element, +are present. + +7 + +Our samples were melted at high temperatures and kept at temperature for several hours. The melts +are characterized by relatively low viscosity which ensures complete homogenization. Once melted, +the structure of the melt does not depend on the starting material. + +In two cases (ICP-HCTa (Stock) and RaB (VdK)), we re-heated the starting material at 1500°C in an +attempt to remove crystals, which have formed during quenching in the first procedure. +In order to prepare grain size fractions, bulk glass material was ground in steel and agate mortars. +The powder was cleaned in acetone and dry sieved for one hour to generate four size fractions: 0-25 +µm, 25-63 µm, 63-125 µm, and 125-250 µm, by using an automatic Retsch Tap Sieve. In order to +remove clinging fines, the larger two fractions were again cleaned in acetone. In addition, polished +thick sections were prepared for microscopic investigation from the pure glass sample. + +2.2. Optical Microscopy +Polarized light microscopy provides fast information about the crystalline or amorphous character of +the single components. It also enables first mineral identification in the samples (Fig. 1), which is +important for the subsequent Raman investigation to avoid mixed measurements on an +inhomogeneous sample location. The first overview images of all polished thick sections were +obtained with a KEYENCE Digital Microscope VHX-500F under normal light conditions and under +crossed polarizers. + +2.3 Raman Spectroscopy +All Raman measurements were conducted using an Ocean Optics IDR-Micro Raman system (IfP, +Münster), operating with an OneFocus optical system equipped with a 40 x objective. The laser +excitation is 532 nm and the spectral resolution is about 7 cm-1. Spectra were obtained with a laser + +8 + +power of 1.8 mW starting at wavenumbers around 200 cm-1. The spot size on the sample is +approximately 2 µm in diameter. Every spectrum is the result of one measurement at 15 seconds +acquisition time (Fig. 2a,b). +All spectra are automatically background and baseline subtracted and have not been smoothed. As +usual, all spectra are given with arbitrary units, because the height of a signal only corresponds to the +quality of the individual Raman scatterer itself (e.g. the double peak in olivine appears because the +two SiO4-stretching (v1 and v3) modes are active.) In addition, the Raman spectra of the glass are +affected by fluorescence, which adds intensity in the form of an underlying continuum. + +2.4 Electron Microprobe Analysis +Backscattered electron (BSE) images (Fig. 4a-c) show crystalline phases (brighter phases) formed +during quenching. Detailed quantitative analyses of the glass and the olivines crystallized during the +quenching process were made with a JEOL JXA-8530F Hyperprobe electron probe micro analyzer +(EPMA) equipped with five wavelength dispersive spectrometers (WDS) (Fig.3). For the glass +analyses, the probe was operated at an excitation voltage of 15 kV and a beam current of 5 nA. The +beam diameter was defocused to 5 µm. The counting time was 5 seconds on the peak and 2 seconds +on the background of each element, respectively. For mineral analyses we used an excitation voltage +of 15 kV and a beam current of 15 nA with a slightly defocused beam diameter of 2 µm. The counting +time for Mg, Al, Si, Ca, Fe, Ti, Cr, and Mn was 15 seconds on the peak and 5 seconds on the +background. And, in order to avoid loss of the volatile elements Na and K, the counting time was +reduced to 5 seconds on the peak and 2 seconds on the background for these two elements. The +following natural and synthetic minerals with well-known compositions were used as standards: +Jadeite (Na2O), SanCarlos Olivine (MgO), Disthene (Al2O3), Hypersthene (SiO2), Sanidine (K2O), +Diopside (CaO), Fayalite (FeO), Rutile (TiO2), Cr2O3 (Cr2O3), and Rhodonite (MnO). + + +9 + +2.5 Bi-directional Diffuse Reflectance FTIR +For the bi-directional analyses of the sieved bulk powder size fractions we used aluminum sample +cups with 1 cm diameter. The surface was gently flattened with a spatula following a procedure +analog to that described by Mustard and Hayes (1997). For the bulk powder analyses in the mid- +infrared from 2.5-18 µm, we used a Bruker Vertex 70 infrared system at IRIS laboratory. +We used a cooled MCT detector to ensure a high signal to noise ratio of the spectra. All analyses +were made under low pressure (10-3 bar). We accumulated 512 scans for each size fraction at al +spectral resolution of 0.02 µm (compared to 0.2 µm for MERTIS; Hiesinger et al., 2010). The machine +background was removed using a diffuse gold standard (INFRAGOLDTM). We obtained analyses with a +variable geometry stage (Bruker A513) in order to emulate various observational geometries of an +orbiter. The data presented in this study were obtained at 30° incidence (i) and 30° emergence angle +(e). +The spectra returned from MERTIS will be emissivity data. For the comparison with remote sensing +data in thermal infrared, reflectance and emission data have to be compared. This is usually done +using Kirchhoff’s law: ε = 1 – R (R=Reflectance, ε = Emission) (Nicodemus, 1965). For a direct +comparison using Kirchhoff’s law, the reflected light in all directions has to be collected. This relation +works best for the comparison of directional emissivity and directional hemispherical reflectance. In +our study, a bi-directional, variable mirror set-up was used, without a hemisphere integrating all +reflected light. This has to be kept in mind when comparing the results in a quantitative manner with +emission data (Salisbury et al., 1991; Hapke, 1993; Thomson and Salisbury, 1993; Salisbury et al., +1994; Christensen et al., 2001). +The spectral range of the MERTIS spectrometer is from 7-14 µm. Features at shorter and longer +wavelengths can be of interest for other studies, we therefore present powder spectra from 6-18 µm +(Fig. 5). The spectra are presented in reflectance, i.e., 0-1. + +10 + +The width of infrared bands was determined using the FWHM (Full Width at Half Maximum) of the +dominant spectral features. To derive this parameter from the often asymmetric, non-Gaussian +bands, we used the Origin software. For the fitting, GCAS (Gram-Charlier peak function) and CCE +(Chesler-Cram) fitting functions were applied to the spectra of the 125-250 µm size fraction, which +were normalized to the same intensity. +The spectra presented in this study are accessible via an online database at the Institut für +Planetologie in Münster (http://www.uni- +muenster.de/Planetology/en/ifp/ausstattung/iris_spectra_database.html), and the Berlin Emissivity +Database (BED). + +2.6 In Situ FTIR Microscope +For in situ analyses, we used a Bruker Hyperion 2000 IR microscope attached to the external port of a +Bruker Vertex 70v at the Hochschule Emden/Leer. Here we used a 256×256 µm2 sized aperture to +obtain analyses of small features with in situ reflectance spectroscopy on polished thin sections. For +each spectrum, 128 scans were integrated. A gold mirror was used for background calibration (Fig. +6a,b). + +3. Results +3.1 Optical Microscopy +The polished thick sections show a great diversity in color from nearly transparent samples, such as +NC (Pep) (Fig. 1), to a brownish glass such as PD (VdK) as the darkest endmember. Samples exhibiting +crystallites tend to be more brownish. + +11 + +Several samples show heterogeneity in the form of clearly identifiable crystals, which are embedded +in a still glassy matrix. These occur in the RaB (VdK) sample, while HMR (VdK), HMC (Pep), and HMR- +CaS (VdK) are examples displaying high crystallinity (Fig. 1). In these samples, olivine crystals were +identified by their habit and by their color in polarized light (Fig. 1). + +3.2 Raman +Raman spectra of the pure glasses show typical shifts, two main broad signals, which cannot be +attributed to single peaks, between 400 cm-1 - 700 cm-1 indicative of a silicate framework and +between 850 cm-1 - 1250 cm-1, which is indicative of tetrahedrally coordinated cations (Fig. 2a; +DiGenova et al., 2015). Raman shifts caused by glass are characterized by broad features lacking a +crystal structure. The spectra in Fig. 2a are displayed with lower Mg contents on top and increasingly +higher Mg contents. G2 (Char) is an outlier within this sequence, which might be a hint for a +transitional sample showing incipient crystallization. Detailed Raman spectroscopy on the more +heterogeneous samples, which include crystals confirms the existence of forsteritic olivine with a +typical Raman double peak (DB) at 823 cm-1 and 856 cm-1 (Fig. 2b, blue line; Chopelas, 1991). In +addition, the shift at around 700 cm-1 (Fig.2b, green line) in the spectra of HMC (Pep) can be +attributed to spinel (Downs, 2006). + +3.3 EMPA +Results of the detailed quantitative analyses of the glasses as well as the sample weight of the +starting oxides are given in Table 1. The analyses are average values of 20 – 70 measurements on +each glass depending on the amount of olivine crystals. Analyses of the crystallized olivines, including +their Fo-content, are listed in Table 2. The BSE image in Fig. 4a-c shows the dendritic growth of +olivine as a result of quenching. In addition, as can be seen in Tables 1 and 2, the formation of olivine + +12 + +during quenching decreases with the decreasing MgO content in the original sample weight. The +lower the MgO content the more identical are the quantitative analyses with the original values +(Fig.3). At a MgO content higher than 23 wt.% (Tab.1), more and more olivine crystals are visible in +the sample (Fig. 1). Also, chemical differences between the original sample (oxide mixture) and the +glass increase above this threshold (Fig.3). However, a few crystallites are already visible at lower +MgO contents like in Hal (VdK) (Fig.2a). +Normative calculations by vander Kaaden et al.(2016) also show high olivine contents for these +samples. +Furthermore, BSE images combined with WDS analyses give hints on critical chemical thresholds for +the formation of further crystalline phases. As visible in Figure 4c and 2b, spinel is another phase +inside of an olivine grain with remnants of MgO around the spinel rim. Spinel and remnants of MgO +occur in G1 (Char), HMR (VdK), HMR-CaS (VdK), and HMC (Pep). The threshold value for Mg for the +crystallization of spinel and MgO remnants are less than 9 wt.% Al and more than 25 wt.% Mg(Tab.1). + +3.4 Diffuse Reflectance FTIR +Spectra of the different size fractions are presented in the 6-18 µm range in Fig. 3, arranged +according to increasing Mg-contents in the starting materials (Fig. 5, 6, Tab. 3). Features below 6 µm +(2.9-3.5 µm) are either features of adsorbed volatiles on the starting materials or in the furnace and +not shown. +For better comparison, the spectra of the largest size fraction (125-250 µm) are presented together +in Fig. 6a. +A series of 12 samples with low Mg contents show spectra which are dominated by a single RB +between 9.5 µm (NP LMg VdK) and 10.7 µm (ICP-HCTa Stock). Endmembers for the CF are 7.9 µm (NC +Pep) and 8.3 µm (PD VdK), for the TF NP-LMg (12.4 µm VdK) with 11.8 µm and RaB (VdK) with 12.2 +µm (Fig. 5). One exception is sample G2 (Char), while the sample has the single RB characteristic for + +13 + +this group, the feature (10.2-10.6 µm) is clearly shifted towards longer wavelength in comparison +(Fig.6a). +The other exception among the low-Mg samples is NVPa (Stock), which shows a shoulder at 9.5-9.7 +µm and the main band at 10.5-10.7 µm (Tab.3). This spectrum was grouped with the remaining four +high-Mg samples HMR, HMR-CaS (VdK), HMC (Pep), and G1 (Char) that exhibit broader main RB +features and show clear crystalline features at 9.5-9.9 µm, 10.1 µm, 10.5 - 11.9 µm (Fig.5). These are +all typical forsterite bands (Hamilton, 2010). +The CF of these high-Mg samples ranges from 7.9 µm to 8.4 µm, and the TFs are found between 11.7 +and 12.3 µm. Additional weak bands are located between 16.2 and 16.7 µm (Tab.3). Of the original +experimental run, RaB (VdK) and ICP-HCTa (Stock) also exhibited crystalline features, which +disappeared after repeated melting/quenching at temperatures over 1700°C. + +3.5 Micro-FTIR +The FTIR spectra of the ‘pure’ glass samples are basically identical to those from the equivalent +powder analyses of the same samples (Fig. 6a; Tab. 4). We also attempted to obtain spectra of glassy +spots in samples showing crystallization (Fig. 6b; Tab. 4). +The CFs of pure glass and glass-spots range from 7.9 µm (NP-LMG; CBC and NC Pep) to 8.3 µm (G1, +Char), the main feature is between 9.6 µm (NP-LMG VdK) to 10.1 µm (HMR-CaS) (Tab.4). +Spots from individual crystals and high-Mg samples show more variation. Features with varying +intensity at ~9.6 µm, ~10.2 µm, 10.5-11.0µm, and 11.9 µm are typical olivine features (Hamilton, +2010; Lane et al., 2011), beginning with the HMR (VdK) sample (Tab.4). + +4. Discussion +4.1. Correlation of spectral and chemical features + +14 + +The aim of this study is to correlate spectral features of the laboratory analyses to remote sensing +data obtained from the surface of Mercury. The spectra from observations of Mercury available +often show low spectral contrast (e.g. Sprague et al., 1998). Therefore, it is useful to identify +potential characteristic features that allow us to derive mineralogical and chemical information from +remote sensing data even if the observational data is of comparatively low quality. +The CF, as a reflectance minimum, is a feature that can be easily identified in remote sensing data. +Hence, the band position of the CF is commonly used as a proxy for the bulk chemical composition of +samples, like the SiO2 content (Salisbury 1993). Analyses of bulk powders and micro-FTIR are basically +identical, and are consistent with the trend line for earlier studies (Morlok et al., 2016b) (Fig. 7). The +analyses from an earlier study on natural impact glass (Morlok et al., 2016b) show a much wider +variation from the trend line. This points towards a high purity of the synthetic material used in the +current study, in contrast to the natural materials, which often contained fragments of unshocked or +unmelted material (Morlok et al., 2016a and b). +If a strong RB is observable in remote sensing data, it also allows for characterizing the material (Fig. +8). Again, our band positions of the strong main RB features with respect to the SiO2 contents fall +close to observations in earlier studies (Lee et al., 2010). They demonstrate a better agreement than +the natural impact glasses from our earlier study (Morlok et al., 2016b). At lower SiO2 contents +(below 60 wt.%) the results of this study show a similar trend than those for basaltic glass from +DuFresne et al. (2009), where the trend line becomes much shallower. However, we cannot +distinguish amorphous from crystalline material with this type of comparison. +The SCFM index (SiO2/(SiO2+CaO+FeO+MgO)) is a way to determine the degree of polymerization of a +material, based on the increased interconnection of the SiO4 tetrahedra (Walter and Salisbury, 1989). +Earlier studies (Morlok et al., 2016b) have indicated that amorphous material like impact glasses plot +off the trend line for crystalline terrestrial material (Cooper et al., 2002). The synthetic glasses in this +study show a similar and in fact clearer behavior, continuing the trend line for basaltic compositions +(Fig. 9). + +15 + +A further way to correlate polymerization with spectral features is to use the bandwidth of the +dominating silicate feature in the glass with the ratio between network forming cations (Si and Al) +and network modifying anions (Fe, Ca, Mg, Na, K) (King et al., 2004; Dufresne et al., 2009; Speck et +al., 2011). A comparison of this ratio to the FWHM (Full Width at Half Maximum) of only the glassy +samples could help to obtain information about the degree of polymerization from remote sensing +data. However, the correlation between these two parameters of the glass in our study is low with a +correlation factor R (Pearson) of only 0.34 (Fig.10). + +4.2. Comparison with Astronomical Observations of Mercury +Earth based mid-infrared observations of Mercury integrate large surface areas of up to 106 km2 and +indicate a surface of mainly plagioclase with minor pyroxene (Donaldson-Hanna et al. 2007; Sprague +et al., 1994, 2000, 2002, 2007; Sprague and Roush, 1998; Emery et al., 1998,; Cooper et al., 2001). +We compare our results with an astronomical spectrum (Fig. 11), obtained by the Mid-Infrared Array +Camera (MIRAC) at the Kitt Peak Observatory. The spectrum is chosen due to its high signal to noise +ratio and strong features in the spectral range of MERTIS (7-14 µm). The observed spectrum is the +average of several observations of an area centered on ~210-250° longitude (Sprague et al., 2000). +The baselined spectrum, re-calculated from emissivity to reflectance, shows a CF-like feature at 8.5 +µm, strong RBs at 9.3 µm, 9.9 µm, and 11.0 µm, and a potential TF at 12.4 µm. Similarly, potential TFs +in the 12.0-12.7 µm region are also observed in various other spectra of Mercury (Cooper et al., +2001; Sprague et al., 2007). +A comparison with results from this study demonstrates that glass from the 125-250 µm fraction of +High Aluminium regions (HAl; VdK) and the 0-25 µm fraction of the High-Magnesium region HMC +(VdK) (which already shows abundant crystallites) are the best to be compared to (Fig. 11). +The glassy HAl (VdK) analog sample has a main RB at the same position as in the astronomical +spectrum (9.9 µm), although it is much broader. The HMC (VdK) analog sample has a TF at 12.3 µm, + +16 + +close to the band in the remote sensing spectrum (12.4 µm). Also, the CF is relatively close to the +astronomical data, which show the CF at 8.4 µm. The broad RB feature of the Hal (VdK) analog +sample around 11.2 µm is rather close to the (much narrower) 11.0 µm band in the Mercury +spectrum. No equivalent for the 9.3 µm feature in the astronomical spectrum was found in the +glasses of this study. This feature could indicate a pyroxene component (e.g., Hamilton, 2000; +Sprague et al., 2000) in the surface regolith of Mercury. +However, we compare here laboratory analyses of samples with very distinct compositions to those +of large surface areas of Mercury, effectively a ‘bulk’ spectra averaging many different mineral +species. Future remote sensing data of smaller, chemically and mineralogically homogeneous areas +of the surface of Mercury returned from the MERTIS instrument on BepiColombo (Benkhoff et al., +2010; Hiesinger et al., 2010) will allow to use the laboratory spectra much more efficiently. +Glass will probably mostly be contained in impactites and regolith, where it can be expected to be +part of a mixture with other, crystalline materials (Tompkins and Peters, 2010; Morlok et al., 2016a +and b). The large volcanic structures like smooth plains and pyroclastic deposits could be a source for +crystalline material, which will make the detection of glasses difficult (Denevi et al., 2013; Goudge et +al., 2014; vander Kaaden et al., 2016). +Visible and Near-Infrared studies of planetary glasses by Cannon et al. (2016) have shown that +identifying glass in spectral mixtures is challenging even at mass abundances up to 70%. Especially +mafic glasses are difficult to be identified ‘by eye’, but can be separated in spectral deconvolution. +Studies in the mid-infrared range by Ramsey and Christensen (1998) were able to reproduce mixtures +of crystalline minerals even for components below 10%. Still, this could lead to challenges identifying +the glass in the surface regolith of Mercury. Surface regolith studies show up to 45% agglutinitic +glass, much higher than 2-5% observed for glass on the lunar surface (Warrell et al., 2010). + +17 + +Also, space weathering has to be taken into account in this context, which can be expected much +more intense closer to the sun and also produce glassy materials (Papike et al., 1982; Hapke, 2001, +Sprague et al., 2007). + +5. Summary and Conclusions +The study of a series of synthetic glasses based on surface regions of Mercury displays mostly spectra +typical for amorphous materials, characterized by a single RB between 9.5 µm and 10.7 µm. Several +spectra show features of crystalline olivine species in both the powder and micro-FTIR spectra. +FTIR features of crystalline species at 9.5-10.2 µm, 10.4-11.2 µm, and at 11.9 µm in the powder and +micro-FTIR spectra are mostly characteristic of forsterite, appearing in a smaller number of samples +at higher MgO contents, beginning with the HMR (VdK) sample. Optical, Raman and EMPA data have +signs of crystallization starting with the RaB (VdK) material. This points towards a threshold for +crystallization starting at MgO contents higher than 23wt.% MgO. +However, in Raman spectra as well as in FTIR G2 (Char) is an outlier regarding the position of the +strongest feature. This might be indicative of a starting crystallization at a lower MgO concentration +in this sample. +The RBs, CFs, and TFs shift, depending on the SiO2 and MgO contents and exhibit similarity to earlier +studies. We also confirm a shift of the CF position of amorphous material compared with the +respective crystalline SCFM index, which could help distinguishing crystalline and amorphous +material in remote sensing data in the mid-IR. +A comparison between the degree of polymerization of the glass and the width of the characteristic +strong silicate feature shows only slight correlation. However, here sample heterogeneity – such as +incipient crystallization – probably affects the spectral features. + +18 + +A comparison with a high-quality mid-IR spectrum of Mercury shows some similarity to the results of +this study, but does not explain all features. A series of analyses of distinct size fractions is needed to +distinguish between genuine effects and effects induced by mixed grain size fractions in the natural +regolith. However, since regolith is also an intimate mixture of many phases, spectral unmixing +modelling will also have to take many other grain-size fractions of other potential mineral phases +into account. + +6. Acknowledgements +We thank Isabelle Dittmar (Emden) for analytical support, Ulla Heitmann (Münster) for thin section +preparation. This work was partly supported by DLR grant 50 QW 0901 in the framework of the +BepiColombo mission. + + + + + + + + + + + +19 + +References +Basilevsky A.T., Yakovlev O.I., Fisenko A.V., Semjonova L.F., Moroz L.V., Pieters C.M., Hiroi T., +Zinovieva N.G., Keller H.U., Semenova A.S., Barsukova L.D., Roshchina I.A., Galuzinskaya A.K. +Stroganov I.A. (2000) Simulation of Impact Melting Effect on optical Properties of Martian Regolith. +31st Annual Lunar and Planetary Science Conference, Abstract no. 1214. +Benkhoff J., van Casteren J., Hayakawa H. Fujimoto M., Laakso H., Novara M., Ferri P., Middleton H. +R., Ziethe R. 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(2015) Evidence for geochemical terranes on Mercury: Global mapping of major elements +with MESSENGER's X-Ray Spectrometer. Earth and Planetary Science Letters 416, 109-120. +Wünnemann K., Collins G. S., Osinski G. R. (2008) Numerical modelling of impact melt production in +porous rocks. Earth and Planetary Science Letters 269, 530-539. + + + + + + + + + + + +28 + +Figure Captions +Figure 1: Overview of optical images of polished blocks of the samples, arranged according to +increasing Mg contents. Increasing abundance of Mg is correlated with increasing appearance of +crystallites in the quenched melt glass. The sample for NVPa (Stock) was only available in powdered +form. Red squares are 256×256 µm sized areas analyzed with micro-FTIR. Red scale bars in the +images for NVPa (Stock), ICP-HCTa (Stock), and RaB (VdK) represent 256 µm. +Figure 2a: Raman spectra for the glasses, in the range from 300 cm-1 to 1300 cm-1 where the Mg- +abundances increase from the top to the bottom. The broad peak between 400 cm-1 and 700-1 is a +result from the extant silicate framework and between 850 cm-1 and 1250 cm-1 tetrahedra +coordinated cations are present. Intensity in graphs is normalized on the strongest signal. The +spectra are shifted vertically for a better view. +Figure 2b: Raman spectra for the crystalline phases formed during quenching in the range from 400 +cm-1 to 1600 cm-1. The blue line represents a typical olivine spectrum with the DB at 823 cm-1 and 856 +cm-1. Beyond this the green line shows an additional peak at 700 cm-1 fitting to spinel. Intensity in +graphs is normalized on the strongest signal. The spectra are shifted vertically for a better view. +Figure 3: Changes in main element abundances (SiO2, Al2O3, MgO, CaO, FeO) between starting +material (Initial) and quenched glass (EMPA data). Data is ordered by their increasing Mg content. +Most samples with lower Mg abundance show only slight variation. Only high-Mg samples that +produced crystallites show increasing differences. +Figure 4: Representative SEM BSE images of crystallites. +a) Example of the dendritic growth of olivine (Ol) as result of quenching in G1 (Char). +b) Example of the dendritic growth of olivine (Ol) as result of quenching in HAI (VdK). +c) Example of a spinel (light grey; Sp) as another present phase inside of an olivine grain (dark grey; +Ol) with remnants of MgO (white) around the spinel rim in G1 (Char). + +29 + + +Figure 5: FTIR reflectance spectra of powdered glass in four size fractions (0-25 µm, 25-63 µm, 63-125 +µm, 125-250 µm), sorted by increasing Mg-content. The crystalline features appearing at higher Mg- +contents are forsterite bands (Hamilton, 2010). +Figure 6a: Comparison of FTIR reflectance powder spectra of the 125-250 µm size fraction of all +samples. Mg-abundance is increasing from top to bottom. Vertical lines denote important features: +Christiansen Feature (CF), and Reststrahlenband (RB). A shift of these features to longer wavelengths +is correlated with increasing Mg abundance. Circle: olivine features in high-Mg samples (Hamilton, +2010). +Figure 6b: Micro-FTIR reflectance spectra of polished samples. Spectra are sorted by increasing Mg- +content. Vertical lines denote important features: Christiansen Feature (CF) and Reststrahlenband +(RB). A shift of these features to loner wavelengths is correlated with increasing Mg abundance. Left +column: Spectra of glass or glassy regions in samples showing crystallization. Right column: Spectra of +areas with high contents of crystallites. Vertical lines show typical forsterite RB bands. +Figure 7: SiO2 contents of glasses (in wt.%) compared with the position of the Christiansen Feature (in +µm). The samples of this study fall on the dashed trend line defined by earlier studies on impact +glasses (Morlok et al., 2016) and crystalline terrestrial rocks (Cooper et al., 2002). +Figure 8: SiO2 contents of glasses (in wt.%) compared with the position of the strongest Reststrahlen- +Band (in µm). Glasses from this study are similar to earlier studies (Cooper et al., 2002), but also +diverge from the trend lower SiO2 contents similar to basaltic glasses in DuFresne et al., 2009. +Figure 9: Comparison of the SCFM index (SiO2/(SiO2+CaO+FeO+MgO); Walter and Salisbury, 1989): +with the position of the CF (in µm). The results of the glasses in this study confirm the findings in +Morlok et al. (2016), where glassy material plotted below the trend line for crystalline materials. The +trend line is for analyses of terrestrial rocks (Cooper et al., 2002). + +30 + +Figure 10: Comparison of the Full Width at Half Maximum (FWHM) of the strong silicate feature in +glasses with the degree in polymerization, based on a ratio between network building ions (Si, Al) +and network modifiers (Fe, Ca, Mg, Na, K). +Figure 11: Comparison of a Mercury mid-infrared spectrum obtained by ground based observation +(Sprague et al., 2000) with the two most similar spectra from this study, a high-Al region (HAl VdK) +and the finest size fraction of a high Mg-region (HMC VdK). There is a general similarity for the CF and +TF as well as the RBs at 9.9 µm, 11.2 µm. There is not equivalent for the feature at 9.3 µm. + + + + +NP-LMg (VdK) +NC (Pep) +CBC (Pep) + +Glass +Standard +deviation +Start +Material +Glass +Standard +deviation +Start +Material +Glass +Standard +deviation +Start +Material +Na2O +4.65 +0.14 +4.65 +7.46 +0.21 +7.42 +3.65 +0.21 +3.72 +MgO +11.54 +0.23 +11.73 +11.65 +0.21 +12.00 +12.90 +0.39 +12.81 +Al2O3 +11.12 +0.21 +11.03 +11.60 +0.19 +11.48 +17.27 +0.24 +17.20 +SiO2 +63.34 +0.31 +63.62 +62.14 +0.44 +61.95 +58.60 +0.43 +59.00 +K2O +0.05 +0.03 +0.05 +0.04 +0.04 +0.03 +CaO +5.69 +0.20 +5.72 +5.54 +0.17 +5.67 +5.79 +0.20 +5.79 +FeO +1.81 +0.16 +1.76 +0.76 +0.13 +0.75 +0.81 +0.12 +0.70 +TiO2 +1.56 +0.22 +1.49 +0.60 +0.20 +0.58 +0.51 +0.17 +0.56 +Cr2O3 +0.02 +0.04 +0.02 +0.03 +0.02 +0.03 +MnO +0.02 +0.03 +0.02 +0.03 +0.15 +0.03 +0.04 +0.14 +Total +99.81 +100.00 +99.83 +100.00 +99.62 +100.00 + + +CB (VdK) +NVPa (Stock) +NP-HMg (VdK) + +Glass +Standard +deviation +Start +Material +Glass +Standard +deviation +Start +Material +Glass +Standard +deviation +Start +Material +Na2O +4.02 +0.16 +4.13 +0.06 +0.03 +5.95 +0.15 +6.18 +MgO +13.69 +0.32 +13.17 +16.09 +0.28 +16.51 +17.48 +0.36 +17.65 +Al2O3 +15.98 +0.22 +15.88 +15.42 +0.19 +15.38 +13.01 +0.23 +12.62 +SiO2 +58.13 +0.39 +59.35 +58.17 +0.32 +58.82 +57.01 +0.41 +57.00 +K2O +0.04 +0.04 +0.05 +0.02 +0.06 +0.03 +CaO +5.96 +0.26 +6.01 +4.91 +0.14 +4.84 +5.02 +0.22 +5.22 +FeO +1.09 +0.16 +0.99 +3.65 +0.21 +3.54 +0.09 +0.07 +TiO2 +0.67 +0.21 +0.46 +0.94 +0.20 +0.92 +1.14 +0.22 +1.33 +Cr2O3 +0.03 +0.03 + +0.01 +0.01 +0.02 +0.03 + +MnO +0.02 +0.03 + +0.01 +0.01 +0.03 +0.03 + +Total +99.64 +100.00 +99.31 + +100 +99.81 + +100.00 + + +G2 (Char) +HAI (VdK) +IC (Pep) + +Glass +Standard +deviation +Start +Material +Glass +Standard +deviation +Start +Material +Glass +Standard +deviation +Start +Material +Na2O +0.04 +0.04 +4.19 +0.16 +4.11 +3.39 +0.18 +3.44 +MgO +18.78 +0.30 +19.3 +18.73 +0.53 +19.42 +19.96 +0.40 +19.85 +Al2O3 +13.20 +0.23 +13.2 +15.94 +0.30 +15.79 +14.07 +0.21 +13.95 +SiO2 +56.15 +0.39 +56.6 +53.06 +0.52 +53.00 +53.87 +0.64 +54.46 +K2O +0.04 +0.03 +0.04 +0.03 +0.04 +0.03 +CaO +7.10 +0.22 +6.92 +6.52 +0.21 +6.43 +5.86 +0.18 +5.69 +FeO +3.49 +0.13 +3.37 +0.09 +0.07 +2.12 +0.17 +1.97 +TiO2 +0.68 +0.17 +0.59 +1.29 +0.32 +1.24 +0.54 +0.20 +0.51 +Cr2O3 +0.02 +0.04 +0.02 +0.03 + +0.02 +0.03 +MnO +0.05 +0.04 +0.02 +0.03 + +0.03 +0.04 +0.13 +Total +99.52 +99.98 +99.91 + +100.00 +99.90 + +100.00 + + + + + + + + + + + + + + + + + + +IT (VdK) +PD (VdK) +ICP-HCTa (Stock) + + +Glass +Standard +deviation +Start +Material +Glass +Standard +deviation +Start +Material +Glass +Standard +deviation +Start +Material +Na2O +3.55 +0.16 +3.55 +4.04 +0.21 +4.06 +0.05 +0.03 + +MgO +21.01 +0.30 +21.30 +21.21 +0.24 +21.51 +22.50 +0.26 +22.86 +Al2O3 +14.02 +0.24 +13.99 +11.67 +0.22 +11.55 +12.80 +0.31 +12.62 +SiO2 +52.14 +0.33 +52.62 +51.91 +0.41 +52.34 +53.32 +0.44 +54.21 +K2O +0.04 +0.03 +0.05 +0.04 +0.04 +0.03 +CaO +5.94 +0.22 +5.74 +7.47 +0.21 +7.36 +6.28 +0.21 +6.20 +FeO +1.83 +0.15 +1.58 +2.14 +0.18 +1.95 +3.64 +0.21 +3.26 +TiO2 +1.22 +0.25 +1.23 +1.23 +0.26 +1.22 +0.94 +0.23 +0.85 +Cr2O3 +0.03 +0.04 +0.03 +0.03 + +0.05 +0.04 + +MnO +0.02 +0.03 +0.03 +0.04 + +0.02 +0.02 + +Total +99.80 +100.00 +99.77 +100.00 +99.79 +0.54 +100 + + +RaB (VdK) +HMR (VdK) +G1 (Char) + +Glass +Standard +deviation +Start +Material +Glass +Standard +deviation +Start +Material +Glass +Standard +deviation +Start +Material +Na2O +3.84 +0.17 +3.68 +3.84 +0.16 +3.55 +0.04 +0.04 +0.00 +MgO +24.98 +0.20 +25.47 +20.69 +0.66 +25.70 +24.95 +0.60 +25.90 +Al2O3 +12.09 +0.20 +11.93 +11.34 +0.25 +9.90 +8.35 +0.22 +8.28 +SiO2 +52.29 +0.27 +52.70 +52.38 +0.47 +51.28 +52.98 +0.45 +53.60 +K2O +0.05 +0.03 +0.05 +0.03 +0.04 +0.03 +0.00 +CaO +5.18 +0.20 +4.98 +7.92 +0.48 +6.49 +10.56 +0.38 +10.30 +FeO +0.24 +0.06 +1.94 +0.16 +1.89 +2.17 +0.17 +1.44 +TiO2 +1.33 +0.21 +1.24 +1.50 +0.26 +1.20 +0.60 +0.17 +0.42 +Cr2O3 +0.04 +0.04 +0.03 +0.04 + +0.03 +0.03 + +MnO +0.02 +0.03 +0.05 +0.05 + +0.03 +0.03 +0.00 +Total +100.06 +100.00 +99.74 +100.00 +99.75 +99.94 + + +G1(b) (Char) +HMR CaS (VdK) +HMC (Pep) + +Glass +Standard +deviation +Start +Material +Glass +Standard +deviation +Start +Material +Glass +Standard +deviation +Start +Material +Na2O +0.04 +0.04 +4.11 +0.29 +3.46 +4.88 +0.36 +3.23 +MgO +24.75 +0.34 +25.92 +20.49 +1.10 +26.91 +17.68 +2.42 +29.82 +Al2O3 +8.71 +0.18 +8.28 +10.70 +0.39 +9.07 +6.32 +0.57 +4.53 +SiO2 +53.44 +0.35 +53. +52.12 +0.46 +49.94 +55.18 +1.12 +51.31 +K2O +0.05 +0.03 +0.05 +0.04 +0.05 +0.03 +0.00 +CaO +10.97 +0.28 +10.3 +9.63 +0.41 +7.79 +11.59 +0.71 +7.38 +FeO +1.44 +0.16 +1.44 +1.50 +0.14 +1.66 +3.71 +0.25 +3.12 +TiO2 +0.41 +0.15 +0.42 +1.42 +0.33 +1.17 +0.05 +0.07 +0.49 +Cr2O3 +0.02 +0.03 +0.03 +0.04 + +0.03 +0.03 +0.00 +MnO +0.03 +0.03 +0.02 +0.04 + +0.03 +0.03 +0.13 +Total +99.87 +99.94 +100.07 +100.00 +99.51 +100.00 + +Table 1: Glass: glass compositions determined by electron microprobe (EMPA data for the chemistry +of the glass% wt.). Start Material: nominal composition of starting materials (% wt.), EMPA Analyses: +composition of melt glass (in wt %). + + + + + +G1 +(Char) +G1(b) +(Char) +ICP-HCTa +(Stock) +HMR CaS +(VdK) +RaB +(VdK) +HAI +(VdK) +PD +(VdK) +HMR +(VdK) +HMC +(Pep) + +Na2O +0.01 +0.01 +0.01 +0.01 +0.01 +0.18 +0.02 +0.02 +0.03 +MgO +57.44 +57.22 +57.36 +57.79 +57.94 +57.28 +58.17 +57.97 +57.66 +Al2O3 +0.05 +0.22 +0.13 +0.06 +0.84 +0.74 +0.04 +0.05 +0.03 +SiO2 +42.41 +42.49 +42.33 +42.22 +41.85 +42.58 +42.40 +42.24 +42.50 +K2O +0.00 +0.01 +0.01 +0.01 +0.01 +0.00 +0.02 +0.01 +0.00 +CaO +0.30 +0.44 +0.15 +0.34 +0.18 +0.27 +0.25 +0.25 +0.37 +FeO +0.61 +0.46 +1.09 +0.35 +0.03 +0.03 +0.33 +0.45 +0.76 +TiO2 +0.02 +0.05 +0.03 +0.01 +0.08 +0.09 +0.00 +0.03 +0.04 +Cr2O3 +0.00 +0.01 +0.01 +0.01 +0.01 +0.07 +0.02 +0.00 +0.00 +MnO +0.01 +0.00 +0.00 +0.01 +0.01 +0.00 +0.00 +0.02 +0.01 +Total +100.86 +100.91 +101.13 +100.79 +100.96 +101.25 +101.24 +101.06 +101.39 + +Fo +99.41 +99.55 +98.94 +99.66 +99.97 +99.97 +99.69 +99.57 +99.27 + +Table 2: EMPA data for the minerals or (olivine) inclusion in the glass (in wt.%). Fo = forsterite +content of olivine + + +NP-LMG (VdK) + + + + + + + + + + + + + +0-25 +2.92 +3.42 +3.5 + +7.93 +9.61 + + + +11.81 + + + +25-63 +2.85 +3.42 +3.5 + +7.93 +9.55 + + + + + + + +63-125 +2.82 +3.41 +3.5 + +7.94 +9.54 + + + + + + + +125-250 +2.8 +3.41 +3.5 + +7.92 +9.56 + + + + + + + +NC (Pep) + + + + + + + + + + + + + +0-25 +2.94 +3.42 +3.5 + +7.98 +9.74 + + + +11.87 + + + +25-63 +2.93 +3.41 +3.5 + +7.95 +9.65 + + + + + + + +63-125 +2.83 +3.42 +3.5 + +7.96 +9.67 + + + + + + + +125-250 +2.82 +3.41 +3.5 + +7.91 +9.66 + + + + + + + +CBC (Pep) + + + + + + + + + + + + + +0-25 +2.94 +3.42 +3.5 + +7.98 +9.82 + + + +11.83 + + + +25-63 +2.87 +3.42 +3.5 + +7.95 +9.76 + + + + + + + +63-125 +2.82 +3.42 +3.5 + +7.95 +9.75 + + + + + + + +125-250 +2.82 +3.42 +3.5 + +8 +9.77 + + + + + + + +CB (VdK) + + + + + + + + + + + + + +0-25 +2.95 +3.42 +3.5 + +8 +9.73 + + + +11.87 + + + +25-63 +2.87 +3.42 +3.5 + +7.97 +9.77 + + + + + + + +63-125 +2.84 +3.42 +3.5 + +7.98 +9.76 + + + + + + + +125-250 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(VdK) +NC (Pep) +NP-HMg (VdK) +HAl (VdK) +IC (Pep) +IT (VdK) +PD (VdK) +G2 (Char) +RaB (VdK) +HMR (VdK) +HMR (VdK) +G1 (Char) +G1 (Char) +HMR-Cas (VdK) +HMR-Cas (VdK) +HMC (Pep) +NVPa (Stock) +Relative Reflectance +6 +8 +10 +12 +14 +16 +18 +µm +Relative Reflectance +6 +8 +10 +12 +14 +16 +18 +µm +Fig. 6b +CF RB +RB + +50 +60 +70 +80 +90 +100 +40 +7.20 +7.40 +7.60 +7.80 +8.00 +8.20 +8.40 +Micro-FTIR synthetic glass +Powder FTIR synthetic glass +Impact Glass (Morlok et al., 2016) +Fig. 7 +CF in µm +SiO2 (wt%) + +40 +50 +60 +70 +80 +90 +100 +8.8 +9.0 +9.2 +9.4 +9.6 +9.8 +10.0 +10.2 +10.4 +10.6 +10.8 +Micro-FTIR synthetic glass +Powder FTIR synthetic glass +Impact Glass (Morlok et al., 2016) +Synthetic Glass (DuFresne et al., 2009) +Synthetic Glass (Lee et al., 2010) +Fig. 8 +Strongest RB in µm +SiO2 (wt%) + +Micro-FTIR synthetic glass +Powder FTIR synthetic glass +Impact Glass (Morlok et al., 2016) +SCFM vs. CF (Glass) +CF in µm +SCFM +SiO2/(SiO2+CaO+FeO+MgO) +Fig. 9 +0.40 +0.50 +0.60 +0.70 +0.80 +0.90 +1.00 +7.5 +7.6 +7.7 +7.8 +7.9 +8.0 +8.1 +8.2 +8.3 +8.4 +8.5 + +Fig. 10 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +0.84 +0.86 +0.88 +0.9 +0.92 +0.94 +0.96 +0.98 +1.0 +Full Width at Half Maximum (FWHM) +(Si+Al)/(Fe+Ca+Mg+Na+K) + +Arbitrary Reflectance +7 +8 +9 +10 +11 +12 +13 +14 +µm +Composition High Al-Region +(HAl VdK) +Composition High Mg-Region +(HMC VdK) +Surface Mercury +(Sprague et al., 2000) +Fig. 11 + diff --git a/etFRT4oBgHgl3EQfVTd7/content/tmp_files/load_file.txt b/etFRT4oBgHgl3EQfVTd7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fbabc5a941555c0fd4a8a9cfa280f9b2e7a2a158 --- /dev/null +++ b/etFRT4oBgHgl3EQfVTd7/content/tmp_files/load_file.txt @@ -0,0 +1,2756 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf,len=2755 +page_content='1 IR Spectroscopy of Synthetic Glasses with Mercury Surface Composition: Analogs for Remote Sensing Corresponding Author: Andreas Morlok, Institut für Planetologie, Wilhelm- Klemm-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 10, 48149 Münster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Email: morlokan@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='de, Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' +49-251-83-39069 Stephan Klemme, Institut für Mineralogie, Corrensstraße 24, 48149 Münster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Email: stephan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='klemme@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='de Iris Weber, Institut für Planetologie, Wilhelm-Klemm-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 10, 48149 Münster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Email: sonderm@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='de Aleksandra Stojic, Institut für Planetologie, Wilhelm-Klemm-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 10, 48149 Münster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Email: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='stojic@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Martin Sohn, Hochschule Emden/Leer, Constantiaplatz 4, 26723 Emden, Germany, Email: martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='sohn@hs-emden-leer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='de Harald Hiesinger, Institut für Planetologie, Wilhelm-Klemm-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 10, 48149 Münster, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Email: hiesinger@uni-muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='de © 2017 This manuscript version is made available under the CC-BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='0 2 Abstract In a study to provide ground-truth data for mid-infrared observations of the surface of Mercury with the MERTIS (Mercury Radiometer and Thermal Infrared Spectrometer) instrument onboard the ESA/JAXA BepiColombo mission, we have studied 17 synthetic glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' These samples have the chemical compositions of characteristic Hermean surface areas based on MESSENGER data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The samples have been characterized using optical microscopy, EMPA and Raman spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Mid- infrared spectra have been obtained from polished thin sections using Micro-FTIR, and of powdered size fractions of bulk material (0-25, 25-63, 93-125 and 125-250 μm) in the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5-18 µm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The synthetic glasses display mostly spectra typical for amorphous materials with a dominating, single Reststrahlen Band (RB) at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5 µm - 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' RB Features of crystalline forsterite are found in some cases at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 µm, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='4-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 µm, and at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Dendritic crystallization starts at a MgO content higher than 23 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='% MgO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The Reststrahlen Bands, Christiansen Features (CF), and Transparency Features (TF) shift depending on the SiO2 and MgO contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Also a shift of the Christiansen Feature of the glasses compared with the SCFM (SiO2/(SiO2+CaO+FeO+MgO)) index is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' This shift could potentially help distinguish crystalline and amorphous material in remote sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' A comparison between the degree of polymerization of the glass and the width of the characteristic strong silicate feature shows a weak positive correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' A comparison with a high-quality mid-IR spectrum of Mercury shows some moderate similarity to the results of this study, but does not explain all features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Introduction Infrared spectroscopy allows determining the mineralogical composition of planetary surfaces via remote sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The Mercury Radiometer and Thermal Infrared Spectrometer (MERTIS) spectrometer onboard the future ESA/JAXA BepiColombo mission to Mercury will allow such remote sensing observations by mapping spectral features of the Hermean surface in the 7-14 µm range, with a spatial resolution of ~500 m (Benkhoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Hiesinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' In order to correctly interpret remote sensing data, laboratory spectra of suitable analog material are of vital importance (Helbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Maturilli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The IRIS (InfraRed and Raman for Interplanetary Spectroscopy) laboratory in Münster therefore generates spectra from analog material similar to those materials expected to occur on the surface of Mercury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Surface regolith and exposed rocks of terrestrial planets and their moons are modified by impact events throughout their lifetimes (Hörz and Cintala, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The investigation of how these related processes affect the spectral properties of the rocks is important for the correct interpretation of infrared data from planetary bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Higher impact shock, for example, results in amorphous phases produced in solid state transformation (such as maskelynite), or melt glass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Stöffler, 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Wünnemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Osinski and Pierrazo, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Jaret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Under shock metamorphic conditions, minerals transform from crystalline to a solid amorphous state including diaplectic glasses like maskelynite at pressures of ~25 - ~40 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Melting of feldspar starts at ~35 to ~45 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Over 60 GPa rocks melt completely, which may result in quenched melt glass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Stöffler, 1966, 1971, 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Chao, 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' von Engelhardt and Stöffler, 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Stöffler and Langenhorst, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' French, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Johnson, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Impact glass lacks a far-range order of its atomic constituents and represents the amorphous building block of a material, typically generated in events involving high shock pressure and temperatures (French, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Speck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' In our study, we present the first mid-infrared reflectance data for synthetic glasses as analogs for melt glass based on the respective chemical compositions derived from remote sensing and model 4 data for the surface of Mercury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Using synthetic materials allows us to produce more realistic analogs for Mercury surface rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' To date, there are no Hermean meteorites we know of (Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Goodrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2017), chemical remote sensing data based on X-ray Spectrometer (XRS) and the Gamma-Ray and Neutron Spectrometer (GNRS) is the best information source available so far to deduce the surface composition of Mercury (Weider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' vander Kaaden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' We expect the surface rocks and regolith not only to consist of glassy material, a mixture of components of various shock stages seems more likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Every respective shock stage will have different spectral characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016a, 2016b, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Therefore, the spectra of the synthetic glasses produced in this study will serve as the endmember for studies of glass mixtures where amorphous and crystalline components are mixed to varying degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Also, areas that underwent more recent volcanism could be less affected by impact alteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Such areas comprise large areas of the Mercurian surface (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' They could provide crystalline minerals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Deneva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Goudge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' vander Kaaden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' To produce the synthetic glasses, we use the average chemical composition of surface regions identified in the MESSENGER data Compositions G1 and G2 (Charlier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2013), which are the average compositions for larger areas in the equatorial region and the southern hemisphere of Mercury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' They are distinguished by their variation in the Ca and Al contents (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='1) and cover both high-reflectance volcanic plains and low-reflectance rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Stockstill-Cahill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2013) and Weider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2012) present average compositions for the Mg poor, alkali-rich northern volcanic plains (NVPa) area, and the Mg-rich intercrater plains and heavily cratered terrain (IcP-HCTa) (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Peplowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2015) present compositions of the high-Al and low-Mg Interior plains (CBC), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', the area inside the young Caloris impact crater and the low-Al and Mg-Northern Terrane (NC), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', northernmost part of Mercury above 60° northern latitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Further areas are High-Mg Terranes (HMC) and an Intermediate composition (IC) (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Compositional data presented in vander Kaaden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2015) and Weider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2015) are the high- Mg (and low Al) region (HMR) and a sub region of the HMR with Ca and S enrichments (HMR-CaS), the low-Mg plains of the Caloris basin (CB), Mg-rich and poor parts of the northern volcanic plains 5 (NP-HMg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' NP-LMg, respectively), the Rachmaninoff basin (RaB), an area with high-Mg near the high- Al northern plains (HAl), a large pyroclastic deposit (PD), and the average of inter crater plains and various, cratered terrains (IT) (Weider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' We analyze four size fractions (0-25, 25-63, 63-125, 125-250 µm), motivated to better account for the high porosity and large grain size variations of surface regolith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Variation in grain size causes changes in the intensity of the characteristic Reststrahlen Bands (RB), fundamental mode absorption features in the 7-14 µm region, resulting in a loss of spectral contrast with decreasing grain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' An earlier study in the visible and near-infrared range (Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2007) indicated a high abundance of grains smaller 30 µm in size comprising the surface regolith on Mercury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Therefore, the corresponding RBs are expected to be weak in the remote sensing data of Mercury (Salisbury and Eastes, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Salisbury and Wald, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Mustard and Hayes, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' In addition, the transparency feature (TF), a characteristic additional spectral feature for small grain sizes, appears around 11-13 µm in the smallest grain size fractions below 50 µm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Salisbury, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Potential TF features have been observed in ground based infrared observation of Mercury, indicating a high abundance of such fine-grained material in the regolith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' This motivates the need for spectral data especially of the fine- grained size fractions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Earlier reflectance and emission studies in the mid-infrared of synthetic glasses as analogues for impact melt glass were made by Byrnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2007) and Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' They analyzed synthetic quartzofeldspathic glasses and found correlations between the band positions of characteristic dominant features and SiO2 contents or Si/O ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Comparable results for synthetic glass with basaltic (low SiO2) to intermediate (high SiO2) composition were obtained by DuFresne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2009), Minitti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2002), and Minitti and Hamilton (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' McMillan and Piriou (1982), Speck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2011) and King et al (2004) provide additional overview of the infrared properties of silicate glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Glasses from laser pulse experiments with a Martian soil analog JSC Mars-1 were analyzed by Basilevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2000), Moroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2009), and Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2000), resulting in spectra dominated by a strong single band in the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5 µm wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Earlier reflectance and emission studies of natural impact melt glass formed during impacts in the mid-infrared were performed by Thomson 6 and Schultz (2002), Gucsik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2004), Faulques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2001), Fröhlich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2013) and Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2016a and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Spectra of these samples are dominated by a broad RB in the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3 µm range, with only few other features in the mid-infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' A complementary study of silicate glasses with Mercurian and other planetary compositions was made by Cannon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Samples and Techniques 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='1 Sample Compositions and Preparation of Glasses The respective chemical composition used for the synthetic glass analogs of surface areas on Mercury are based on Charlier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2013 (Char), Stockstill-Cahill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2013 (Stock), Peplowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2015 (Pep), and vander Kaaden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2015 (VdK) (comparable to those in vander Kaaden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The various studies and models did not always present the same range of oxide components, we therefore limited the composition used in this study to SiO2, TiO2, Al2O3, Fe2O3, MgO, CaCO3, Na2O, and K2O for better comparability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Components below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='% were omitted for individual mixtures for simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Starting material compositions are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The oxide and carbonate starting mixtures were finely ground to a powder in an agate mortar under acetone and then dried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The resulting mixtures were placed in medium sized Pt crucibles in which they were slowly heated to 1000°C to de-carbonate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Subsequently, the mixtures were heated and melted in a conventional box furnace at 1450°C for 2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' They were quenched immediately after complete liquefaction, the crucibles were swiftly taken out of the furnace and submerged in water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The samples were vitrified within 10 secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The samples were melted in a box furnace in air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Oxygen fugacity was not controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' This may affect phase equilibria slightly, but only when high amounts of Fe, the only redox sensitive major element, are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 7 Our samples were melted at high temperatures and kept at temperature for several hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The melts are characterized by relatively low viscosity which ensures complete homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Once melted, the structure of the melt does not depend on the starting material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' In two cases (ICP-HCTa (Stock) and RaB (VdK)), we re-heated the starting material at 1500°C in an attempt to remove crystals, which have formed during quenching in the first procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' In order to prepare grain size fractions, bulk glass material was ground in steel and agate mortars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The powder was cleaned in acetone and dry sieved for one hour to generate four size fractions: 0-25 µm, 25-63 µm, 63-125 µm, and 125-250 µm, by using an automatic Retsch Tap Sieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' In order to remove clinging fines, the larger two fractions were again cleaned in acetone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' In addition, polished thick sections were prepared for microscopic investigation from the pure glass sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Optical Microscopy Polarized light microscopy provides fast information about the crystalline or amorphous character of the single components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' It also enables first mineral identification in the samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 1), which is important for the subsequent Raman investigation to avoid mixed measurements on an inhomogeneous sample location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The first overview images of all polished thick sections were obtained with a KEYENCE Digital Microscope VHX-500F under normal light conditions and under crossed polarizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3 Raman Spectroscopy All Raman measurements were conducted using an Ocean Optics IDR-Micro Raman system (IfP, Münster), operating with an OneFocus optical system equipped with a 40 x objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The laser excitation is 532 nm and the spectral resolution is about 7 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Spectra were obtained with a laser 8 power of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='8 mW starting at wavenumbers around 200 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The spot size on the sample is approximately 2 µm in diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Every spectrum is the result of one measurement at 15 seconds acquisition time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 2a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' All spectra are automatically background and baseline subtracted and have not been smoothed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' As usual, all spectra are given with arbitrary units, because the height of a signal only corresponds to the quality of the individual Raman scatterer itself (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' the double peak in olivine appears because the two SiO4-stretching (v1 and v3) modes are active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=') In addition, the Raman spectra of the glass are affected by fluorescence, which adds intensity in the form of an underlying continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='4 Electron Microprobe Analysis Backscattered electron (BSE) images (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 4a-c) show crystalline phases (brighter phases) formed during quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Detailed quantitative analyses of the glass and the olivines crystallized during the quenching process were made with a JEOL JXA-8530F Hyperprobe electron probe micro analyzer (EPMA) equipped with five wavelength dispersive spectrometers (WDS) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' For the glass analyses, the probe was operated at an excitation voltage of 15 kV and a beam current of 5 nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The beam diameter was defocused to 5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The counting time was 5 seconds on the peak and 2 seconds on the background of each element, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' For mineral analyses we used an excitation voltage of 15 kV and a beam current of 15 nA with a slightly defocused beam diameter of 2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The counting time for Mg, Al, Si, Ca, Fe, Ti, Cr, and Mn was 15 seconds on the peak and 5 seconds on the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' And, in order to avoid loss of the volatile elements Na and K, the counting time was reduced to 5 seconds on the peak and 2 seconds on the background for these two elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The following natural and synthetic minerals with well-known compositions were used as standards: Jadeite (Na2O), SanCarlos Olivine (MgO), Disthene (Al2O3), Hypersthene (SiO2), Sanidine (K2O), Diopside (CaO), Fayalite (FeO), Rutile (TiO2), Cr2O3 (Cr2O3), and Rhodonite (MnO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5 Bi-directional Diffuse Reflectance FTIR For the bi-directional analyses of the sieved bulk powder size fractions we used aluminum sample cups with 1 cm diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The surface was gently flattened with a spatula following a procedure analog to that described by Mustard and Hayes (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' For the bulk powder analyses in the mid- infrared from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5-18 µm, we used a Bruker Vertex 70 infrared system at IRIS laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' We used a cooled MCT detector to ensure a high signal to noise ratio of the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' All analyses were made under low pressure (10-3 bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' We accumulated 512 scans for each size fraction at al spectral resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='02 µm (compared to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 µm for MERTIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Hiesinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The machine background was removed using a diffuse gold standard (INFRAGOLDTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' We obtained analyses with a variable geometry stage (Bruker A513) in order to emulate various observational geometries of an orbiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The data presented in this study were obtained at 30° incidence (i) and 30° emergence angle (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The spectra returned from MERTIS will be emissivity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' For the comparison with remote sensing data in thermal infrared, reflectance and emission data have to be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' This is usually done using Kirchhoff’s law: ε = 1 – R (R=Reflectance, ε = Emission) (Nicodemus, 1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' For a direct comparison using Kirchhoff’s law, the reflected light in all directions has to be collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' This relation works best for the comparison of directional emissivity and directional hemispherical reflectance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' In our study, a bi-directional, variable mirror set-up was used, without a hemisphere integrating all reflected light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' This has to be kept in mind when comparing the results in a quantitative manner with emission data (Salisbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Hapke, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Thomson and Salisbury, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Salisbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The spectral range of the MERTIS spectrometer is from 7-14 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Features at shorter and longer wavelengths can be of interest for other studies, we therefore present powder spectra from 6-18 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The spectra are presented in reflectance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 10 The width of infrared bands was determined using the FWHM (Full Width at Half Maximum) of the dominant spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' To derive this parameter from the often asymmetric, non-Gaussian bands, we used the Origin software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' For the fitting, GCAS (Gram-Charlier peak function) and CCE (Chesler-Cram) fitting functions were applied to the spectra of the 125-250 µm size fraction, which were normalized to the same intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The spectra presented in this study are accessible via an online database at the Institut für Planetologie in Münster (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='uni- muenster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='de/Planetology/en/ifp/ausstattung/iris_spectra_database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='html), and the Berlin Emissivity Database (BED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='6 In Situ FTIR Microscope For in situ analyses, we used a Bruker Hyperion 2000 IR microscope attached to the external port of a Bruker Vertex 70v at the Hochschule Emden/Leer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Here we used a 256×256 µm2 sized aperture to obtain analyses of small features with in situ reflectance spectroscopy on polished thin sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' For each spectrum, 128 scans were integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' A gold mirror was used for background calibration (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 6a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='1 Optical Microscopy The polished thick sections show a great diversity in color from nearly transparent samples, such as NC (Pep) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 1), to a brownish glass such as PD (VdK) as the darkest endmember.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Samples exhibiting crystallites tend to be more brownish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 11 Several samples show heterogeneity in the form of clearly identifiable crystals, which are embedded in a still glassy matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' These occur in the RaB (VdK) sample, while HMR (VdK), HMC (Pep), and HMR- CaS (VdK) are examples displaying high crystallinity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' In these samples, olivine crystals were identified by their habit and by their color in polarized light (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 Raman Raman spectra of the pure glasses show typical shifts, two main broad signals, which cannot be attributed to single peaks, between 400 cm-1 - 700 cm-1 indicative of a silicate framework and between 850 cm-1 - 1250 cm-1, which is indicative of tetrahedrally coordinated cations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 2a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' DiGenova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Raman shifts caused by glass are characterized by broad features lacking a crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 2a are displayed with lower Mg contents on top and increasingly higher Mg contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' G2 (Char) is an outlier within this sequence, which might be a hint for a transitional sample showing incipient crystallization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Detailed Raman spectroscopy on the more heterogeneous samples, which include crystals confirms the existence of forsteritic olivine with a typical Raman double peak (DB) at 823 cm-1 and 856 cm-1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 2b, blue line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Chopelas, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' In addition, the shift at around 700 cm-1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2b, green line) in the spectra of HMC (Pep) can be attributed to spinel (Downs, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3 EMPA Results of the detailed quantitative analyses of the glasses as well as the sample weight of the starting oxides are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The analyses are average values of 20 – 70 measurements on each glass depending on the amount of olivine crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Analyses of the crystallized olivines, including their Fo-content, are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The BSE image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 4a-c shows the dendritic growth of olivine as a result of quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' In addition, as can be seen in Tables 1 and 2, the formation of olivine 12 during quenching decreases with the decreasing MgO content in the original sample weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The lower the MgO content the more identical are the quantitative analyses with the original values (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' At a MgO content higher than 23 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='% (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='1), more and more olivine crystals are visible in the sample (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Also, chemical differences between the original sample (oxide mixture) and the glass increase above this threshold (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' However, a few crystallites are already visible at lower MgO contents like in Hal (VdK) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Normative calculations by vander Kaaden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2016) also show high olivine contents for these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Furthermore, BSE images combined with WDS analyses give hints on critical chemical thresholds for the formation of further crystalline phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' As visible in Figure 4c and 2b, spinel is another phase inside of an olivine grain with remnants of MgO around the spinel rim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Spinel and remnants of MgO occur in G1 (Char), HMR (VdK), HMR-CaS (VdK), and HMC (Pep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The threshold value for Mg for the crystallization of spinel and MgO remnants are less than 9 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='% Al and more than 25 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='% Mg(Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='4 Diffuse Reflectance FTIR Spectra of the different size fractions are presented in the 6-18 µm range in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 3, arranged according to increasing Mg-contents in the starting materials (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 5, 6, Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Features below 6 µm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5 µm) are either features of adsorbed volatiles on the starting materials or in the furnace and not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' For better comparison, the spectra of the largest size fraction (125-250 µm) are presented together in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' A series of 12 samples with low Mg contents show spectra which are dominated by a single RB between 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5 µm (NP LMg VdK) and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='7 µm (ICP-HCTa Stock).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Endmembers for the CF are 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9 µm (NC Pep) and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3 µm (PD VdK), for the TF NP-LMg (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='4 µm VdK) with 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='8 µm and RaB (VdK) with 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' One exception is sample G2 (Char), while the sample has the single RB characteristic for 13 this group, the feature (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='6 µm) is clearly shifted towards longer wavelength in comparison (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='6a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The other exception among the low-Mg samples is NVPa (Stock), which shows a shoulder at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='7 µm and the main band at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='7 µm (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' This spectrum was grouped with the remaining four high-Mg samples HMR, HMR-CaS (VdK), HMC (Pep), and G1 (Char) that exhibit broader main RB features and show clear crystalline features at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9 µm, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='1 µm, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5 - 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' These are all typical forsterite bands (Hamilton, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The CF of these high-Mg samples ranges from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9 µm to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='4 µm, and the TFs are found between 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='7 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Additional weak bands are located between 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 and 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='7 µm (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Of the original experimental run, RaB (VdK) and ICP-HCTa (Stock) also exhibited crystalline features, which disappeared after repeated melting/quenching at temperatures over 1700°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5 Micro-FTIR The FTIR spectra of the ‘pure’ glass samples are basically identical to those from the equivalent powder analyses of the same samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 6a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' We also attempted to obtain spectra of glassy spots in samples showing crystallization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 6b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The CFs of pure glass and glass-spots range from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9 µm (NP-LMG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' CBC and NC Pep) to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3 µm (G1, Char), the main feature is between 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='6 µm (NP-LMG VdK) to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='1 µm (HMR-CaS) (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Spots from individual crystals and high-Mg samples show more variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Features with varying intensity at ~9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='6 µm, ~10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 µm, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='0µm, and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9 µm are typical olivine features (Hamilton, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Lane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2011), beginning with the HMR (VdK) sample (Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Correlation of spectral and chemical features 14 The aim of this study is to correlate spectral features of the laboratory analyses to remote sensing data obtained from the surface of Mercury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The spectra from observations of Mercury available often show low spectral contrast (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Therefore, it is useful to identify potential characteristic features that allow us to derive mineralogical and chemical information from remote sensing data even if the observational data is of comparatively low quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The CF, as a reflectance minimum, is a feature that can be easily identified in remote sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Hence, the band position of the CF is commonly used as a proxy for the bulk chemical composition of samples, like the SiO2 content (Salisbury 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Analyses of bulk powders and micro-FTIR are basically identical, and are consistent with the trend line for earlier studies (Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016b) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The analyses from an earlier study on natural impact glass (Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016b) show a much wider variation from the trend line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' This points towards a high purity of the synthetic material used in the current study, in contrast to the natural materials, which often contained fragments of unshocked or unmelted material (Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016a and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' If a strong RB is observable in remote sensing data, it also allows for characterizing the material (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Again, our band positions of the strong main RB features with respect to the SiO2 contents fall close to observations in earlier studies (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' They demonstrate a better agreement than the natural impact glasses from our earlier study (Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' At lower SiO2 contents (below 60 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='%) the results of this study show a similar trend than those for basaltic glass from DuFresne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2009), where the trend line becomes much shallower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' However, we cannot distinguish amorphous from crystalline material with this type of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The SCFM index (SiO2/(SiO2+CaO+FeO+MgO)) is a way to determine the degree of polymerization of a material, based on the increased interconnection of the SiO4 tetrahedra (Walter and Salisbury, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Earlier studies (Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016b) have indicated that amorphous material like impact glasses plot off the trend line for crystalline terrestrial material (Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The synthetic glasses in this study show a similar and in fact clearer behavior, continuing the trend line for basaltic compositions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 15 A further way to correlate polymerization with spectral features is to use the bandwidth of the dominating silicate feature in the glass with the ratio between network forming cations (Si and Al) and network modifying anions (Fe, Ca, Mg, Na, K) (King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Dufresne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Speck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' A comparison of this ratio to the FWHM (Full Width at Half Maximum) of only the glassy samples could help to obtain information about the degree of polymerization from remote sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' However, the correlation between these two parameters of the glass in our study is low with a correlation factor R (Pearson) of only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='34 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Comparison with Astronomical Observations of Mercury Earth based mid-infrared observations of Mercury integrate large surface areas of up to 106 km2 and indicate a surface of mainly plagioclase with minor pyroxene (Donaldson-Hanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 1994, 2000, 2002, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Sprague and Roush, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Emery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 1998,;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' We compare our results with an astronomical spectrum (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 11), obtained by the Mid-Infrared Array Camera (MIRAC) at the Kitt Peak Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The spectrum is chosen due to its high signal to noise ratio and strong features in the spectral range of MERTIS (7-14 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The observed spectrum is the average of several observations of an area centered on ~210-250° longitude (Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The baselined spectrum, re-calculated from emissivity to reflectance, shows a CF-like feature at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5 µm, strong RBs at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3 µm, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9 µm, and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='0 µm, and a potential TF at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Similarly, potential TFs in the 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='0-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='7 µm region are also observed in various other spectra of Mercury (Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' A comparison with results from this study demonstrates that glass from the 125-250 µm fraction of High Aluminium regions (HAl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' VdK) and the 0-25 µm fraction of the High-Magnesium region HMC (VdK) (which already shows abundant crystallites) are the best to be compared to (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The glassy HAl (VdK) analog sample has a main RB at the same position as in the astronomical spectrum (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9 µm), although it is much broader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The HMC (VdK) analog sample has a TF at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3 µm, 16 close to the band in the remote sensing spectrum (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='4 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Also, the CF is relatively close to the astronomical data, which show the CF at 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The broad RB feature of the Hal (VdK) analog sample around 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 µm is rather close to the (much narrower) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='0 µm band in the Mercury spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' No equivalent for the 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3 µm feature in the astronomical spectrum was found in the glasses of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' This feature could indicate a pyroxene component (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Hamilton, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2000) in the surface regolith of Mercury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' However, we compare here laboratory analyses of samples with very distinct compositions to those of large surface areas of Mercury, effectively a ‘bulk’ spectra averaging many different mineral species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Future remote sensing data of smaller, chemically and mineralogically homogeneous areas of the surface of Mercury returned from the MERTIS instrument on BepiColombo (Benkhoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Hiesinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2010) will allow to use the laboratory spectra much more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Glass will probably mostly be contained in impactites and regolith, where it can be expected to be part of a mixture with other, crystalline materials (Tompkins and Peters, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016a and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The large volcanic structures like smooth plains and pyroclastic deposits could be a source for crystalline material, which will make the detection of glasses difficult (Denevi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Goudge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' vander Kaaden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Visible and Near-Infrared studies of planetary glasses by Cannon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2016) have shown that identifying glass in spectral mixtures is challenging even at mass abundances up to 70%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Especially mafic glasses are difficult to be identified ‘by eye’, but can be separated in spectral deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Studies in the mid-infrared range by Ramsey and Christensen (1998) were able to reproduce mixtures of crystalline minerals even for components below 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Still, this could lead to challenges identifying the glass in the surface regolith of Mercury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Surface regolith studies show up to 45% agglutinitic glass, much higher than 2-5% observed for glass on the lunar surface (Warrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 17 Also, space weathering has to be taken into account in this context, which can be expected much more intense closer to the sun and also produce glassy materials (Papike et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Hapke, 2001, Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Summary and Conclusions The study of a series of synthetic glasses based on surface regions of Mercury displays mostly spectra typical for amorphous materials, characterized by a single RB between 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5 µm and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Several spectra show features of crystalline olivine species in both the powder and micro-FTIR spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' FTIR features of crystalline species at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 µm, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='4-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 µm, and at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9 µm in the powder and micro-FTIR spectra are mostly characteristic of forsterite, appearing in a smaller number of samples at higher MgO contents, beginning with the HMR (VdK) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Optical, Raman and EMPA data have signs of crystallization starting with the RaB (VdK) material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' This points towards a threshold for crystallization starting at MgO contents higher than 23wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='% MgO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' However, in Raman spectra as well as in FTIR G2 (Char) is an outlier regarding the position of the strongest feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' This might be indicative of a starting crystallization at a lower MgO concentration in this sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The RBs, CFs, and TFs shift, depending on the SiO2 and MgO contents and exhibit similarity to earlier studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' We also confirm a shift of the CF position of amorphous material compared with the respective crystalline SCFM index, which could help distinguishing crystalline and amorphous material in remote sensing data in the mid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' A comparison between the degree of polymerization of the glass and the width of the characteristic strong silicate feature shows only slight correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' However, here sample heterogeneity – such as incipient crystallization – probably affects the spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 18 A comparison with a high-quality mid-IR spectrum of Mercury shows some similarity to the results of this study, but does not explain all features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' A series of analyses of distinct size fractions is needed to distinguish between genuine effects and effects induced by mixed grain size fractions in the natural regolith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' However, since regolith is also an intimate mixture of many phases, spectral unmixing modelling will also have to take many other grain-size fractions of other potential mineral phases into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Acknowledgements We thank Isabelle Dittmar (Emden) for analytical support, Ulla Heitmann (Münster) for thin section preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' This work was partly supported by DLR grant 50 QW 0901 in the framework of the BepiColombo mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 19 References Basilevsky A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Yakovlev O.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='1– 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='5 µm) imaging of Mercury.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Icarus 147, 421–432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Sprague A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Emery J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Orsini S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Milillo A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2007) Mercury’s Surface Composition and Character as Measured by Ground-Based Observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Space Science Reviews 132, 399-431 26 Stockstill-Cahill K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', McCoy T.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Joy K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Crowther, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Jastrzebski N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Gilmour J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Clay P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2016) Cosmochemical and spectroscopic properties of Northwest Africa 7325—A consortium study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Meteoritics & Planetary Science 51, 3-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Weider, Shoshana Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Nittler, Larry R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Starr, Richard D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' McCoy, Timothy J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Stockstill-Cahill, Karen R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Byrne, Paul K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Denevi, Brett W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Head, James W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Solomon, Sean C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=" (2012) Chemical heterogeneity on Mercury's surface revealed by the MESSENGER X-Ray Spectrometer." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Journal of Geophysical Research (Planets) 117, E00L05 Weider, Shoshana Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Nittler, Larry R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Starr, Richard D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Crapster-Pregont, Ellen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Peplowski, Patrick N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Denevi, Brett W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Head, James W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Byrne, Paul K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Hauck, Steven A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Ebel, Denton S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Solomon, Sean C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=" (2015) Evidence for geochemical terranes on Mercury: Global mapping of major elements with MESSENGER's X-Ray Spectrometer." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Earth and Planetary Science Letters 416, 109-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Wünnemann K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Collins G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', Osinski G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2008) Numerical modelling of impact melt production in porous rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Earth and Planetary Science Letters 269, 530-539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 28 Figure Captions Figure 1: Overview of optical images of polished blocks of the samples, arranged according to increasing Mg contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Increasing abundance of Mg is correlated with increasing appearance of crystallites in the quenched melt glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The sample for NVPa (Stock) was only available in powdered form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Red squares are 256×256 µm sized areas analyzed with micro-FTIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Red scale bars in the images for NVPa (Stock), ICP-HCTa (Stock), and RaB (VdK) represent 256 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Figure 2a: Raman spectra for the glasses, in the range from 300 cm-1 to 1300 cm-1 where the Mg- abundances increase from the top to the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The broad peak between 400 cm-1 and 700-1 is a result from the extant silicate framework and between 850 cm-1 and 1250 cm-1 tetrahedra coordinated cations are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Intensity in graphs is normalized on the strongest signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The spectra are shifted vertically for a better view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Figure 2b: Raman spectra for the crystalline phases formed during quenching in the range from 400 cm-1 to 1600 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The blue line represents a typical olivine spectrum with the DB at 823 cm-1 and 856 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Beyond this the green line shows an additional peak at 700 cm-1 fitting to spinel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Intensity in graphs is normalized on the strongest signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The spectra are shifted vertically for a better view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Figure 3: Changes in main element abundances (SiO2, Al2O3, MgO, CaO, FeO) between starting material (Initial) and quenched glass (EMPA data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Data is ordered by their increasing Mg content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Most samples with lower Mg abundance show only slight variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Only high-Mg samples that produced crystallites show increasing differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Figure 4: Representative SEM BSE images of crystallites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' a) Example of the dendritic growth of olivine (Ol) as result of quenching in G1 (Char).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' b) Example of the dendritic growth of olivine (Ol) as result of quenching in HAI (VdK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' c) Example of a spinel (light grey;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Sp) as another present phase inside of an olivine grain (dark grey;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Ol) with remnants of MgO (white) around the spinel rim in G1 (Char).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 29 Figure 5: FTIR reflectance spectra of powdered glass in four size fractions (0-25 µm, 25-63 µm, 63-125 µm, 125-250 µm), sorted by increasing Mg-content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The crystalline features appearing at higher Mg- contents are forsterite bands (Hamilton, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Figure 6a: Comparison of FTIR reflectance powder spectra of the 125-250 µm size fraction of all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Mg-abundance is increasing from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Vertical lines denote important features: Christiansen Feature (CF), and Reststrahlenband (RB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' A shift of these features to longer wavelengths is correlated with increasing Mg abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Circle: olivine features in high-Mg samples (Hamilton, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Figure 6b: Micro-FTIR reflectance spectra of polished samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Spectra are sorted by increasing Mg- content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Vertical lines denote important features: Christiansen Feature (CF) and Reststrahlenband (RB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' A shift of these features to loner wavelengths is correlated with increasing Mg abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Left column: Spectra of glass or glassy regions in samples showing crystallization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Right column: Spectra of areas with high contents of crystallites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Vertical lines show typical forsterite RB bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Figure 7: SiO2 contents of glasses (in wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='%) compared with the position of the Christiansen Feature (in µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The samples of this study fall on the dashed trend line defined by earlier studies on impact glasses (Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016) and crystalline terrestrial rocks (Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Figure 8: SiO2 contents of glasses (in wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='%) compared with the position of the strongest Reststrahlen- Band (in µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Glasses from this study are similar to earlier studies (Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2002), but also diverge from the trend lower SiO2 contents similar to basaltic glasses in DuFresne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Figure 9: Comparison of the SCFM index (SiO2/(SiO2+CaO+FeO+MgO);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Walter and Salisbury, 1989): with the position of the CF (in µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The results of the glasses in this study confirm the findings in Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' (2016), where glassy material plotted below the trend line for crystalline materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' The trend line is for analyses of terrestrial rocks (Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 30 Figure 10: Comparison of the Full Width at Half Maximum (FWHM) of the strong silicate feature in glasses with the degree in polymerization, based on a ratio between network building ions (Si, Al) and network modifiers (Fe, Ca, Mg, Na, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' Figure 11: Comparison of a Mercury mid-infrared spectrum obtained by ground based observation (Sprague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2000) with the two most similar spectra from this study, a high-Al region (HAl VdK) and the finest size fraction of a high Mg-region (HMC VdK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' There is a general similarity for the CF and TF as well as the RBs at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9 µm, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' There is not equivalent for the feature at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='3 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' NP LMg (VdK) NC (Pep) CBC (Pep) Glass Standard deviation Start Material Glass Standard deviation Start Material Glass Standard deviation Start Material Na2O 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='65 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='21 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='72 MgO 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='23 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='73 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='65 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='43 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='81 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='88 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='55 RaB (VdK) NC (Pep) 1 glass?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='15 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='99 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='94 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='73 2 glass?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='11 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='98 CBC (Pep) 3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='12 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='95 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='14 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='94 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='89 4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='94 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='13 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='44 2 7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='98 3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='15 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='7 NVPa (Stock) 4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='15 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='16 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='47 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='92 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='58 G1 (Char) 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='92 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='52 1 glass ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='22 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='11 N HMG (VdK) 2 glass ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='25 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='1 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='04 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='82 3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='16 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='12 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='07 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='4 HMR CaS (VdK) HAI (VdK) 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='25 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='16 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='71 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='02 2 glass ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='23 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='13 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='13 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='96 3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='28 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='15 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='77 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='87 IC (Pep) HMC (Pep) 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='14 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='94 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='22 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='63 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='15 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='89 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='11 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='95 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='15 11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='9 IT (VdK) 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='13 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='99 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='18 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='12 PD (VdK) 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='21 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='96 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='06 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='21 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='98 Table 4: Band positions of selected spots analyzed in-situ with micro-FTIR (aperture 256×256µm) on polished sections of the samples HMC (Pep) NC (Pep) CBC (Pep) CB (VdK) NP LMg (VdK) NP HMg (VdK) HAl (VdK) G2 (Char) NVPa (Stock) IcP HCTa (Stock) PD (VdK) IC (Pep) IT (VdK) RaB (VdK) HMR (VdK) G1 (Char) HMR CaS (VdK) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 1 1 2 1 2 1 2 1 2 1 1 1 2 1 2 1 2 1 2 1 2 4 3 1 3 1 2 2 3 1 2 500 µm 500 µm 500 µm 500 µm 500 µm 500 µm 500 µm 500 µm 500 µm 500 µm 500 µm 500 µm 500 µm 500 µm 500 µm 500 µm 500 µm 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='00 umFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 2a NC Pep NP LMg vdK CBC Pep CB vdK NP HMg vdK W HAI vdK G2 Char Intensity [arbitrary units IC Pep IT Pep tetraedra silicate coordinated framework cations 400 600 800 1000 1200 Raman Shift [cm1]Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 2b Glassy matrix+ Olivine 823 cm 1 857 cm 1 724 cm 1 Intensity (arbritary units) Olivine Wy Spinel + Olivine 400 600 800 1000 1200 1400 1600 Raman Shift (cm 1)Glass/Initial Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='80 NP LMg (VdK) NC (Pep) CBC (Pep) CB (VdK) NVPa (Stock) NP HMg (VdK) G2 (Char) HAI (VdK) IC (Pep) IT (VdK) PD (VdK) IcP HCTa (Stock) RB (VdK) HMR (VdK) G1 (Char) HMR CaS (VdK) HMC (Pep) MgO Al2O3 SiO2 CaO FeO Increasing Mg content Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 4 OI 10μm O1 10μm OI Sp Mgo 10μm0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='08 Reflectance 6 8 10 12 14 16 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='00 µm G1 (Char) 6 8 10 12 14 16 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='02 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='08 Reflectance µm IT (VdK) NC (Pep) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} 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PD (VdK) HMR (VdK) G1 (Char) HMR Cas (VdK) HMC (Pep) Relative Reflectance 6 8 10 12 14 16 18 µm G2 (Char) NVPa (Stock) ICP HCT(Stock) RaB (VdK) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 6a CF RB NP LMg (VdK) CBC (Pep) CB (VdK) NC (Pep) NP HMg (VdK) HAl (VdK) IC (Pep) IT (VdK) PD (VdK) G2 (Char) RaB (VdK) HMR (VdK) HMR (VdK) G1 (Char) G1 (Char) HMR Cas (VdK) HMR Cas (VdK) HMC (Pep) NVPa (Stock) Relative Reflectance 6 8 10 12 14 16 18 µm Relative Reflectance 6 8 10 12 14 16 18 µm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 6b CF RB RB 50 60 70 80 90 100 40 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='20 7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 7 CF in µm SiO2 (wt%) 40 50 60 70 80 90 100 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content='6 9.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016) Synthetic Glass (DuFresne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2009) Synthetic Glass (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2010) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 8 Strongest RB in µm SiO2 (wt%) Micro-FTIR synthetic glass Powder FTIR synthetic glass Impact Glass (Morlok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=', 2016) SCFM vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/etFRT4oBgHgl3EQfVTd7/content/2301.13539v1.pdf'} +page_content=' 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Alessandro Pieropan2 and Patric Jensfelt1 +Abstract— Visual localization allows autonomous robots to +relocalize when losing track of their pose by matching their +current observation with past ones. However, ambiguous scenes +pose a challenge for such systems, as repetitive structures +can be viewed from many distinct, equally likely camera +poses, which means it is not sufficient to produce a single +best pose hypothesis. In this work, we propose a probabilistic +framework that for a given image predicts the arbitrarily +shaped posterior distribution of its camera pose. We do this +via a novel formulation of camera pose regression using +variational inference, which allows sampling from the predicted +distribution. Our method outperforms existing methods on +localization in ambiguous scenes. Code and data will be released +at github.com/efreidun/vapor. +I. INTRODUCTION +Visual localization is the task of inferring the ego pose +of a camera from its image. It enables mobile robots to +localize themselves in an environment, which is crucial for +their navigation. Regardless of the paradigm that is followed +to solve this task, the proposed methods revolve around +detection of visual features that are unique to different +regions of the environment and the camera poses that view +them. Some methods do this by retrieving the most similar +image to a query image from a database of images previously +collected in the scene [1], [2], [3], [4]; some establish point +correspondences between the salient features of the query +image and a pre-built 3D feature map, and use projective +geometry relations to estimate the camera pose [5], [6], [7], +[8], [9]; and some delegate this estimation problem to end- +to-end learning-based solutions that regress the camera pose +from what it views [10], [11], [12], [13], [14]. +As long as there are unique identifying features in the +images, there exist numerous solutions that can accurately +estimate the camera pose [6], [15]. However, the same cannot +be said when the scene is ambiguous [16], that is, when it +contains distinct regions that are visually indistinguishable. +Examples of this include identical doors, identical chairs +arranged around a table, or the flights of stairs in a staircase, +as illustrated in Fig. 1. A desired solution in these cases +is one that produces multiple pose hypotheses, capturing +the repetitive patterns of the scene, rather than attempting +to produce a single best hypothesis. This calls for a multi- +hypothesis localization framework, which we address in this +* This work was partially supported by the Wallenberg AI, Autonomous +Systems and Software Program (WASP) funded by the Knut and Alice +Wallenberg Foundation. Authors thank Thien-Minh Nguyen for his help in +recording and obtaining ground-truth poses for the new image sequence. +1 Authors are with the division of Robotics, Perception and Learn- +ing, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden. +{fzk,leonardb,patric}@kth.se +2 Authors are with Univrses AB, SE-118 26 Stockholm, Sweden. +{firstname.lastname}@univrses.com +Variational +Pose Regressor +(a) Ambiguous images +(b) Pose posterior +(c) Samples from pose posterior +Position +Orientation +x +y +Fig. 1. +(a) Visually similar images taken from three different flights +of stairs, (b) camera pose distribution predicted for the right image, and +(c) samples drawn from this posterior. The distribution is visualized by a +position heatmap on the xy-plane (marginalizing height) and an elliptic +orientation heatmap2. We show the drawn samples in a 3D reconstruction +of the scene by small camera frusta in purple. The ground-truth camera +poses are shown by color-coded circles in (b) and camera frusta in (c). +work. We focus on inference of the camera pose distribution +from a single image, and refer to the rich literature on robot +localization for how to accumulate evidence and maintain +such a distribution over time [17], [18], [19]. +We propose a probabilistic framework that allows inferring +the posterior distribution over camera poses for a given +image. We represent this distribution by an arbitrary number +of samples drawn from it, which in theory can model distri- +butions with any number of modes and of any shape. Samples +from this distribution can be used in downstream tasks, such +as motion planning or active localization. We formulate our +solution following the paradigm of end-to-end camera pose +regression, and employ variational inference [21], [22] to +model the visual features of images used for localization. +We show that camera pose regression, despite its limitations +in generalization and accuracy compared to structure-based +methods [23], when combined with variational inference +2We use the Mollweide projection for the surface of the 2-sphere +component of SO(3) obtained through Hopf-fibration (marginalizing the +fibers), inspired by Murphy et al. [20]. +arXiv:2301.02086v1 [cs.CV] 5 Jan 2023 + +V +Vgives rise to a simple, yet powerful solution for pose posterior +prediction from an observed image. +We summarize our contributions as the following: (1) We +lay out a novel formulation of camera pose regression +using variational inference, which allows sampling from an +arbitrarily shaped pose distribution for a given image. (2) We +propose a novel sampling-based Winners-Take-All optimiza- +tion scheme, which allows learning multimodal distributions. +(3) We record a sequence of real-world camera images +capturing a case of severe visual ambiguity for evaluation +of localization solutions. (4) We show that our formulation +outperforms existing methods on ambiguous scenes. +II. RELATED WORK +Regression-based approaches aim to solve the pose esti- +mation problem in a single step by finding a function that +directly maps an image to its pose, promising improved +performance in feature-less environments or under motion +blur [12]. In early work, Shotton et al. [24] proposed re- +gressing 3D scene coordinates for each pixel in an image. In +combination with depth data, this allows robust estimation +of the 6D pose of the camera by employing RANSAC +with Kabsch’s algorithm [25]. The first end-to-end approach +for image-based pose regression was PoseNet proposed by +Kendall et al. [10]. Specifically, they proposed to train a deep +neural network to directly regress the 6D camera pose from +the image features extracted by a pre-trained backbone. +Following this early work, various improvements orthog- +onal to our work have subsequently been proposed. Naseer +and Burgard [26] showed that RGB-D data can be exploited +to generate additional views from the limited training images +to improve performance. Recently, Ng et al. [27] and Moreau +et al. [15] extended this idea to RGB data. Other works +propose to use additional information often available in +robotic applications [11], [13]. +More closely related to our work, several works inves- +tigate how to model uncertainty for pose regressors. In +[28], the authors apply Bayesian deep learning to PoseNet. +This allows one to gauge the uncertainty in the prediction, +although the ability to learn more complicated distributions +remains limited, as noted by [16]. While [28] focused on +epistemic uncertainty, Kendall and Cipolla [29] considered +homoscedastic aleatoric uncertainty by modifying the loss, +and Moreau et al. [30] modeled heteroscedastic aleatoric +uncertainty instead by predicting an uncertainty measure. +Deng et al. [16] further extend the idea of representing +uncertainty by predicting a mixture of multiple unimodal +distributions. In principle, this allows the network to correctly +predict multiple modes for ambiguous queries. A downside +to this mixture-based approach is the difficulty of picking +the correct number of modes, and training the network so +that it actually predicts different modes. To handle the latter +issue, the authors propose a Winner-Takes-All scheme that +only gives supervision to the best predicted mode. Our +work follows a similar idea, but instead of employing a +mixture model with a fixed number of components, we +x +y +p(z) = N (z; 0, I) +ˆy +Encoder +gθ(·) +PoseMap +fφ(·) +∼ +q(z | x) +z +Winners-Take-All Prediction Error +KL Divergence +Fig. 2. +Our pipeline for inference of the camera pose distribution for an +image x ∈ RH×W ×3 with ground-truth pose label y ∈ SE(3). We can +simulate the posterior distribution p(y | x) by drawing samples z ∼ q(z | +x), z ∈ Rd and applying the mapping fφ(z) to get ˆy ∼ p(y | x), ˆy ∈ +SE(3). The loss terms used in the learning objective are shown in purple. +follow a variational approach, which, in principle, can learn +to produce arbitrarily shaped pose distributions. +In another related line of work, Murphy et al. [20] build +on the recent success of neural fields [31] by employing an +MLP that predicts the probability density for a given rotation, +allowing representation of arbitrary distributions in SO(3). +Our work also aims to learn arbitrary distributions, but we +propose a sampling-based approach, in which a sample from +a latent space is transformed to a pose in SE(3). This +simplifies inference, as it does not require dense querying +of the support to find the modes of the distribution; instead, +our approach allows direct sampling from it. +III. METHOD +We propose to perform visual localization for an image in +two steps: (1) infer a distribution in the latent space capturing +the visual features that are useful for localization within +the scene; (2) perform a random variable transformation to +obtain a distribution of camera poses for the query image. +Fig. 2 visualizes our proposed pipeline. +A. Formulation +Let x ∈ RH×W ×3 be a color image taken from camera +pose y ∈ SE(3). In localization, where the scene is known +beforehand, one can in theory infer the posterior distribution +of visual features as seen in the observed image p(z | x). +Here, z ∈ Rd is the latent variable corresponding to the +visual features that the scene comprises. With this definition +of the latent variable, visually similar images result in similar +posterior distributions in the latent space, even if the images +are taken from distinct camera poses, as in ambiguous scenes. +Having full knowledge of the scene, the posterior dis- +tribution of visual features should contain the information +needed to infer the posterior distribution of camera poses +given the observed image p(y | x). This can be formulated +as a transformation of densities from visual features in Rd to +camera pose in SE(3), which can be achieved by applying +a deterministic mapping f +: Rd → SE(3) to samples +drawn from the posterior distribution in the latent space: +y = f(z), +z ∼ p(z | x). +B. Modeling via learning +In +the +proposed +formulation, +there +are +two +scene- +dependent operations that model the scene for the purpose + +of visual localization, namely the inference of the posterior +distribution in the latent visual features’ space p(z | ·), and +the mapping to camera pose f(·). We parameterize these in +the weights of two deep neural networks and learn them +from data samples collected from the scene. We refer to +the two networks as Encoder gθ(·) and PoseMap fφ(·), +parameterized by θ and φ, respectively. +Encoder gθ(·) is an inference network with a Gaussian +inference model that for an input image x outputs µ ∈ Rd +and σ ∈ Rd defining the posterior distribution q(z | x) = +N(z; µ, diag(σ2)) in the latent space. This follows the vari- +ational principle, where an unknown posterior distribution is +modeled by optimizing the parameters of a convenient family +of distributions such as Gaussians to best resemble the true +posterior. Akin to Variational Auto-Encoders (VAEs) [21], +[22], we amortize this per-image optimization at inference +time by optimizing the Encoder weights at training time to +directly predict the distribution parameters. +PoseMap fφ(·) is a fully connected network that, for +an input sample from the latent space z, outputs a camera +pose y. This means that the posterior distribution of the +camera pose p(y | x) can be approximated by simulating +the inferred posterior distribution in the latent space z ∼ +q(z | x) via reparameterization trick and passing the drawn +samples through the mapping y = fφ(z) to obtain samples +y ∼ p(y | x). The output of the network y comprises +a translation vector t ∈ R3 and a 6D representation for +rotation r ∈ R6. The rotation parameterization choice is the +continuous representation for rotations in 3D introduced by +Zhou et al. [32], where a rotation matrix is retrieved from the +6D representation following a Gram-Schmidt-like process. +C. Learning scheme +The network weights θ, φ that represent a scene are learned +from a dataset of images and camera poses D = {(xi, yi) | +i = 1, . . . , N} collected in that scene. For this, we lay out an +optimization scheme that enables learning multimodal pose +distributions as is desired in ambiguous scenes. +1) Objective terms: Prediction error measures the qual- +ity of a predicted pose ˆy ∈ SE(3) against its ground truth +y. We define the prediction error as the weighted sum of +a translation error term defined on R3 and a rotation error +term defined on SO(3). The translation error is the Euclidean +distance between the translation components ˆt, t ∈ R3 of +predicted and ground-truth poses. For the rotation error, we +opt for the chordal distance between the rotation compo- +nents ˆR, R ∈ SO(3), for its more favorable gradients in +gradient-based optimization than, for example, the geodesic +distance’s. The prediction error is thus defined as +dpose(ˆy, y) = λt∥ˆt − t∥2 + λr∥ ˆR − R∥F, +(1) +where λt and λr are tunable constants, balancing the scales +of the two terms. +Kullback–Leibler divergence DKL(q(z | x) ∥ p(z)) mea- +sures how different an inferred latent posterior distribution +q(z | x) is from a prior distribution defined on the latent +variable p(z). This is an integral part of the variational +principle, which together with the prediction error forms +the evidence lower bound (ELBO) optimized in variational +approaches. As is common practice, we assume a standard +Gaussian prior p(z) = N(z; 0, I) for its simplicity in +computing the KL divergence. +2) Evidence lower bound (ELBO): In variational ap- +proaches, the ELBO objective that is typically maximized +is a combination of negative KL divergence and expected +log-likelihood of predictions Eq(z|x)[log pφ(y | z)]. The +latter expectation is generally computed by Monte Carlo +simulation of q(z | x). With our choice of pose prediction +error, the variational optimization objective can be written as +min +θ,φ +� +xi,yi∈D +� +DKL(q(z | xi) ∥ p(z)) ++ +1 +|Zi| +� +zj∈Zi +dpose(fφ(zj), yi) +� +, +(2) +where Zi = {zj ∼ q(z | xi) | j = 1, . . . , M} is the Monte +Carlo sample set and |Zi| its cardinality. +We argue that minimizing this objective, and specifically +the expected prediction error, is counterproductive in our +setting, where the camera pose posterior p(y | z) can be +multimodal in ambiguous scenes. In such scenarios, two +visually similar images xi and xj (i ̸= j) are encoded to +similar latent posterior distributions p(z | xi) and p(z | xj). +However, these images can be taken from two distinct poses +yi and yj in the scene, in which case the true posterior +distributions of the camera pose p(y | xi) and p(y | xj) +are both bimodal. Minimizing the expected prediction error +results in a compromised solution in the form of a unimodal +inferred distribution between the two true modes. We propose +a modification of the expected error term to address this. +3) Winners-Take-All optimization: We propose to confine +the computed mean prediction error to a subset of Monte +Carlo samples ˆZi ⊆ Zi, whose image through the mapping +fφ(·) is within a certain distance δ of the true mode yi, that +is, ˆZi = {zj ∈ Zi | dpose(fφ(zj), yi) < δ}. This ensures +that pose samples can concentrate around individual modes +during optimization without influence from other modes. +However, the true posterior is unknown and different modes +can have different shapes, rendering the choice of δ non- +trivial. Moreover, random initialization of the parameters θ +and φ does not guarantee that there will be pose samples +within any δ distance of the modes at the start of the +optimization. This calls for an adaptive selection of δ at every +iteration and for every mode. +At every iteration and for a ground-truth pose yi we pick +δi,α as the radius of the smallest ball centered at yi containing +a fraction α of samples in Zi. In other words, our adaptive +δi,α, defined as +δi,α = inf +� +δ ∈ R+ +��� +�� ˆZi +�� = ⌊α · |Zi|⌋ +� +, +(3) +results in minimizing the prediction error for only the closest +fraction α of Monte Carlo samples per ground-truth pose yi. + +Our proposed optimization objective is +min +θ,φ +� +xi,yi∈D +� +β DKL(q(z | xi) ∥ p(z)) ++ +1 +| ˆZi,α| +� +zj∈ ˆ +Zi,α +dpose(fφ(zj), yi) +� +, +(4) +where ˆZi,α = {zj ∈ Zi | dpose(fφ(zj), yi) < δi,α}. α and β +are tunable constants, the latter being the balancing weight +for the KL divergence term. +This is in spirit similar to the Winner-Takes-All multi- +hypothesis optimization scheme used for learning mixture +models, where the closest mixture component is optimized +per label [16], [33]. However, our proposed solution is in +a different setting, as we represent posteriors by samples +instead of mixture models. We therefore refer to our method +as Winners-Take-All to acknowledge this similarity, while +reflecting the fact that it is used for optimizing sample sets +rather than individual mixture components. +IV. EXPERIMENTS +A. Implementation details +We implement our method using the PyTorch library [34]. +We use ResNet-18 [35] as the backbone of the Encoder +to extract 2048-dimensional feature vectors, followed by a +linear layer to predict d-dimensional µ and log σ2 vectors +for the latent posterior. The PoseMap is implemented with a +fully connected network taking the input vector through the +dimensionality transformation d → 128(→ 128)×nlayers → +3 + 6 with ReLU activations in-between. The minimum +number of hidden layers nlayers depends on the complexity of +the target pose distributions in the scene. In nearly all tested +scenes we achieved favorable performance with as few as +nlayers = 3, which, unless otherwise stated, is used across all +experiments. The final layer corresponds to the prediction +of translation and rotation vectors, where the former goes +through a sigmoid activation, followed by a fixed affine +transformation that shifts and scales the predictions to the +metric ranges of the scene. +We train our networks using Adam optimizer [36] with +initial learning rate of 1 × 10−4 and an exponential learning +rate decay of 0.8, applied every nlr-decay epochs for 10 +occurences. Following the pose regression literature, we +first resize each image such that its smallest edge is 256, +then randomly crop 224 × 224 regions for input to the +Encoder. We also augment the data with color/brightness +jittering and Gaussian blur to account for lighting changes +and motion blur between images. Unless otherwise stated, +we let α = 0.20, β = 0.01, use a d = 16-dimensional latent +space, and represent distributions with 1000 Monte Carlo +samples in all experiments, since we found this to produce +good predictions in our setting. Other hyperparameters are +reported in Table I, tuned to reflect the number of images +and metric scales of different datasets, which range from +small indoor to large outdoor scenes. Note that we found +these settings without a major hyperparameter search, and +TABLE I +HYPERPARAMETERS USED IN TRAINING +Dataset +λt +λr +Batch Size +# Epochs +nlr-decay +7-Scenes [24] +5 +10 +64 +100 +10 +Cambridge Land. [10] +5 +100 +64 +500 +50 +Ambiguous Reloc. [16] +5 +2 +4 +500 +50 +Ceiling +5 +2 +4 +2000 +50 +Synthetic +5 +2 +4 +500 +50 +one may improve the performance by a thorough search of +the optimal hyperparameters. +B. Datasets and metrics +We evaluate our method on the Ambiguous Relocalization +dataset [16] as an existing benchmark with real-world image +sequences of ambiguous environments. For each scene in the +dataset there are separate training and test image sequences +recorded from their own unique camera trajectories, but with +generally similar views. We found that despite the apparent +ambiguity to the human eye, a large fraction of frames in +this dataset contain unique identifying features, which an +expressive feature detector can infer the pose from. This +results in unimodal predicted posteriors for a large number of +frames, which hinders the evaluation of a method’s capability +in forming multimodal distributions. To address this, we +complement the dataset by recording a new real-world se- +quence of a ceiling with machine-fabricated panels, capturing +a case of severe visual ambiguity. We record the training and +test sequences with a calibrated LiDAR-IMU-camera rig, and +obtain ground-truth camera poses using MILIOM [37]. We +also render image sequences of two synthetic scenes from +3D Warehouse3, which contain symmetries by design, and +use them to investigate our method in a controlled setting. +We use recall as the metric to evaluate pose distributions +in ambiguous scenes. For a query image, we draw samples +from its posterior distribution, and consider it a true positive +if at least a fraction γ of the samples are within a distance of +the ground-truth pose (and a false negative otherwise). We +argue that for a distribution with well-separated equally likely +modes, setting γ inversely proportional to the number of +modes gives an estimate of whether the distribution contains +sufficient density around the ground-truth pose. We report +recall with γ = 0.1 for all tested scenes except for the ceiling +scene, where we use γ = 0.05. +To validate the performance of our method as a general +pose regressor on unambiguous scenes, we evaluate it on the +visual localization benchmarks 7-Scenes [24] and Cambridge +Landmarks [10]. As is commonly reported by pose regres- +sion works, we use median error for evaluation on these +datasets. We obtain a point prediction from the Monte Carlo +samples of each predicted distribution using the arithmetic +and chordal L2 [38] means for translation and orientation, +respectively. The median of this estimate’s error compared +to the ground-truth pose is reported across each scene. +3https://3dwarehouse.sketchup.com/ + +0 m +2 m +4 m +6 m +8 m +10 m +12 m +14 m +16 m +0 m +1 m +Fig. 3. +Marginal posterior distributions along x-axis (top left) and y-axis (bottom right) predicted by our method ( +), by Bingham MDN [16] ( +), the +prediction by MapMet [13] ( +), and the ground truth ( +) for a query image (top right) from the ceiling scene. The heatmap shows the 2D histogram +predicted by our method overlaid on top of stitched images of the scene. Note that our method successfully captures all six modes of the distribution while +MapNet only predicts a single estimate at a wrong location, and Bingham MDN method assigns large probabilities in visually dissimilar locations. +TABLE II +MEASURED RECALL IN AMBIGUOUS SCENES (HIGHER IS BETTER) +Scene +Threshold +PN [10] +MN [13] +BMDN [16] +Abl. +Ours +Blue Chairs +0.1m/10◦ +0.08 +0.05 +0.41 +0.32 +0.45 +0.2m/15◦ +0.40 +0.33 +0.83 +0.89 +0.99 +0.3m/20◦ +0.56 +0.46 +0.89 +0.97 +1.00 +Meeting Table +0.1m/10◦ +0.00 +0.00 +0.09 +0.03 +0.06 +0.2m/15◦ +0.02 +0.03 +0.27 +0.24 +0.35 +0.3m/20◦ +0.02 +0.07 +0.33 +0.34 +0.43 +Staircase +0.1m/10◦ +0.00 +0.07 +0.24 +0.12 +0.30 +0.2m/15◦ +0.01 +0.17 +0.48 +0.44 +0.62 +0.3m/20◦ +0.01 +0.29 +0.69 +0.63 +0.72 +Staircase Ext. +0.1m/10◦ +0.00 +0.01 +0.11 +0.03 +0.12 +0.2m/15◦ +0.00 +0.03 +0.43 +0.24 +0.53 +0.3m/20◦ +0.01 +0.07 +0.60 +0.44 +0.71 +Seminar Room +0.1m/10◦ +0.00 +0.09 +0.38 +0.17 +0.43 +0.2m/15◦ +0.02 +0.37 +0.79 +0.53 +0.90 +0.3m/20◦ +0.10 +0.53 +0.91 +0.80 +0.97 +Ceiling† +0.1m/10◦ +0.00 +0.02 +0.08 +0.00 +0.09 +0.2m/15◦ +0.03 +0.05 +0.19 +0.02 +0.31 +0.3m/20◦ +0.06 +0.09 +0.30 +0.04 +0.44 +† We train the independently recorded ceiling scene with α = 0.05 and nlayers = +9, reflecting the richer presence of ambiguities in the scene. +C. Evaluation on benchmark datasets +We report the results on the ambiguous scenes in Table +II. We can see that our method, outperforms Bingham +MDN [16] as the method closest to ours that predicts a +distribution of poses aimed at localization in ambiguous +scenes. We considered two settings of their approach with +10 and 50 components in their mixture model, and evaluated +the metric based on samples drawn from them. As the 10- +component setting consistently performed better, we report +its results as a representative in the table (marked BMDN). +Fig. 3 shows an example of the predicted posterior given +a query image from the ceiling scene, where we can see +posterior predicted by our method better captures the am- +biguous structure of the scene. We also evaluate PoseNet +[10] and its Bayesian variant [28], as well as MapNet [13]. +However, we see that these single estimate methods fail to +achieve comparable performance on the ambiguous scenes. +To our surprise, vanilla PoseNet performed comparatively +better than Bayesian PoseNet, so we include its results as +representative (marked PN) alongside MapNet (marked MN). +In order to investigate whether our method’s improved +performance stems from our novel formulation with varia- +tional inference, we perform an ablation, in which we modify +our pipeline to produce a single pose for an input image. +TABLE III +MEDIAN ERROR (M / ◦) IN UNAMBIGUOUS SCENES (LOWER IS BETTER) +Scene +PN [10] +MN† [13] +BPN [28] +BMDN [16] +Ours +Chess +0.32/8.12 +0.08/3.25 +0.37/7.24 +0.10/6.47 +0.17/6.90 +Fire +0.47/14.4 +0.27/11.7 +0.43/13.7 +0.26/14.8 +0.30/14.1 +Heads +0.29/12.0 +0.18/13.3 +0.31/12.0 +0.13/13.4 +0.17/14.5 +Office +0.48/7.68 +0.17/5.15 +0.48/8.04 +0.19/9.73 +0.24/9.30 +Pumpkin +0.47/8.42 +0.22/4.02 +0.61/7.08 +0.20/9.40 +0.30/8.33 +Kitchen +0.59/8.64 +0.23/4.93 +0.58/7.54 +0.19/10.9 +0.26/10.2 +Stairs +0.47/13.8 +0.30/12.1 +0.48/13.1 +0.34/14.1 +0.47/15.5 +College +1.92/5.40 +1.07/1.89 +1.74/4.06 +1.51/2.14 +1.65/2.88 +Street +3.67/6.50 +− +2.96/6.00 +16.3/25.2 +17.2/23.8 +Hospital +2.31/5.38 +1.94/3.91 +2.57/5.14 +2.25/3.93 +2.06/4.33 +Fac¸ade +1.46/8.08 +1.49/4.22 +1.25/7.54 +3.52/5.41 +1.02/6.03 +Church +2.65/8.48 +2.00/4.53 +2.11/8.38 +2.16/5.99 +1.80/5.90 +† Results of MapNet on Cambridge Landmarks taken from Sattler et al. [23]. +We remove the KL divergence term from the objective, +modify the Encoder to predict a single point, and obtain +a single pose prediction by passing the encoder’s prediction +through PoseMap. All else equal, we evaluate this ablative +variant of our method that is in principle very similar to +PoseNet. We can see in Table II that this variant, marked +Abl., while performing better than PoseNet due to its more +recent feature extractor network, falls short of the unablated +variant, validating the merit of our proposed formulation. +For completeness, we report our results on the unambigu- +ous 7-Scenes and Cambridge Landmarks datasets in Table +III. We include results of PoseNet and MapNet as single +pose regressor baselines, and Bayesian PoseNet and Bingham +MDN as methods that, in principle, can predict multimodal +distributions. While our method does not perform the best, it +is not far from the top-performers. This experiment merely +serves as a sanity check of our approach’s performance in a +minimal pipeline, without any particular mechanism aimed +at improving accuracy in unambiguous scenes. As seen in +Table II, the better performing methods on unambiguous +scenes show poor performance on ambiguous scenes, which +is the problem that our method targets to solve. An interesting +direction for future work is to apply our proposed formula- +tion, aimed at handling ambiguous scenes, in tandem with +techniques for improved unambiguous pose regression. +D. A closer look +Fig. 4 (top) shows the predicted distribution by our method +for an example query image from the Ambiguous Relocaliza- +tion dataset. Although the scene, made up of identical chairs, +is arguably ambiguous to the human eye, we can see that + +(a) Query image +(b) Position posterior +(c) Samples +No Weight Decay +Weight Decay +Fig. 4. +Posteriors predicted by our method in full capacity (top) and in a +constrained learning mode by L2 weight decay of λ = 0.1 (bottom). The +latter captures multiple modes whereas the former mode predicts a single +mode at the correct pose. +the predicted posterior identifies and concentrates its density +around the correct pose. We hypothesize that a sufficiently +expressive Encoder can distinguish a seemingly ambiguous +image taken in real life by its smallest of details, such as +the chair’s background in this example. However, a less +expressive Encoder for the data is unable to learn every detail +and can give in to the ambiguities. We test this hypothesis by +adding a penalty term on the L2 norm of the Encoder weights +during training. We can see in Fig. 4 (bottom) that this setting +results in the predicted posterior assigning probabilities to +poses viewing two additional chairs. We argue that when +there exists a domain gap between the training data and the +operation conditions, it is desirable for the model to trade +off confidence in predictions for better generalization, which +can be achieved via deliberate learning constraints. We leave +the study of such learning constraints to future work. +We study the effect of α in the Winners-Take-All optimiza- +tion scheme in two synthetic scenes, where the camera circles +around a round table with four legs, resulting in four modes +in the pose distribution of an image, as well as a rectangular +dinner table that results in bimodal distributions. We report +the statistics over 10 training runs for the 0.1m/10◦ recall +evaluated at the end of training with different α values in +Fig. 6. We can see that in these scenes the highest recall +is achieved with α in a range of values greater than zero +but less than 1/# modes. Fig. 5 shows the predicted camera +position posterior for three choices of α. We can see that +a too large α, as discussed in Section III-C.2, results in a +compromised posterior, and a too small α predicts close-to- +uniform densities across the span of the training data. We +hypothesize that α must be smaller than 1/# modes for +the Winners-Take-All optimization to converge and capture +all modes in the distribution, and must be sufficiently larger +than zero to overcome the noise as a result of mini-batch +optimization. There is a trade-off between training speed +and the quality of the learned distribution within this range +of α values, as a smaller α results in optimization of fewer +samples at every iteration, hence a slower training, but is less +Scene +α = 0.01 +α = 0.20 +α = 1.00 +Round Table +Dinner Table +Fig. 5. +Predicted position posterior on xy-plane for various choices of α +in the synthetic scenes. The round table and dinner table have four and two +modes in their true distributions, respectively. The optimization successfully +converges to predict the correct modes when 0 ≪ α < 1/# modes. For +example α = 0.20 produces good predictions for both scenes. +0.01 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +0.5 +1 +α +(a) Round Table +0.01 +0.2 +0.4 +0.6 +0.8 +1.0 +α +(b) Dinner Table +Fig. 6. +Measured recall with threshold 0.1m/10◦ for various choices of α +in Winners-Take-All optimization of the synthetic scenes. For each choice +of α, the statistics over 10 training runs is shown by a box extending from +the lower to upper quartile recall values, a purple line at the median recall, +and whiskers that extend to the minimum and maximum values. +susceptible to the noise induced by Monte Carlo sampling. +We leave the study of finding the optimal α to future work. +E. Run-time evaluation +We measure the time taken for a forward pass of one query +image through our pipeline for 1000 Monte Carlo samples, +on a desktop computer with an Intel Core i7-8700K CPU and +an NVIDIA GeForce GTX 1080 Ti GPU. We repeat each +measurement 100 times and we find that a forward pass on +average takes 14.88 ± 0.75 ms on CPU and 2.26 ± 0.10 ms +on GPU, that is, our pipeline can run in real time. +V. CONCLUSION +In this work, we addressed the task of visual localization +in ambiguous scenes. 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Li, “Rotation averaging,” +International Journal of Computer Vision, vol. 103, no. 3, pp. 267– +305, 2013. + diff --git a/lNA0T4oBgHgl3EQfJP8k/content/tmp_files/load_file.txt b/lNA0T4oBgHgl3EQfJP8k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..be3f5521207ea29d09afa11a353d3b60e8858113 --- /dev/null +++ b/lNA0T4oBgHgl3EQfJP8k/content/tmp_files/load_file.txt @@ -0,0 +1,803 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf,len=802 +page_content='A Probabilistic Framework for Visual Localization in Ambiguous Scenes Fereidoon Zangeneh1,2, Leonard Bruns1, Amit Dekel2, Alessandro Pieropan2 and Patric Jensfelt1 Abstract— Visual localization allows autonomous robots to relocalize when losing track of their pose by matching their current observation with past ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' However, ambiguous scenes pose a challenge for such systems, as repetitive structures can be viewed from many distinct, equally likely camera poses, which means it is not sufficient to produce a single best pose hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' In this work, we propose a probabilistic framework that for a given image predicts the arbitrarily shaped posterior distribution of its camera pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We do this via a novel formulation of camera pose regression using variational inference, which allows sampling from the predicted distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Our method outperforms existing methods on localization in ambiguous scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Code and data will be released at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='com/efreidun/vapor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' INTRODUCTION Visual localization is the task of inferring the ego pose of a camera from its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' It enables mobile robots to localize themselves in an environment, which is crucial for their navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Regardless of the paradigm that is followed to solve this task, the proposed methods revolve around detection of visual features that are unique to different regions of the environment and the camera poses that view them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Some methods do this by retrieving the most similar image to a query image from a database of images previously collected in the scene [1], [2], [3], [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' some establish point correspondences between the salient features of the query image and a pre-built 3D feature map, and use projective geometry relations to estimate the camera pose [5], [6], [7], [8], [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' and some delegate this estimation problem to end- to-end learning-based solutions that regress the camera pose from what it views [10], [11], [12], [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' As long as there are unique identifying features in the images, there exist numerous solutions that can accurately estimate the camera pose [6], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' However, the same cannot be said when the scene is ambiguous [16], that is, when it contains distinct regions that are visually indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Examples of this include identical doors, identical chairs arranged around a table, or the flights of stairs in a staircase, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' A desired solution in these cases is one that produces multiple pose hypotheses, capturing the repetitive patterns of the scene, rather than attempting to produce a single best hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' This calls for a multi- hypothesis localization framework, which we address in this This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Authors thank Thien-Minh Nguyen for his help in recording and obtaining ground-truth poses for the new image sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 1 Authors are with the division of Robotics, Perception and Learn- ing, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' {fzk,leonardb,patric}@kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='se 2 Authors are with Univrses AB, SE-118 26 Stockholm, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' {firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='lastname}@univrses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='com Variational Pose Regressor (a) Ambiguous images (b) Pose posterior (c) Samples from pose posterior Position Orientation x y Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' (a) Visually similar images taken from three different flights of stairs, (b) camera pose distribution predicted for the right image, and (c) samples drawn from this posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The distribution is visualized by a position heatmap on the xy-plane (marginalizing height) and an elliptic orientation heatmap2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We show the drawn samples in a 3D reconstruction of the scene by small camera frusta in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The ground-truth camera poses are shown by color-coded circles in (b) and camera frusta in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We focus on inference of the camera pose distribution from a single image, and refer to the rich literature on robot localization for how to accumulate evidence and maintain such a distribution over time [17], [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We propose a probabilistic framework that allows inferring the posterior distribution over camera poses for a given image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We represent this distribution by an arbitrary number of samples drawn from it, which in theory can model distri- butions with any number of modes and of any shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Samples from this distribution can be used in downstream tasks, such as motion planning or active localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We formulate our solution following the paradigm of end-to-end camera pose regression, and employ variational inference [21], [22] to model the visual features of images used for localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We show that camera pose regression, despite its limitations in generalization and accuracy compared to structure-based methods [23], when combined with variational inference 2We use the Mollweide projection for the surface of the 2-sphere component of SO(3) obtained through Hopf-fibration (marginalizing the fibers), inspired by Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='02086v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='CV] 5 Jan 2023 V Vgives rise to a simple, yet powerful solution for pose posterior prediction from an observed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We summarize our contributions as the following: (1) We lay out a novel formulation of camera pose regression using variational inference, which allows sampling from an arbitrarily shaped pose distribution for a given image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' (2) We propose a novel sampling-based Winners-Take-All optimiza- tion scheme, which allows learning multimodal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' (3) We record a sequence of real-world camera images capturing a case of severe visual ambiguity for evaluation of localization solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' (4) We show that our formulation outperforms existing methods on ambiguous scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' RELATED WORK Regression-based approaches aim to solve the pose esti- mation problem in a single step by finding a function that directly maps an image to its pose, promising improved performance in feature-less environments or under motion blur [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' In early work, Shotton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' [24] proposed re- gressing 3D scene coordinates for each pixel in an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' In combination with depth data, this allows robust estimation of the 6D pose of the camera by employing RANSAC with Kabsch’s algorithm [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The first end-to-end approach for image-based pose regression was PoseNet proposed by Kendall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Specifically, they proposed to train a deep neural network to directly regress the 6D camera pose from the image features extracted by a pre-trained backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Following this early work, various improvements orthog- onal to our work have subsequently been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Naseer and Burgard [26] showed that RGB-D data can be exploited to generate additional views from the limited training images to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Recently, Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' [27] and Moreau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' [15] extended this idea to RGB data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Other works propose to use additional information often available in robotic applications [11], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' More closely related to our work, several works inves- tigate how to model uncertainty for pose regressors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' In [28], the authors apply Bayesian deep learning to PoseNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' This allows one to gauge the uncertainty in the prediction, although the ability to learn more complicated distributions remains limited, as noted by [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' While [28] focused on epistemic uncertainty, Kendall and Cipolla [29] considered homoscedastic aleatoric uncertainty by modifying the loss, and Moreau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' [30] modeled heteroscedastic aleatoric uncertainty instead by predicting an uncertainty measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' [16] further extend the idea of representing uncertainty by predicting a mixture of multiple unimodal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' In principle, this allows the network to correctly predict multiple modes for ambiguous queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' A downside to this mixture-based approach is the difficulty of picking the correct number of modes, and training the network so that it actually predicts different modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' To handle the latter issue, the authors propose a Winner-Takes-All scheme that only gives supervision to the best predicted mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Our work follows a similar idea, but instead of employing a mixture model with a fixed number of components, we x y p(z) = N (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 0, I) ˆy Encoder gθ(·) PoseMap fφ(·) ∼ q(z | x) z Winners-Take-All Prediction Error KL Divergence Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Our pipeline for inference of the camera pose distribution for an image x ∈ RH×W ×3 with ground-truth pose label y ∈ SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We can simulate the posterior distribution p(y | x) by drawing samples z ∼ q(z | x), z ∈ Rd and applying the mapping fφ(z) to get ˆy ∼ p(y | x), ˆy ∈ SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The loss terms used in the learning objective are shown in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' follow a variational approach, which, in principle, can learn to produce arbitrarily shaped pose distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' In another related line of work, Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' [20] build on the recent success of neural fields [31] by employing an MLP that predicts the probability density for a given rotation, allowing representation of arbitrary distributions in SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Our work also aims to learn arbitrary distributions, but we propose a sampling-based approach, in which a sample from a latent space is transformed to a pose in SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' This simplifies inference, as it does not require dense querying of the support to find the modes of the distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' instead, our approach allows direct sampling from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' METHOD We propose to perform visual localization for an image in two steps: (1) infer a distribution in the latent space capturing the visual features that are useful for localization within the scene;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' (2) perform a random variable transformation to obtain a distribution of camera poses for the query image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 2 visualizes our proposed pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Formulation Let x ∈ RH×W ×3 be a color image taken from camera pose y ∈ SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' In localization, where the scene is known beforehand, one can in theory infer the posterior distribution of visual features as seen in the observed image p(z | x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Here, z ∈ Rd is the latent variable corresponding to the visual features that the scene comprises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' With this definition of the latent variable, visually similar images result in similar posterior distributions in the latent space, even if the images are taken from distinct camera poses, as in ambiguous scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Having full knowledge of the scene, the posterior dis- tribution of visual features should contain the information needed to infer the posterior distribution of camera poses given the observed image p(y | x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' This can be formulated as a transformation of densities from visual features in Rd to camera pose in SE(3), which can be achieved by applying a deterministic mapping f : Rd → SE(3) to samples drawn from the posterior distribution in the latent space: y = f(z), z ∼ p(z | x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Modeling via learning In the proposed formulation, there are two scene- dependent operations that model the scene for the purpose of visual localization, namely the inference of the posterior distribution in the latent visual features’ space p(z | ·), and the mapping to camera pose f(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We parameterize these in the weights of two deep neural networks and learn them from data samples collected from the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We refer to the two networks as Encoder gθ(·) and PoseMap fφ(·), parameterized by θ and φ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Encoder gθ(·) is an inference network with a Gaussian inference model that for an input image x outputs µ ∈ Rd and σ ∈ Rd defining the posterior distribution q(z | x) = N(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' µ, diag(σ2)) in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' This follows the vari- ational principle, where an unknown posterior distribution is modeled by optimizing the parameters of a convenient family of distributions such as Gaussians to best resemble the true posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Akin to Variational Auto-Encoders (VAEs) [21], [22], we amortize this per-image optimization at inference time by optimizing the Encoder weights at training time to directly predict the distribution parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' PoseMap fφ(·) is a fully connected network that, for an input sample from the latent space z, outputs a camera pose y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' This means that the posterior distribution of the camera pose p(y | x) can be approximated by simulating the inferred posterior distribution in the latent space z ∼ q(z | x) via reparameterization trick and passing the drawn samples through the mapping y = fφ(z) to obtain samples y ∼ p(y | x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The output of the network y comprises a translation vector t ∈ R3 and a 6D representation for rotation r ∈ R6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The rotation parameterization choice is the continuous representation for rotations in 3D introduced by Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' [32], where a rotation matrix is retrieved from the 6D representation following a Gram-Schmidt-like process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Learning scheme The network weights θ, φ that represent a scene are learned from a dataset of images and camera poses D = {(xi, yi) | i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' , N} collected in that scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' For this, we lay out an optimization scheme that enables learning multimodal pose distributions as is desired in ambiguous scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 1) Objective terms: Prediction error measures the qual- ity of a predicted pose ˆy ∈ SE(3) against its ground truth y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We define the prediction error as the weighted sum of a translation error term defined on R3 and a rotation error term defined on SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The translation error is the Euclidean distance between the translation components ˆt, t ∈ R3 of predicted and ground-truth poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' For the rotation error, we opt for the chordal distance between the rotation compo- nents ˆR, R ∈ SO(3), for its more favorable gradients in gradient-based optimization than, for example, the geodesic distance’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The prediction error is thus defined as dpose(ˆy, y) = λt∥ˆt − t∥2 + λr∥ ˆR − R∥F, (1) where λt and λr are tunable constants, balancing the scales of the two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Kullback–Leibler divergence DKL(q(z | x) ∥ p(z)) mea- sures how different an inferred latent posterior distribution q(z | x) is from a prior distribution defined on the latent variable p(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' This is an integral part of the variational principle, which together with the prediction error forms the evidence lower bound (ELBO) optimized in variational approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' As is common practice, we assume a standard Gaussian prior p(z) = N(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 0, I) for its simplicity in computing the KL divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 2) Evidence lower bound (ELBO): In variational ap- proaches, the ELBO objective that is typically maximized is a combination of negative KL divergence and expected log-likelihood of predictions Eq(z|x)[log pφ(y | z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The latter expectation is generally computed by Monte Carlo simulation of q(z | x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' With our choice of pose prediction error, the variational optimization objective can be written as min θ,φ � xi,yi∈D � DKL(q(z | xi) ∥ p(z)) + 1 |Zi| � zj∈Zi dpose(fφ(zj), yi) � , (2) where Zi = {zj ∼ q(z | xi) | j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' , M} is the Monte Carlo sample set and |Zi| its cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We argue that minimizing this objective, and specifically the expected prediction error, is counterproductive in our setting, where the camera pose posterior p(y | z) can be multimodal in ambiguous scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' In such scenarios, two visually similar images xi and xj (i ̸= j) are encoded to similar latent posterior distributions p(z | xi) and p(z | xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' However, these images can be taken from two distinct poses yi and yj in the scene, in which case the true posterior distributions of the camera pose p(y | xi) and p(y | xj) are both bimodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Minimizing the expected prediction error results in a compromised solution in the form of a unimodal inferred distribution between the two true modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We propose a modification of the expected error term to address this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 3) Winners-Take-All optimization: We propose to confine the computed mean prediction error to a subset of Monte Carlo samples ˆZi ⊆ Zi, whose image through the mapping fφ(·) is within a certain distance δ of the true mode yi, that is, ˆZi = {zj ∈ Zi | dpose(fφ(zj), yi) < δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' This ensures that pose samples can concentrate around individual modes during optimization without influence from other modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' However, the true posterior is unknown and different modes can have different shapes, rendering the choice of δ non- trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Moreover, random initialization of the parameters θ and φ does not guarantee that there will be pose samples within any δ distance of the modes at the start of the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' This calls for an adaptive selection of δ at every iteration and for every mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' At every iteration and for a ground-truth pose yi we pick δi,α as the radius of the smallest ball centered at yi containing a fraction α of samples in Zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' In other words, our adaptive δi,α, defined as δi,α = inf � δ ∈ R+ ��� �� ˆZi �� = ⌊α · |Zi|⌋ � , (3) results in minimizing the prediction error for only the closest fraction α of Monte Carlo samples per ground-truth pose yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Our proposed optimization objective is min θ,φ � xi,yi∈D � β DKL(q(z | xi) ∥ p(z)) + 1 | ˆZi,α| � zj∈ ˆ Zi,α dpose(fφ(zj), yi) � , (4) where ˆZi,α = {zj ∈ Zi | dpose(fφ(zj), yi) < δi,α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' α and β are tunable constants, the latter being the balancing weight for the KL divergence term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' This is in spirit similar to the Winner-Takes-All multi- hypothesis optimization scheme used for learning mixture models, where the closest mixture component is optimized per label [16], [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' However, our proposed solution is in a different setting, as we represent posteriors by samples instead of mixture models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We therefore refer to our method as Winners-Take-All to acknowledge this similarity, while reflecting the fact that it is used for optimizing sample sets rather than individual mixture components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Implementation details We implement our method using the PyTorch library [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We use ResNet-18 [35] as the backbone of the Encoder to extract 2048-dimensional feature vectors, followed by a linear layer to predict d-dimensional µ and log σ2 vectors for the latent posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The PoseMap is implemented with a fully connected network taking the input vector through the dimensionality transformation d → 128(→ 128)×nlayers → 3 + 6 with ReLU activations in-between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The minimum number of hidden layers nlayers depends on the complexity of the target pose distributions in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' In nearly all tested scenes we achieved favorable performance with as few as nlayers = 3, which, unless otherwise stated, is used across all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The final layer corresponds to the prediction of translation and rotation vectors, where the former goes through a sigmoid activation, followed by a fixed affine transformation that shifts and scales the predictions to the metric ranges of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We train our networks using Adam optimizer [36] with initial learning rate of 1 × 10−4 and an exponential learning rate decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='8, applied every nlr-decay epochs for 10 occurences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Following the pose regression literature, we first resize each image such that its smallest edge is 256, then randomly crop 224 × 224 regions for input to the Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We also augment the data with color/brightness jittering and Gaussian blur to account for lighting changes and motion blur between images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Unless otherwise stated, we let α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='20, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='01, use a d = 16-dimensional latent space, and represent distributions with 1000 Monte Carlo samples in all experiments, since we found this to produce good predictions in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Other hyperparameters are reported in Table I, tuned to reflect the number of images and metric scales of different datasets, which range from small indoor to large outdoor scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Note that we found these settings without a major hyperparameter search, and TABLE I HYPERPARAMETERS USED IN TRAINING Dataset λt λr Batch Size # Epochs nlr-decay 7-Scenes [24] 5 10 64 100 10 Cambridge Land.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' [10] 5 100 64 500 50 Ambiguous Reloc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' [16] 5 2 4 500 50 Ceiling 5 2 4 2000 50 Synthetic 5 2 4 500 50 one may improve the performance by a thorough search of the optimal hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Datasets and metrics We evaluate our method on the Ambiguous Relocalization dataset [16] as an existing benchmark with real-world image sequences of ambiguous environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' For each scene in the dataset there are separate training and test image sequences recorded from their own unique camera trajectories, but with generally similar views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We found that despite the apparent ambiguity to the human eye, a large fraction of frames in this dataset contain unique identifying features, which an expressive feature detector can infer the pose from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' This results in unimodal predicted posteriors for a large number of frames, which hinders the evaluation of a method’s capability in forming multimodal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' To address this, we complement the dataset by recording a new real-world se- quence of a ceiling with machine-fabricated panels, capturing a case of severe visual ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We record the training and test sequences with a calibrated LiDAR-IMU-camera rig, and obtain ground-truth camera poses using MILIOM [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We also render image sequences of two synthetic scenes from 3D Warehouse3, which contain symmetries by design, and use them to investigate our method in a controlled setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We use recall as the metric to evaluate pose distributions in ambiguous scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' For a query image, we draw samples from its posterior distribution, and consider it a true positive if at least a fraction γ of the samples are within a distance of the ground-truth pose (and a false negative otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We argue that for a distribution with well-separated equally likely modes, setting γ inversely proportional to the number of modes gives an estimate of whether the distribution contains sufficient density around the ground-truth pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We report recall with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='1 for all tested scenes except for the ceiling scene, where we use γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' To validate the performance of our method as a general pose regressor on unambiguous scenes, we evaluate it on the visual localization benchmarks 7-Scenes [24] and Cambridge Landmarks [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' As is commonly reported by pose regres- sion works, we use median error for evaluation on these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We obtain a point prediction from the Monte Carlo samples of each predicted distribution using the arithmetic and chordal L2 [38] means for translation and orientation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The median of this estimate’s error compared to the ground-truth pose is reported across each scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 3https://3dwarehouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='sketchup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='com/ 0 m 2 m 4 m 6 m 8 m 10 m 12 m 14 m 16 m 0 m 1 m Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Marginal posterior distributions along x-axis (top left) and y-axis (bottom right) predicted by our method ( ), by Bingham MDN [16] ( ), the prediction by MapMet [13] ( ), and the ground truth ( ) for a query image (top right) from the ceiling scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The heatmap shows the 2D histogram predicted by our method overlaid on top of stitched images of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Note that our method successfully captures all six modes of the distribution while MapNet only predicts a single estimate at a wrong location, and Bingham MDN method assigns large probabilities in visually dissimilar locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' TABLE II MEASURED RECALL IN AMBIGUOUS SCENES (HIGHER IS BETTER) Scene Threshold PN [10] MN [13] BMDN [16] Abl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Ours Blue Chairs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='1m/10◦ 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='44 † We train the independently recorded ceiling scene with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='05 and nlayers = 9, reflecting the richer presence of ambiguities in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Evaluation on benchmark datasets We report the results on the ambiguous scenes in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We can see that our method, outperforms Bingham MDN [16] as the method closest to ours that predicts a distribution of poses aimed at localization in ambiguous scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We considered two settings of their approach with 10 and 50 components in their mixture model, and evaluated the metric based on samples drawn from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' As the 10- component setting consistently performed better, we report its results as a representative in the table (marked BMDN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 3 shows an example of the predicted posterior given a query image from the ceiling scene, where we can see posterior predicted by our method better captures the am- biguous structure of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We also evaluate PoseNet [10] and its Bayesian variant [28], as well as MapNet [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' However, we see that these single estimate methods fail to achieve comparable performance on the ambiguous scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' To our surprise, vanilla PoseNet performed comparatively better than Bayesian PoseNet, so we include its results as representative (marked PN) alongside MapNet (marked MN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' In order to investigate whether our method’s improved performance stems from our novel formulation with varia- tional inference, we perform an ablation, in which we modify our pipeline to produce a single pose for an input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' TABLE III MEDIAN ERROR (M / ◦) IN UNAMBIGUOUS SCENES (LOWER IS BETTER) Scene PN [10] MN† [13] BPN [28] BMDN [16] Ours Chess 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='41 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='02/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='03 Church 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='65/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='00/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='11/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='16/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='80/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='90 † Results of MapNet on Cambridge Landmarks taken from Sattler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We remove the KL divergence term from the objective, modify the Encoder to predict a single point, and obtain a single pose prediction by passing the encoder’s prediction through PoseMap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' All else equal, we evaluate this ablative variant of our method that is in principle very similar to PoseNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We can see in Table II that this variant, marked Abl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=', while performing better than PoseNet due to its more recent feature extractor network, falls short of the unablated variant, validating the merit of our proposed formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' For completeness, we report our results on the unambigu- ous 7-Scenes and Cambridge Landmarks datasets in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We include results of PoseNet and MapNet as single pose regressor baselines, and Bayesian PoseNet and Bingham MDN as methods that, in principle, can predict multimodal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' While our method does not perform the best, it is not far from the top-performers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' This experiment merely serves as a sanity check of our approach’s performance in a minimal pipeline, without any particular mechanism aimed at improving accuracy in unambiguous scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' As seen in Table II, the better performing methods on unambiguous scenes show poor performance on ambiguous scenes, which is the problem that our method targets to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' An interesting direction for future work is to apply our proposed formula- tion, aimed at handling ambiguous scenes, in tandem with techniques for improved unambiguous pose regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' A closer look Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 4 (top) shows the predicted distribution by our method for an example query image from the Ambiguous Relocaliza- tion dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Although the scene, made up of identical chairs, is arguably ambiguous to the human eye, we can see that (a) Query image (b) Position posterior (c) Samples No Weight Decay Weight Decay Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Posteriors predicted by our method in full capacity (top) and in a constrained learning mode by L2 weight decay of λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='1 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The latter captures multiple modes whereas the former mode predicts a single mode at the correct pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' the predicted posterior identifies and concentrates its density around the correct pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We hypothesize that a sufficiently expressive Encoder can distinguish a seemingly ambiguous image taken in real life by its smallest of details, such as the chair’s background in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' However, a less expressive Encoder for the data is unable to learn every detail and can give in to the ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We test this hypothesis by adding a penalty term on the L2 norm of the Encoder weights during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 4 (bottom) that this setting results in the predicted posterior assigning probabilities to poses viewing two additional chairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We argue that when there exists a domain gap between the training data and the operation conditions, it is desirable for the model to trade off confidence in predictions for better generalization, which can be achieved via deliberate learning constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We leave the study of such learning constraints to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We study the effect of α in the Winners-Take-All optimiza- tion scheme in two synthetic scenes, where the camera circles around a round table with four legs, resulting in four modes in the pose distribution of an image, as well as a rectangular dinner table that results in bimodal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We report the statistics over 10 training runs for the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='1m/10◦ recall evaluated at the end of training with different α values in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We can see that in these scenes the highest recall is achieved with α in a range of values greater than zero but less than 1/# modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 5 shows the predicted camera position posterior for three choices of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We can see that a too large α, as discussed in Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='2, results in a compromised posterior, and a too small α predicts close-to- uniform densities across the span of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We hypothesize that α must be smaller than 1/# modes for the Winners-Take-All optimization to converge and capture all modes in the distribution, and must be sufficiently larger than zero to overcome the noise as a result of mini-batch optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' There is a trade-off between training speed and the quality of the learned distribution within this range of α values, as a smaller α results in optimization of fewer samples at every iteration, hence a slower training, but is less Scene α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='01 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='20 α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='00 Round Table Dinner Table Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Predicted position posterior on xy-plane for various choices of α in the synthetic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The round table and dinner table have four and two modes in their true distributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' The optimization successfully converges to predict the correct modes when 0 ≪ α < 1/# modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' For example α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='20 produces good predictions for both scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='5 1 α (a) Round Table 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='0 α (b) Dinner Table Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Measured recall with threshold 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='1m/10◦ for various choices of α in Winners-Take-All optimization of the synthetic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' For each choice of α, the statistics over 10 training runs is shown by a box extending from the lower to upper quartile recall values, a purple line at the median recall, and whiskers that extend to the minimum and maximum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' susceptible to the noise induced by Monte Carlo sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We leave the study of finding the optimal α to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Run-time evaluation We measure the time taken for a forward pass of one query image through our pipeline for 1000 Monte Carlo samples, on a desktop computer with an Intel Core i7-8700K CPU and an NVIDIA GeForce GTX 1080 Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We repeat each measurement 100 times and we find that a forward pass on average takes 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='75 ms on CPU and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content='10 ms on GPU, that is, our pipeline can run in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' CONCLUSION In this work, we addressed the task of visual localization in ambiguous scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We proposed a novel formulation of camera pose regression with variational inference, which allows learning and sampling from the distribution over all camera poses given an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' This is done by first encoding the images to predict a posterior distribution over the latent space of visual features present in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Drawing samples from this distribution and passing them through a learned mapping produces a set of pose samples that represent the posterior distribution over camera poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' We show that our formulation outperforms existing methods on localization in ambiguous scenes, and propose directions for future work to further investigate our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' OO风REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Torii, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Arandjelovic, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNA0T4oBgHgl3EQfJP8k/content/2301.02086v1.pdf'} +page_content=' Sivic, M.' metadata={'source': 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Slobodeniuk1, ∗ and Maciej R. Molas2, † +1Department of Condensed Matter Physics, Faculty of Mathematics +and Physics, Charles University, CZ-121 16 Prague, Czech Republic +2Institute of Experimental Physics, Faculty of Physics, University of Warsaw, 02-093 Warsaw, Poland +The high-quality structures containing semiconducting transition metal dichalcogenides (S-TMDs) +monolayer (MLs) required for optical and electrical studies are achieved by their encapsulation in +hexagonal BN (hBN) flakes. To examine the effect of hBN thickness in these systems, we consider +a model with an S-TMD ML placed between a semi-infinite in the out-of-plane direction substrate +and complex top cover layers: a layer of finite thickness, adjacent to the ML, and a semi-infinite in +the out-of-plane direction top part. We obtain the expression for the Coulomb potential for such a +structure. Using this result we demonstrate that the energies of excitonic s states in the structure +with WSe2 ML change significantly for the top hBN with thickness <30 layers. We observed that +the excitonic binding energy (Eb) is reduced by almost 40% in the transition from the sample +without the top hBN layer (Eb=256 meV) to the one with an infinite thickness of the top hBN layer +(Eb=165 meV). +I. +INTRODUCTION +The properties of excitons, electron-hole (e-h) pairs +bounded by Coulomb force, in two-dimensional (2D) +monolayers (MLs) of semiconducting transition metal +dichalcogenides (S-TMDs) are remarkably modified due +to significant change of the Coulomb interaction between +charge carriers in such 2D crystals [1–3]. The excitons +are characterized by the energy spectrum, composed in +analogy to the hydrogen series as of the ground (1s) and +excited (2s, 2p, 3s . . . ) states. Although excitonic states +of the s-type are observable in the linear optical spec- +tra of S-TMD MLs, i.e., photoluminescence [4–8], trans- +mission [9–11], and reflectance contrast [6, 12, 13], the +excitonic states of the p- and d-types can be seen in non- +linear experiments performed on S-TMD MLs, i.e., sec- +ond harmonic generation or two-photon absorption [14– +17]. It turns out that the energy spectrum of s-type states +in these atomically-thin semiconductors does not repro- +duce the conventional Rydberg series of a 2D hydrogen +atom [18, 19]. The main reason for that is the dielectric +inhomogeneity of the S-TMD structures, i.e., MLs sur- +rounded by dielectric materials. While the Coulomb in- +teraction scales as ∝ 1/εr with the dielectric response of +the surrounding medium ε at large e-h distances r, it ap- +pears to be significantly weakened at short e-h distances +due to exceptionally strong dielectric screening within +the ML plane. Consequently, the energy spectrum of the +excitons in S-TMD MLs and hence their binding energy, +defined as the energy difference between the electronic +band gap and the ground 1s state, can be strongly modi- +fied by the used surrounding media of different dielectric +responses. Whereas several scientific papers have focused +on the effect of surrounding dielectrics on the excitonic +∗ aslobodeniuk@karlov.mff.cuni.cz +† maciej.molas@fuw.edu.pl +ladder in MLs [20–25], there is a lack of analogous inves- +tigations of the thickness influence of the media enclosing +S-TMD MLs. However, it is of utmost importance, as the +highest quality MLs are obtained by their encapsulation +in flakes of hexagonal BN (hBN), leading to a narrow- +ing of excitonic resonances approaching the homogeneous +linewidth limit [26–28]. +In this work, we investigate theoretically the energy +spectrum of free excitons in S-TMD MLs encapsulated +in between a semi-infinite in the out-of-plane direction +bottom substrate and complex top cover layers consist- +ing of two parts: a layer of finite thickness L, adjacent +to the ML, and semi-infinite in the out-of-plane direction +top part with the aid of generalization of the Rytova- +Keldysh potential. We demonstrate that the energies of +the excitonic s states in such a system with the WSe2 ML +are strongly modified when the thickness of the top hBN +layers decreases below about 30 layers. In addition, it +results in a significant reduction in excitonic binding en- +ergy (Eb) of almost 40% in the transition from the sample +without the top hBN layer (Eb=256 meV) to the one with +infinite thickness of the top hBN layer (Eb=165 meV). +II. +COULOMB POTENTIAL IN THE +NON-HOMOGENEOUS SYSTEM: GENERAL +CASE +Let us consider the S-TMD ML encapsulated in be- +tween a semi-infinite bottom substrate (1-st layer) and +complex top cover layers consisting of two parts: +2- +nd layer of finite thickness L, adjacent to the ML, and +semi-infinite in the out-of-plane direction 3-rd part. A +schematic illustration of the studied system is presented +in Fig. 1. +The ML is arranged in the xy plane and +is centered in the out-of-plane direction (z = 0). The +bottom substrate, 1-st layer, belongs to the domain +z ∈ (−∞, −δ], and is characterized by the in-plane ε1,∥ +and out-of-plane ε1,⊥ dielectric constants. The top (2- +arXiv:2301.03245v1 [cond-mat.mes-hall] 9 Jan 2023 + +2 +z +r +δ +-δ +-∞ +∞ +𝑳 +𝜀1,∥ 𝜀1,⊥ +𝜀2,∥ 𝜀2,⊥ +𝜀3,∥ 𝜀3,⊥ +FIG. 1. The schematic illustration of the S-TMD monolayer +encapsulated in between a semi-infinite bottom substrate (1-st +layer) and a complex top cover layers consisting of two parts: +2-nd layer of the finite thickness, adjacent to the ML and +semi-infinite in the out-of-plane direction 3-rd part. +nd) layer, next to the ML, unfolds in the range z ∈ [δ, L] +with the in-plane ε2,∥ and out-of-plane ε2,⊥ dielectric con- +stants. Finally, the 3-rd top layer spreads over the dis- +tance z ∈ [L, ∞) and is described by the in-plane ε3,∥ +and out-of-plane ε3,⊥ dielectric constants. +In order to find the potential energy between two +charges in S-TMD MLs, we solve the following elec- +trostatic problem. We investigate the point-like charge +Q at the point r += +(ρ, z) += +(0, 0, 0) and calcu- +late the electric potential in such a system following +Refs. [29, 30]. Namely, we analyze four regions: bottom +(z ∈ (−∞, −δ]), ML (z ∈ [−δ, δ]), top finite (z ∈ [δ, L)) +and overtop (z ∈ [L, ∞)) media, with potentials Φ1(ρ, z), +Φ(ρ, z), Φ2(ρ, z), and Φ3(ρ, z), respectively. These po- +tentials are defined by the Maxwell equations. As the +studied problem has cylindrical symmetry, we present the +potentials in the form +Φj(ρ, z) = +1 +(2π)2 +� +d2keikρΦj(k, z), +(1) +Φ(ρ, z) = +1 +(2π)2 +� +d2keikρΦ(k, z). +(2) +The Φj(ρ, z) potentials for the j-th region, where j = +1, 2, 3, satisfy Maxwell’s equation div Dj(ρ, z) = 0, which +can be written as +− εj,∥k2Φj(k, z) + εj,⊥ +d2Φj(k, z) +dz2 += 0. +(3) +The solutions of these equations are +Φ1(k, z) = B1eκ1z +for z ∈ (−∞, −δ], +(4) +Φ2(k, z) = A2e−κ2z + B2eκ2z +for z ∈ [δ, L], +(5) +Φ3(k, z) = A3e−κ3z +for z ∈ [L, ∞), +(6) +where κj = |k| +� +εj,∥/εj,⊥ = k +� +εj,∥/εj,⊥. +Maxwell’s equation in the ML domain, i.e., z ∈ [−δ, δ], +reads div D(ρ, z) = 4πQδ(ρ)δ(z). It gives the equation +for the potential Φ(r, z) +� +∆∥ + d2 +dz2 +� +Φ(ρ, z) = −4π[Qδ(ρ)δ(z) − ϱind(ρ, z)], (7) +where ∆∥ is 2D Laplace operator. The first term in the +right-hand-side of Eq. (7) is the charge density of the +charge Q, localized in the ML plane. The second term +represents the polarization charge density ϱind(ρ, z), in- +duced in the ML by point charge Q, which is given by +ϱind(ρ, z) = div P(ρ, z). +(8) +Following Ref. [29], we present the polarization in the +form +P(ρ, z) = δ(z)P∥(ρ, z = 0). +(9) +Using the proportionality between the induced polariza- +tion P∥(ρ, 0) and the in-plane component of the electric +field E∥(ρ, 0), P∥(ρ, 0) = χ2DE∥(ρ, 0), we obtain the ex- +pression for the induced charge +ϱind(ρ, z) = −χ2Dδ(z)∆∥Φ(ρ, 0). +(10) +Here χ2D is the 2D polarizability of the ML [29, 31]. +Taking the Fourier transformation of Eq. (7) with the +introduced induced charge in Eq. (10), one gets +� +k2 − d2 +dz2 +� +Φ(k, z) = 4πQδ(z) − 4πχ2Dk2δ(z)Φ(k, 0). +(11) +This is a linear non-homogeneous differential equation of +the second order, which solution can be presented as a +sum of the general solution of the homogeneous equa- +tion and a particular solution of the non-homogeneous +equation (see Ref. [32] and Supplementary Material) +Φ(k, z) = Ψe−k|z| + Ae−kz + Bekz. +(12) +The k-dependent parameters of the potential Ψ, A, and +B are not independent. The relations between them are +defined from Eq. (7). Integrating it over z in the domain +z ∈ [−ϵ, ϵ] and then taking the limit ϵ → 0, one obtains +[1 + r0k]Ψ + r0k[A + B] = 2πQ +k +, +(13) +where we introduced the screening length r0 = 2πχ2D. +Using the continuity of the potential and z component +of the displacement field D(ρ, z) on the boundary of two +adjusted domains, one obtains the set of equations for +the parameters Ψ, A, B, B1, A2, B2, A3. The boundary +conditions for the 1-st and ML domains give relations +B1e−κ1δ = Ψe−kδ + Aekδ + Be−kδ, +(14) +ε1B1e−κ1δ = Ψe−kδ − Aekδ + Be−kδ. +(15) +The boundary conditions between the ML and 2-nd do- +mains are described by equations +A2e−κ2δ + B2eκ2δ = Ψe−kδ + Ae−kδ + Bekδ, (16) +ε2A2e−κ2δ − ε2B2eκ2δ = Ψe−kδ + Ae−kδ − Bekδ. (17) + +3 +Finally, the boundary conditions between the 2-nd and +3-rd domains give +A2e−κ2L + B2eκ2L = A3e−κ3L, +(18) +ε2A2e−κ2L − ε2B2eκ2L = ε3A3e−κ3L. +(19) +Here, we introduce εj = √εj,⊥εj,∥ for j = 1, 2, 3. Solving +these equations together with Eq. (13), we obtain the +values of the Ψ, A, and B parameters. +Let us consider the particular case δ = 0, which corre- +sponds to the situation of the infinitely thin layer inves- +tigated first by N. S. Rytova [30] and L. V. Keldysh [33]. +The expression of the Coulomb potential in the in-plane +direction reads Φ(k, z = 0) = (2πQ/k)v(k), with +v(k) = +1 + ε2−ε3 +ε2+ε3 e−2κ2L +kr0 + ε1+ε2 +2 ++ +� +kr0 + ε1−ε2 +2 +� +ε2−ε3 +ε2+ε3 e−2κ2L . (20) +We can check that this expression provides the correct +answer for the Coulomb potential for various limit cases, +such as L → 0, L → ∞, r0 → 0, ε1 → ε2, ε2 → ε3, as +well as their combinations. +Note that the expression for the in-plane potential con- +tains only the combinations εj = √εj,∥εj,⊥ and κ2 = +k +� +ε2,∥/ε2,⊥. That means, for example, that substrates +with different values of in-plane ε∥ and out-of-plane ε⊥ +dielectric constants can give the same shape of the po- +tential in S-TMD MLs. +The interesting limit is ε3 → ∞, which corresponds +to the case of a metallic plane placed horizontally at the +distance L from the ML. The related potential describes +the modified Rytova-Keldysh potential suppressed by the +metal substrate. +III. +THE COULOMB POTENTIAL IN S-TMD +SAMPLE: EFFECT OF FINITE THICKNESS OF +HBN TOP LAYER +We examine the particular case of an S-TMD ML en- +capsulated in hBN layers, i.e., ε1,∥ = ε2,∥ = εhBN,∥, +ε1,⊥ = ε2,⊥ = εhBN,⊥, ε3,∥ = ε3,⊥ = 1. +Following +the values available in the literature, we use ε1 = ε2 = +εhBN = 4.5 [9], and κ2 = +� +εhBN,∥/εhBN,⊥ ≈ 1.36k [34]. +Note that ε3,∥ = ε3,⊥ = 1 resemble typical experimental +conditions, i.e., the sample is placed in air, vacuum, or +gaseous helium. Introducing the dimensionless momen- +tum x = kr0/εhBN and length l = εhBNL/r0 parameters, +we obtain +v(x) = +1 +εhBN +1 + εhBN−1 +εhBN+1 exp +� +− 2 +� εhBN,∥ +εhBN,⊥ xl +� +1 + x +� +1 + εhBN−1 +εhBN+1 exp +� +− 2 +� εhBN,∥ +εhBN,⊥ xl +��. +(21) +The corresponding potential as a function of the dimen- +sionless distance ξ = ρεhBN/r0 is Φ(ξ) = (Q/r0)φ(ξ, l), +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +∞ +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Potential function, � (� , l) +Distance parameter, � + l=0 + l=1 + l=5 + l=∞ +FIG. 2. Potential function φ(ξ, l) for four values of dimen- +sionless length parameter, l: l=0, l=1, l=5, and l=∞ as a +function of a dimensionless distance parameter, ξ. +where +φ(ξ, l) = +� ∞ +0 +dx +J0(xξ) +� +1 + εhBN−1 +εhBN+1 exp +� +− 2 +� εhBN,∥ +εhBN,⊥ xl +�� +1 + x +� +1 + εhBN−1 +εhBN+1 exp +� +− 2 +� εhBN,∥ +εhBN,⊥ xl +�� . +(22) +Note that in the limit l → ∞, i.e., an ML encapsulated +in semi-infinite hBN layers, the potential expression is +simplified and gives the well-known Rytova-Keldysh po- +tential [30, 33] +φ(ξ, ∞) = +� ∞ +0 +dxJ0(xξ) +1 + x = π +2 +� +H0(ξ) − Y0(ξ) +� +. +(23) +The other limit l → 0 corresponds to the case of an ML +deposited on a semi-infinite hBN substrate, uncovered +from the top. The related potential also has the Rytova- +Keldysh form +φ(ξ, 0) = π +2 +� +H0 +�εhBN + 1 +2εhBN +ξ +� +− Y0 +�εhBN + 1 +2εhBN +ξ +�� +. +(24) +The evolution of the φ(ξ, l) potential as a function of +a ξ parameter is presented in Fig. 2 for four l values. As +can be seen in the Figure, the strongest φ(ξ, l) potential +is apparent for l = 0, while the weakest potential is for +the case l = ∞, the potential for l > 0 lies in the region in +between of two former potentials. The obtained results +are in full agreement with previously reported results [9, +20, 25], where it was demonstrated that the increase of +the average dielectric constant of the media surrounding +the ML leads to the decrease of the confining potential. + +4 +0 +10 20 30 40 50 60 70 80 90 100 +∞ +-240 +-210 +-180 +-60 +-30 +0 +1s +Energy (meV) +Number of top hBN layers +5s +4s +3s +2s +¥ +FIG. 3. Energy spectrum of s excitonic state in the WSe2 +ML encapsulated in hBN layers as a function of the thickness +of the top hBN layer (colored online). The full points corre- +spond to the case of a semi-infinite thickness of top hBN. The +colored open points represent the energy spectrum of excitons +calculated individually for the mono- and bilayer top hBN (see +SM for details). Note that the thickness of the bottom hBN +layer is semi-infinite in both cases. The gray-shaded region +represents the infinity of states above the bandgap energy. +IV. +EXCITONIC SPECTRUM IN +NON-HOMOGENEOUS SYSTEM +Using the φ(ξ, l) potential, expressed by Eq. (22), we +can evaluate the energy spectrum of excitons in the in- +vestigated structure composed of S-TMD ML as a func- +tion of the parameter l. The corresponding equation for +eigenvalues is given by +� +b2 1 +ξ +d +dξ +� +ξ d +dξ +� ++ 2bφ(ξ, l) + ϵ +� +ψ(ξ) = 0, +(25) +where we introduced b = ℏ2ε2 +hBN/(µe2r0) and E = Ry∗ϵ. +Ry∗ = µe4/(2ℏ2ε2 +hBN) is an effective Rydberg energy and +ψ(ξ) represents the wave function of an exciton. +µ = +memh/(me + mh) is the reduced mass of the exciton (e- +h pair) with the effective electron (me) and hole (mh) +masses and e represents electron’s charge. +Let +us +examine +the +case +of +WSe2 +ML +with +r0 = 4.5 nm [31] and µ = 0.21 m0 [6], where m0 is elec- +tron’s mass. It gives b ≈ 1.13357 and Ry∗ ≈ 141 meV. +Note that for WSe2 ML r0 = 4.5 nm, therefore, the di- +mensionless parameter l corresponds to the l nm of the +thickness of the top flake of hBN. Taking into account +that the distance between the layers in hBN (in other +words, the thickness of hBN ML) is dhBN = 0.33 nm [35], +we conclude that l = 1 corresponds to 3 layers of hBN. +For the thinnest hBN flake, the hBN layer demonstrates +its discrete structure, which may be beyond the contin- +uous model of hBN medium considered here. To verify +it, we performed the corresponding analysis for the ML +and BL of top hBN in the Supplementary Material (SM), +which takes into account the discrete nature of the hBN +flake. The calculated energy spectra of an exciton for the +ground (1s) and four excited (2s – 5s) states as a function +of the thickness of the top hBN layer for the aforemen- +tioned continuous model as well as the discrete one for +the thinnest layers (described in the SM) are presented +in Fig. 3. Due to the observed evolutions in the Figure, +three main points can be raised: (i) the continuous model +provides a very good method for the calculation of the +exciton spectrum, even in the case of the extremely thin +top hBN flake of about 1-2 layers, see Fig. 3. The largest +discrepancy is observed between the homogeneous model +and the ML of top hBN for the energy of 1s state of +about 5%. (ii) the energies of excitonic states are sub- +jected to the most significant variations for the thinnest +top hBN layers with thicknesses below about 30 layers. +For thicker top hBN layers, the corresponding excitonic +energies are almost fixed; (iii) the thickness effect of the +top hBN layer is the largest for the ground 1s state of the +exciton with its substantial reduction when the number +of excitonic states is increased. The maximum change of +the 1s energy of about 91 meV between two limits: with- +out the top hBN layer and with its infinite thickness. The +analogous differences of the 2s, 3s, 4s, and 5s states are +of the order of 39 meV, 20 meV, 13 meV, and 8 meV, +respectively. +Finally, we can focus on the analysis of the excitonic +binding energy (Eb, defined as the energy difference be- +tween the electronic bang gap and the ground 1s state. +The dependence of the Eb energy in the WSe2 ML en- +capsulated in the hBN layers for selected numbers of the +top hBN layer is summarized in Table I. The experimen- +tally measured binding energies of excitons in WSe2 MLs +encapsulated in hBN flakes is of about 170 meV [5–7, 9]. +Besides the structures with the hBN encapsulation in- +vestigated experimentally differ from the one analyzed in +this work, e.g. a WSe2 ML sandwiched between 10 nm +thick hBN deposited on the core of a single-mode opti- +cal fiber [9], the theoretically calculated binding energies +are in very good agreement with the experimental ones. +Using our approach, the excitonic binding energy can be +changed by almost 40% in the transition from the sample +without the top hBN layer (Eb=256 meV) to the one with +the infinite thickness of the top hBN layer (Eb=165 meV). +This reveals that the thickness of the surrounding media +TABLE I. Calculated binding energies of excitons (Eb) in +WSe2 ML encapsulated in hBN layers for selected numbers +of the top hBN layer. +Number of +top hBN layers +0 +3 +6 +10 +20 +40 100 ∞ +Eb (meV) +256 207 192 184 175 171 167 165 + +5 +of S-TMD MLs also plays a crucial role in the modifica- +tion of the exciton energy spectrum in S-TMD MLs in +addition to the engineering of the surrounding dielectric +environment, i.e., the encapsulation of an ML in media +characterized by dielectric constants [20]. +V. +SUMMARY +Concluding, we have demonstrated using the general- +ized Rytova-Keldysh potential that the energy spectrum +of excitonic states in S-TMD monolayers placed on a +semi-infinite hBN substrate and covered with a top hBN +layer of finite thickness can be significantly modified due +to the screening effects. For WSe2 ML in such a struc- +ture, the energies of the excitonic states are substantially +adjusted for the thinnest top hBN layers with thicknesses +below about 30 layers. 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Grzeszczyk, +K. Watanabe, T. Taniguchi, A. Babiński, and M. R. Mo- +las, arXiv , 2211.16186 (2022). +[33] N. S. Rytova, Moscow University Physics Bulletin 3, 30 +(1967). +[34] A. Laturia, , M. L. Van de Put, and W. G. Vandenberghe, +npj 2D Materials and Applications 2, 6 (2018). +[35] B. Xu, M. Lv, X. Fan, W. Zhang, Y. Xu, and T. Zhai, +Integrated Ferroelectrics 162, 85 (2015). +[36] M. Baranowski and P. Plochocka, Advanced Energy Ma- +terials 10, 1903659 (2020). + +7 +Supplementary Material for "Exciton spectrum in atomically thin monolayers: +The role of hBN encapsulation" +SI. +THE MODIFIED RYTOVA-KELDYSH +POTENTIAL FOR THE CASE OF ULTRATHIN +TOP HBN FLAKE +In our study, we modeled the top layer of hBN with +its thickness L and the macroscopic in- (εhBN,∥) and out- +of-plane (εhBN,⊥) dielectric constants of hBN, i.e. in the +limit of the continuous hBN medium. However, in the +case of the ultrathin top hBN flake, where it consists of +only a few layers of hBN monolayers, the use of such a +limit is doubtful. In order to find the limits of applica- +bility of the continuous model, we derive the modified +Rytova-Keldysh potential with a few layers of the hBN. +To do it, we consider the modification of the system +studied in the main text. The bottom semi-infinite sub- +strate remains unchanged. Thus, it is characterized by +in- (ε1,∥) and out-of-plane (ε1,⊥) dielectric constants and +occupies the domain z ∈ (−∞, −δ]. The S-TMD mono- +layer is placed in the plane z = 0 with its in-plane po- +larizability χTMD. The first top hBN flake consists of N +layer and has a thickness L = Nd. Here d = 0.33 nm +is the distance between the hBN layers in a bulk hBN +crystal. The jth layer of the hBN flake is placed in the +zj = jd plane. Each hBN layer is characterized by the +in-plane polarizability χhBN. The second top layer with +the in- (ε3,∥) and out-of-plane (ε3,⊥) dielectric constants +occupies the domain z ∈ [L + δ, ∞). +In order to find the potential energy between two +charges in S-TMD monolayer, we solve the following elec- +trostatic problem. We consider the point-like charge Q +at the point (0, 0, z′), where 0 < z′ ̸= zj < L and calcu- +late the potential in such a system following [29]. Simi- +larly to the previous case, we consider 3 regions: bottom +semi-infinite medium with potential Φ1(ρ, z), second top +semi-infinite layer with the potential Φ3(ρ, z), and the +space between them with the potential Φ2(ρ, z), where +we introduced the in-plane vector ρ = (x, y). Using the +cylindrical symmetry of the problem, we present the po- +tentials in the form +Φj(ρ, z) = +1 +(2π)2 +� +d2keikρΦj(k, z) +(26) +The Maxwell’s equation div D1,3 = 0 for 1-st and 3-rd +regions (j = 1, 3) reads +− εj,∥k2Φj(k, z) + εj,⊥ +d2Φj(k, z) +dz2 += 0. +(27) +The solutions to these equations are +Φ1(k, z) = Aeκ1z, +for z ∈ (−∞, −δ], +(28) +Φ3(k, z) = Be−κ3z, +for z ∈ [L + δ, ∞), +(29) +where κj = |k| +� +εj,∥/εj,⊥ = k +� +εj,∥/εj,⊥. +The equation for the potential Φ2(r, z) takes the form +∆∥Φ2(ρ, z)+ϵ⊥ +d2Φ2(ρ, z) +dz2 += +− 4π[Qδ(ρ)δ(z − z′) − ϱind(ρ, z)], +(30) +where ∆∥ is 2D Laplace operator. We introduced the out- +of-plane dielectric constant (ϵ⊥) of the hBN flake with +N > 1. In the case N = 1, one needs to put ϵ⊥ = 1. +The first term on the right-hand side of the equation is +the charge density of the charge Q. +The second term +ϱind(ρ, z) represents the induced charge density due to +the polarization of the hBN and S-TMD layer by charge +Q. Following Ref. [29], we present the induced charge in +the form +ϱind(ρ, z) = −χhBN +N +� +j=1 +δ(z − zj)∆∥Φ2(ρ, zj)− +−χTMDδ(z)∆∥Φ2(ρ, 0). +(31) +Using the Fourier transform (26) of the potential Φ2(ρ, z) +we present Eq. (30) as +� +k2 − ϵ⊥ +d2 +dz2 +� +Φ2(k, z) =4πQδ(z − z′) − 2r0k2δ(z)Φ2(k, 0) +−2R0k2 +N +� +j=1 +δ(z − zj)Φ2(k, zj), +(32) +where we introduced the in-plane screening lengths r0 = +2πχTMD and R0 = 2πχhBN for S-TMD and hBN mono- +layers, respectively. Integrating this equation in the re- +gions z ∈ [−ϵ, ϵ], z ∈ [zj −ϵ, zj +ϵ], and z ∈ [z′ −ϵ, z′ +ϵ] +one obtains the following conditions in the limit ϵ → 0 +ϵ⊥ +dΦ2(k, z) +dz +��� ++0 +−0 = 2r0k2Φ2(k, 0), +(33) +ϵ⊥ +dΦ2(k, z) +dz +��� +zj+0 +zj−0 = 2R0k2Φ2(k, zj), +(34) +ϵ⊥ +dΦ2(k, z) +dz +��� +z′+0 +z′−0 = −4πQ. +(35) +At the points z ̸= 0 , zj , z′ the Eq. (32) is simplified +� +k2 − ϵ⊥ +d2 +dz2 +� +Φ2(k, z) = 0 +(36) +Therefore, the general solution for Φ2(k, z) can be writ- +ten in the form +Φ2(k, z) =Ψ0e−K|z−z′| + +N +� +j=1 +Ψje−K|z−zj|+ ++Ψe−K|z| + αeKz + βe−Kz, +(37) + +8 +where Ψ0,Ψj,Ψ, α and β are unknown functions of K = +k/√ϵ⊥. +The aforementioned boundary conditions to- +gether with the equation give the following restrictions +Ψ = − r0KΦ2(k, 0), +Ψj = − R0KΦ2(k, zj), +Ψ0 =2πQ/ϵ⊥K. +(38) +Considering the boundary conditions for the electrostatic +potential and for the out-of-plane component of the dis- +placement field at z = −δ +Φ1(k, −δ) = Φ2(k, −δ), +(39) +ε1,⊥ +ϵ⊥ +dΦ1(k, z) +dz +��� +z=−δ = dΦ2(k, z) +dz +��� +z=−δ, +(40) +and at z = L + δ +Φ3(k, L + δ) = Φ2(k, L + δ), +(41) +ε3,⊥ +ϵ⊥ +dΦ3(k, z) +dz +��� +z=L+δ = dΦ2(k, z) +dz +��� +z=L+δ, +(42) +we get the following equations +Ae−κ1δ = Ψ0e−K(z′+δ) + +N +� +j=1 +Ψje−K(zj+δ) + Ψe−Kδ + αe−Kδ + βeKδ, +(43) +A ε1 +√ϵ⊥ +e−κ1δ = Ψ0e−K(z′+δ) + +N +� +j=1 +Ψje−K(zj+δ) + Ψe−Kδ + αe−Kδ − βeKδ, +(44) +Be−κ3(L+δ) = Ψ0e−K(L+δ−z′) + +N +� +j=1 +Ψje−K(L+δ−zj) + Ψe−K(L+δ) + αeK(L+δ) + βe−K(L+δ), +(45) +B ε3 +√ϵ⊥ +e−κ3(L+δ) = Ψ0e−K(L+δ−z′) + +N +� +j=1 +Ψje−K(L+δ−zj) + Ψe−K(L+δ) − αeK(L+δ) + βe−K(L+δ). +(46) +where we introduced the short notation εj = √εj,⊥εj,∥ +for j = 1, 3. Introducing a = ε1/√ϵ⊥ and b = ε3/√ϵ⊥, +and considering the general case a, b ̸= 1 we remove the +parameters A and B from the above equations and obtain +Ψ0e−K(z′+δ) + +N +� +j=1 +Ψje−K(zj+δ) + Ψe−Kδ + αe−Kδ + +�a + 1 +a − 1 +� +βeKδ = 0, +(47) +Ψ0e−K(L+δ−z′) + +N +� +j=1 +Ψje−K(L+δ−zj) + Ψe−K(L+δ) + +�b + 1 +b − 1 +� +αeK(L+δ) + βe−K(L+δ) = 0. +(48) +Note that the solution of the aforementioned system of +equations for the case a = 1 and/or b = 1 can be obtained +from the general solution by taking the corresponding +limit. +We solve the general equation in a few steps. First, we +write the system of equations in the following form +e−Kδα + +�a + 1 +a − 1 +� +eKδβ =A, +�b + 1 +b − 1 +� +eK(L+δ)α + e−K(L+δ)β =B, +(49) +where we introduced the following notations A += +−Ψ0e−K(z′+δ) − �N +j=1 Ψje−K(zj+δ) − Ψe−Kδ, +B += +−Ψ0e−K(L+δ−z′) − �N +j=1 Ψje−K(L+δ−zj) − Ψe−K(L+δ). +Solving the system of equations (49) one gets +α = +Ae−K(L+δ) − B +� +a+1 +a−1 +� +eKδ +D +, +(50) +β = +Be−Kδ − A +� +b+1 +b−1 +� +eK(L+δ) +D +, +(51) + +9 +with D = e−K(L+2δ) − +� +a+1 +a−1 +�� +b+1 +b−1 +� +eK(L+2δ). +Substi- +tuting these solutions into (37), we express Φ2(k, z) via +N + 2 parameters Ψ0, Ψj and Ψ. Then, using the latter +expression, we calculate the expressions for the potential +in z = 0, z = z′, and z = zl, l = 1, 2 . . . N points +Φ2(k, 0) =Ψ0e−Kz′ + +N +� +j=1 +Ψje−Kzj + Ψ + α + β, +Φ2(k, z′) =Ψ0 + +N +� +j=1 +Ψje−K(zl−z′) + Ψe−Kz′ + αeKz′ + βe−Kz′, +Φ2(k, zl) =Ψ0e−K(zl−z′) + +N +� +j=1 +Ψje−K|zl−zj| + Ψe−Kzl + αeKzl + βe−Kzl, +(52) +which together with (38) gives the complete set of equa- +tions for the parameters Ψ0, Ψj, and Ψ. In the system +of Eqs. (52), we considered the particular case 0 < z′ < +z1 < z2 < · · · < zN = L for simplicity. Then we sim- +plify our solutions by taking the following limits δ → 0 +and z′ → 0, and obtain Φ2(k, z = 0) = (2πQ/k)v(k) +which defines the electrostatic potential Φ(ρ) in the S- +TMD plane, which reads +Φ(ρ) = Q +� ∞ +0 +dkJ0(kρ)v(k). +(53) +We derive the expressions for the case of monolayer (vm) +and bilayer (vb) hBN flake to demonstrate the aforemen- +tioned algorithm and then estimate the spectrum of ex- +citons for both cases. +First, we consider the case of monolayer hBN, i.e. +N = 1, ϵ⊥ = 1, and hence a = ε1, b = ε3 and K = k. +Following the procedure mentioned above, we obtain the +result for the monolayer (m) +vm(k) = +1 − +� +2kR0+ε3−1 +2kR0+ε3+1 +� +e−2dk +(kr0 + ε1+1 +2 +) − (kr0 + ε1−1 +2 +) +� +2kR0+ε3−1 +2kR0+ε3+1 +� +e−2dk . +(54) +Note that this result looks similar to the general result +in the main text, see Eq. (20). The eigenvalue problem +for this potential has a form +� +�b2 1 +ξ +� +ξ d +dξ +� ++ 2�b +� ∞ +0 +dxJ0(xξ)vm(x) + ϵ +� +ψ(ξ) = 0. +(55) +Here vm(x = kr0) ≡ vm(k), ϵ = E/Ry and ξ = ρ/r0 +are the dimensionless energy and coordinate, respec- +tively, with Ry = µe4/2ℏ2 and �b = ℏ2/(µe2r0). +To +calculate the spectrum of excitons, we consider the par- +ticular case ε1 = εhBN = 4.5, ε3 = 1. +Using for- +mula (4) from Ref. [31] and in-plane dielectric constant +of the monolayer hBN εhBN,∥ = 6.93 from [34] R0 = +(εhBN,∥ − 1)d/2, we obtain R0 = 9.78 Å. Note that this +screening length is much smaller than, for example, in +the WSe2 monolayer r0 = 45 Å [31]. Solving the eigen- +value equation with �b ≈ 0.056 and taking into account +that Ry = 2.856 eV we obtain the following binding en- +ergies: E1 ≈ −242 meV, E2 ≈ −74 meV, E3 ≈ −36 meV, +E4 ≈ −21 meV, E5 ≈ −14 meV. Note that this result is +close to the result of the homogeneous model proposed +in the main text: +E1 ≈ −231 meV, E2 ≈ −73 meV, +E3 ≈ −35 meV, E4 ≈ −21 meV, E5 ≈ −14 meV. The +main difference between the two calculated spectra is the +binding energy of the 1s exciton. The smaller binding +energy of the 1s state in the homogeneous model can be +explained by the less effective screening of the Coulomb +potential than in the model considered here. +The potential for the case of the bilayer flake N = 2 +Φ2(k, z = 0) = (2πQ/k√ϵ⊥)vb(k) with the correspond- +ing out-of-plane dielectric constant ϵ⊥ is +vb(k) = +1 − +2KR0(b+2KR0) +(KR0+1)(b+2KR0+1)e−2dK + (KR0−1)(b+2KR0−1) +(KR0+1)(b+2KR0+1)e−4dK +a+1 +2 ++ Kr0 − [(a+2Kr0)(b+2KR0)−1] +(KR0+1)(b+2KR0+1) KR0e−2dK + (KR0−1)(b+2KR0−1) +(KR0+1)(b+2KR0+1) +� a−1 +2 ++ Kr0 +� +e−4dK . +(56) +Here K = k/√ϵ⊥, a = ε1/√ϵ⊥, and b = ε3/√ϵ⊥. The +eigenvalue problem for the case of the bilayer reads +� +�b2ϵ⊥ +1 +ξ +� +ξ d +dξ +� ++ 2�b +� ∞ +0 +dxJ0(xξ)vb(x) + ϵ +� +ψ(ξ) = 0. +(57) + +10 +Here vb(x = kr0/√ϵ⊥) ≡ vb(k), ϵ = E/Ry and ξ = +ρ√ϵ⊥/r0 are the dimensionless energy and coordinate, +respectively, with Ry = µe4/2ℏ2 and �b = ℏ2/(µe2r0). +For the numerical estimation of the exciton spectrum in +this case, we use ε1 = 4.5 and ε3 = 1 and the result of +Ref. [34] for ϵ⊥ = 3.44. The spectrum of the excitons then +reads E1 ≈ −218 meV, E2 ≈ −69 meV, E3 ≈ −34 meV, +E4 ≈ −20 meV, E5 ≈ −13 meV. It is surprisingly very +close to the result of the homogeneous model, described +in the main text, E1 ≈ −217 meV, E2 ≈ −69 meV, E3 ≈ +−34 meV, E4 ≈ −20 meV, E5 ≈ −13 meV. Therefore, we +conclude that the homogeneous model provides a very +good method for the calculation of the exciton spectrum, +even in the case of the extremely thin top hBN flake of +about 1-2 layers. + diff --git a/odE1T4oBgHgl3EQfhwTG/content/tmp_files/load_file.txt b/odE1T4oBgHgl3EQfhwTG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4f20e7427dde197eaa8cedd6d904a68665ad8239 --- /dev/null +++ b/odE1T4oBgHgl3EQfhwTG/content/tmp_files/load_file.txt @@ -0,0 +1,689 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf,len=688 +page_content='Exciton spectrum in atomically thin monolayers: The role of hBN encapsulation Artur O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Slobodeniuk1, ∗ and Maciej R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Molas2, † 1Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University, CZ-121 16 Prague, Czech Republic 2Institute of Experimental Physics, Faculty of Physics, University of Warsaw, 02-093 Warsaw, Poland The high-quality structures containing semiconducting transition metal dichalcogenides (S-TMDs) monolayer (MLs) required for optical and electrical studies are achieved by their encapsulation in hexagonal BN (hBN) flakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' To examine the effect of hBN thickness in these systems, we consider a model with an S-TMD ML placed between a semi-infinite in the out-of-plane direction substrate and complex top cover layers: a layer of finite thickness, adjacent to the ML, and a semi-infinite in the out-of-plane direction top part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' We obtain the expression for the Coulomb potential for such a structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Using this result we demonstrate that the energies of excitonic s states in the structure with WSe2 ML change significantly for the top hBN with thickness <30 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' We observed that the excitonic binding energy (Eb) is reduced by almost 40% in the transition from the sample without the top hBN layer (Eb=256 meV) to the one with an infinite thickness of the top hBN layer (Eb=165 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' INTRODUCTION The properties of excitons, electron-hole (e-h) pairs bounded by Coulomb force, in two-dimensional (2D) monolayers (MLs) of semiconducting transition metal dichalcogenides (S-TMDs) are remarkably modified due to significant change of the Coulomb interaction between charge carriers in such 2D crystals [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The excitons are characterized by the energy spectrum, composed in analogy to the hydrogen series as of the ground (1s) and excited (2s, 2p, 3s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' ) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Although excitonic states of the s-type are observable in the linear optical spec- tra of S-TMD MLs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=', photoluminescence [4–8], trans- mission [9–11], and reflectance contrast [6, 12, 13], the excitonic states of the p- and d-types can be seen in non- linear experiments performed on S-TMD MLs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=', sec- ond harmonic generation or two-photon absorption [14– 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' It turns out that the energy spectrum of s-type states in these atomically-thin semiconductors does not repro- duce the conventional Rydberg series of a 2D hydrogen atom [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The main reason for that is the dielectric inhomogeneity of the S-TMD structures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=', MLs sur- rounded by dielectric materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' While the Coulomb in- teraction scales as ∝ 1/εr with the dielectric response of the surrounding medium ε at large e-h distances r, it ap- pears to be significantly weakened at short e-h distances due to exceptionally strong dielectric screening within the ML plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Consequently, the energy spectrum of the excitons in S-TMD MLs and hence their binding energy, defined as the energy difference between the electronic band gap and the ground 1s state, can be strongly modi- fied by the used surrounding media of different dielectric responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Whereas several scientific papers have focused on the effect of surrounding dielectrics on the excitonic ∗ aslobodeniuk@karlov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='mff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='cuni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='cz † maciej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='molas@fuw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='pl ladder in MLs [20–25], there is a lack of analogous inves- tigations of the thickness influence of the media enclosing S-TMD MLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' However, it is of utmost importance, as the highest quality MLs are obtained by their encapsulation in flakes of hexagonal BN (hBN), leading to a narrow- ing of excitonic resonances approaching the homogeneous linewidth limit [26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' In this work, we investigate theoretically the energy spectrum of free excitons in S-TMD MLs encapsulated in between a semi-infinite in the out-of-plane direction bottom substrate and complex top cover layers consist- ing of two parts: a layer of finite thickness L, adjacent to the ML, and semi-infinite in the out-of-plane direction top part with the aid of generalization of the Rytova- Keldysh potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' We demonstrate that the energies of the excitonic s states in such a system with the WSe2 ML are strongly modified when the thickness of the top hBN layers decreases below about 30 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' In addition, it results in a significant reduction in excitonic binding en- ergy (Eb) of almost 40% in the transition from the sample without the top hBN layer (Eb=256 meV) to the one with infinite thickness of the top hBN layer (Eb=165 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' COULOMB POTENTIAL IN THE NON-HOMOGENEOUS SYSTEM: GENERAL CASE Let us consider the S-TMD ML encapsulated in be- tween a semi-infinite bottom substrate (1-st layer) and complex top cover layers consisting of two parts: 2- nd layer of finite thickness L, adjacent to the ML, and semi-infinite in the out-of-plane direction 3-rd part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' A schematic illustration of the studied system is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The ML is arranged in the xy plane and is centered in the out-of-plane direction (z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The bottom substrate, 1-st layer, belongs to the domain z ∈ (−∞, −δ], and is characterized by the in-plane ε1,∥ and out-of-plane ε1,⊥ dielectric constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The top (2- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='03245v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='mes-hall] 9 Jan 2023 2 z r δ δ ∞ ∞ 𝑳 𝜀1,∥ 𝜀1,⊥ 𝜀2,∥ 𝜀2,⊥ 𝜀3,∥ 𝜀3,⊥ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The schematic illustration of the S-TMD monolayer encapsulated in between a semi-infinite bottom substrate (1-st layer) and a complex top cover layers consisting of two parts: 2-nd layer of the finite thickness, adjacent to the ML and semi-infinite in the out-of-plane direction 3-rd part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' nd) layer, next to the ML, unfolds in the range z ∈ [δ, L] with the in-plane ε2,∥ and out-of-plane ε2,⊥ dielectric con- stants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Finally, the 3-rd top layer spreads over the dis- tance z ∈ [L, ∞) and is described by the in-plane ε3,∥ and out-of-plane ε3,⊥ dielectric constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' In order to find the potential energy between two charges in S-TMD MLs, we solve the following elec- trostatic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' We investigate the point-like charge Q at the point r = (ρ, z) = (0, 0, 0) and calcu- late the electric potential in such a system following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Namely, we analyze four regions: bottom (z ∈ (−∞, −δ]), ML (z ∈ [−δ, δ]), top finite (z ∈ [δ, L)) and overtop (z ∈ [L, ∞)) media, with potentials Φ1(ρ, z), Φ(ρ, z), Φ2(ρ, z), and Φ3(ρ, z), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' These po- tentials are defined by the Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' As the studied problem has cylindrical symmetry, we present the potentials in the form Φj(ρ, z) = 1 (2π)2 � d2keikρΦj(k, z), (1) Φ(ρ, z) = 1 (2π)2 � d2keikρΦ(k, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (2) The Φj(ρ, z) potentials for the j-th region, where j = 1, 2, 3, satisfy Maxwell’s equation div Dj(ρ, z) = 0, which can be written as − εj,∥k2Φj(k, z) + εj,⊥ d2Φj(k, z) dz2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (3) The solutions of these equations are Φ1(k, z) = B1eκ1z for z ∈ (−∞, −δ], (4) Φ2(k, z) = A2e−κ2z + B2eκ2z for z ∈ [δ, L], (5) Φ3(k, z) = A3e−κ3z for z ∈ [L, ∞), (6) where κj = |k| � εj,∥/εj,⊥ = k � εj,∥/εj,⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Maxwell’s equation in the ML domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=', z ∈ [−δ, δ], reads div D(ρ, z) = 4πQδ(ρ)δ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' It gives the equation for the potential Φ(r, z) � ∆∥ + d2 dz2 � Φ(ρ, z) = −4π[Qδ(ρ)δ(z) − ϱind(ρ, z)], (7) where ∆∥ is 2D Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The first term in the right-hand-side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (7) is the charge density of the charge Q, localized in the ML plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The second term represents the polarization charge density ϱind(ρ, z), in- duced in the ML by point charge Q, which is given by ϱind(ρ, z) = div P(ρ, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (8) Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [29], we present the polarization in the form P(ρ, z) = δ(z)P∥(ρ, z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (9) Using the proportionality between the induced polariza- tion P∥(ρ, 0) and the in-plane component of the electric field E∥(ρ, 0), P∥(ρ, 0) = χ2DE∥(ρ, 0), we obtain the ex- pression for the induced charge ϱind(ρ, z) = −χ2Dδ(z)∆∥Φ(ρ, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (10) Here χ2D is the 2D polarizability of the ML [29, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Taking the Fourier transformation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (7) with the introduced induced charge in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (10), one gets � k2 − d2 dz2 � Φ(k, z) = 4πQδ(z) − 4πχ2Dk2δ(z)Φ(k, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (11) This is a linear non-homogeneous differential equation of the second order, which solution can be presented as a sum of the general solution of the homogeneous equa- tion and a particular solution of the non-homogeneous equation (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [32] and Supplementary Material) Φ(k, z) = Ψe−k|z| + Ae−kz + Bekz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (12) The k-dependent parameters of the potential Ψ, A, and B are not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The relations between them are defined from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Integrating it over z in the domain z ∈ [−ϵ, ϵ] and then taking the limit ϵ → 0, one obtains [1 + r0k]Ψ + r0k[A + B] = 2πQ k , (13) where we introduced the screening length r0 = 2πχ2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Using the continuity of the potential and z component of the displacement field D(ρ, z) on the boundary of two adjusted domains, one obtains the set of equations for the parameters Ψ, A, B, B1, A2, B2, A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The boundary conditions for the 1-st and ML domains give relations B1e−κ1δ = Ψe−kδ + Aekδ + Be−kδ, (14) ε1B1e−κ1δ = Ψe−kδ − Aekδ + Be−kδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (15) The boundary conditions between the ML and 2-nd do- mains are described by equations A2e−κ2δ + B2eκ2δ = Ψe−kδ + Ae−kδ + Bekδ, (16) ε2A2e−κ2δ − ε2B2eκ2δ = Ψe−kδ + Ae−kδ − Bekδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (17) 3 Finally, the boundary conditions between the 2-nd and 3-rd domains give A2e−κ2L + B2eκ2L = A3e−κ3L, (18) ε2A2e−κ2L − ε2B2eκ2L = ε3A3e−κ3L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (19) Here, we introduce εj = √εj,⊥εj,∥ for j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Solving these equations together with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (13), we obtain the values of the Ψ, A, and B parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Let us consider the particular case δ = 0, which corre- sponds to the situation of the infinitely thin layer inves- tigated first by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Rytova [30] and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Keldysh [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The expression of the Coulomb potential in the in-plane direction reads Φ(k, z = 0) = (2πQ/k)v(k), with v(k) = 1 + ε2−ε3 ε2+ε3 e−2κ2L kr0 + ε1+ε2 2 + � kr0 + ε1−ε2 2 � ε2−ε3 ε2+ε3 e−2κ2L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (20) We can check that this expression provides the correct answer for the Coulomb potential for various limit cases, such as L → 0, L → ∞, r0 → 0, ε1 → ε2, ε2 → ε3, as well as their combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Note that the expression for the in-plane potential con- tains only the combinations εj = √εj,∥εj,⊥ and κ2 = k � ε2,∥/ε2,⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' That means, for example, that substrates with different values of in-plane ε∥ and out-of-plane ε⊥ dielectric constants can give the same shape of the po- tential in S-TMD MLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The interesting limit is ε3 → ∞, which corresponds to the case of a metallic plane placed horizontally at the distance L from the ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The related potential describes the modified Rytova-Keldysh potential suppressed by the metal substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' THE COULOMB POTENTIAL IN S-TMD SAMPLE: EFFECT OF FINITE THICKNESS OF HBN TOP LAYER We examine the particular case of an S-TMD ML en- capsulated in hBN layers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=', ε1,∥ = ε2,∥ = εhBN,∥, ε1,⊥ = ε2,⊥ = εhBN,⊥, ε3,∥ = ε3,⊥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Following the values available in the literature, we use ε1 = ε2 = εhBN = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='5 [9], and κ2 = � εhBN,∥/εhBN,⊥ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='36k [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Note that ε3,∥ = ε3,⊥ = 1 resemble typical experimental conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=', the sample is placed in air, vacuum, or gaseous helium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Introducing the dimensionless momen- tum x = kr0/εhBN and length l = εhBNL/r0 parameters, we obtain v(x) = 1 εhBN 1 + εhBN−1 εhBN+1 exp � − 2 � εhBN,∥ εhBN,⊥ xl � 1 + x � 1 + εhBN−1 εhBN+1 exp � − 2 � εhBN,∥ εhBN,⊥ xl ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (21) The corresponding potential as a function of the dimen- sionless distance ξ = ρεhBN/r0 is Φ(ξ) = (Q/r0)φ(ξ, l), 1 2 3 4 5 6 7 8 9 10 ∞ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='2 Potential function, � (� , l) Distance parameter, � l=0 l=1 l=5 l=∞ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Potential function φ(ξ, l) for four values of dimen- sionless length parameter, l: l=0, l=1, l=5, and l=∞ as a function of a dimensionless distance parameter, ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' where φ(ξ, l) = � ∞ 0 dx J0(xξ) � 1 + εhBN−1 εhBN+1 exp � − 2 � εhBN,∥ εhBN,⊥ xl �� 1 + x � 1 + εhBN−1 εhBN+1 exp � − 2 � εhBN,∥ εhBN,⊥ xl �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (22) Note that in the limit l → ∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=', an ML encapsulated in semi-infinite hBN layers, the potential expression is simplified and gives the well-known Rytova-Keldysh po- tential [30, 33] φ(ξ, ∞) = � ∞ 0 dxJ0(xξ) 1 + x = π 2 � H0(ξ) − Y0(ξ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (23) The other limit l → 0 corresponds to the case of an ML deposited on a semi-infinite hBN substrate, uncovered from the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The related potential also has the Rytova- Keldysh form φ(ξ, 0) = π 2 � H0 �εhBN + 1 2εhBN ξ � − Y0 �εhBN + 1 2εhBN ξ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (24) The evolution of the φ(ξ, l) potential as a function of a ξ parameter is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' 2 for four l values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' As can be seen in the Figure, the strongest φ(ξ, l) potential is apparent for l = 0, while the weakest potential is for the case l = ∞, the potential for l > 0 lies in the region in between of two former potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The obtained results are in full agreement with previously reported results [9, 20, 25], where it was demonstrated that the increase of the average dielectric constant of the media surrounding the ML leads to the decrease of the confining potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' 4 0 10 20 30 40 50 60 70 80 90 100 ∞ 240 210 180 60 30 0 1s Energy (meV) Number of top hBN layers 5s 4s 3s 2s ¥ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Energy spectrum of s excitonic state in the WSe2 ML encapsulated in hBN layers as a function of the thickness of the top hBN layer (colored online).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The full points corre- spond to the case of a semi-infinite thickness of top hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The colored open points represent the energy spectrum of excitons calculated individually for the mono- and bilayer top hBN (see SM for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Note that the thickness of the bottom hBN layer is semi-infinite in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The gray-shaded region represents the infinity of states above the bandgap energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' EXCITONIC SPECTRUM IN NON-HOMOGENEOUS SYSTEM Using the φ(ξ, l) potential, expressed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (22), we can evaluate the energy spectrum of excitons in the in- vestigated structure composed of S-TMD ML as a func- tion of the parameter l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The corresponding equation for eigenvalues is given by � b2 1 ξ d dξ � ξ d dξ � + 2bφ(ξ, l) + ϵ � ψ(ξ) = 0, (25) where we introduced b = ℏ2ε2 hBN/(µe2r0) and E = Ry∗ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Ry∗ = µe4/(2ℏ2ε2 hBN) is an effective Rydberg energy and ψ(ξ) represents the wave function of an exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' µ = memh/(me + mh) is the reduced mass of the exciton (e- h pair) with the effective electron (me) and hole (mh) masses and e represents electron’s charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Let us examine the case of WSe2 ML with r0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='5 nm [31] and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='21 m0 [6], where m0 is elec- tron’s mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' It gives b ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='13357 and Ry∗ ≈ 141 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Note that for WSe2 ML r0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='5 nm, therefore, the di- mensionless parameter l corresponds to the l nm of the thickness of the top flake of hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Taking into account that the distance between the layers in hBN (in other words, the thickness of hBN ML) is dhBN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='33 nm [35], we conclude that l = 1 corresponds to 3 layers of hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' For the thinnest hBN flake, the hBN layer demonstrates its discrete structure, which may be beyond the contin- uous model of hBN medium considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' To verify it, we performed the corresponding analysis for the ML and BL of top hBN in the Supplementary Material (SM), which takes into account the discrete nature of the hBN flake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The calculated energy spectra of an exciton for the ground (1s) and four excited (2s – 5s) states as a function of the thickness of the top hBN layer for the aforemen- tioned continuous model as well as the discrete one for the thinnest layers (described in the SM) are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Due to the observed evolutions in the Figure, three main points can be raised: (i) the continuous model provides a very good method for the calculation of the exciton spectrum, even in the case of the extremely thin top hBN flake of about 1-2 layers, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The largest discrepancy is observed between the homogeneous model and the ML of top hBN for the energy of 1s state of about 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (ii) the energies of excitonic states are sub- jected to the most significant variations for the thinnest top hBN layers with thicknesses below about 30 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' For thicker top hBN layers, the corresponding excitonic energies are almost fixed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (iii) the thickness effect of the top hBN layer is the largest for the ground 1s state of the exciton with its substantial reduction when the number of excitonic states is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The maximum change of the 1s energy of about 91 meV between two limits: with- out the top hBN layer and with its infinite thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The analogous differences of the 2s, 3s, 4s, and 5s states are of the order of 39 meV, 20 meV, 13 meV, and 8 meV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Finally, we can focus on the analysis of the excitonic binding energy (Eb, defined as the energy difference be- tween the electronic bang gap and the ground 1s state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The dependence of the Eb energy in the WSe2 ML en- capsulated in the hBN layers for selected numbers of the top hBN layer is summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The experimen- tally measured binding energies of excitons in WSe2 MLs encapsulated in hBN flakes is of about 170 meV [5–7, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Besides the structures with the hBN encapsulation in- vestigated experimentally differ from the one analyzed in this work, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' a WSe2 ML sandwiched between 10 nm thick hBN deposited on the core of a single-mode opti- cal fiber [9], the theoretically calculated binding energies are in very good agreement with the experimental ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Using our approach, the excitonic binding energy can be changed by almost 40% in the transition from the sample without the top hBN layer (Eb=256 meV) to the one with the infinite thickness of the top hBN layer (Eb=165 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' This reveals that the thickness of the surrounding media TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Calculated binding energies of excitons (Eb) in WSe2 ML encapsulated in hBN layers for selected numbers of the top hBN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Number of top hBN layers 0 3 6 10 20 40 100 ∞ Eb (meV) 256 207 192 184 175 171 167 165 5 of S-TMD MLs also plays a crucial role in the modifica- tion of the exciton energy spectrum in S-TMD MLs in addition to the engineering of the surrounding dielectric environment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=', the encapsulation of an ML in media characterized by dielectric constants [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' SUMMARY Concluding, we have demonstrated using the general- ized Rytova-Keldysh potential that the energy spectrum of excitonic states in S-TMD monolayers placed on a semi-infinite hBN substrate and covered with a top hBN layer of finite thickness can be significantly modified due to the screening effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' For WSe2 ML in such a struc- ture, the energies of the excitonic states are substantially adjusted for the thinnest top hBN layers with thicknesses below about 30 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Moreover, we obtained that the thickness effect of the top hBN layer is the largest for the ground 1s state of the exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' It results in a signif- icant reduction in excitonic binding energy that can be changed by almost 40% in the transition from the sam- ple without the top hBN layer to the one with an infinite thickness of the top hBN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The proposed model may be applicable for other 2D layered materials in which the screening effects on the excitonic spectrum play an es- sential role, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=', 2D perovskites [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' ACKNOWLEDGMENTS The work has been supported by the National Science Centre, Poland (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' 2018/31/B/ST3/02111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Cheiwchanchamnangij and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Lambrecht, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' B 85, 205302 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Ramasubramaniam, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' B 86, 115409 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [3] D.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Chang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Lui, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' B 99, 205420 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Lu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Goldstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Tong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Chaves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Kunstmann, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Cavalcante, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Woźniak, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Seifert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Reichman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Taniguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Watanabe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Smirnov, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Yan, Nano Letters 19, 2464 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' R.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Watanabe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Taniguchi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Faugeras, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Potemski, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' B 106, L081409 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Stier, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Wilson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' A.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Mak, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Zhao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Shan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Lett.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Berkelbach, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Nagler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Schüller, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Korn, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Nuckolls, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Hone, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Russo, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Thygesen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' B 102, 201402 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [23] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Vandenberghe, npj 2D Materials and Applications 2, 6 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [35] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Lv, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Fan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Xu, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Zhai, Integrated Ferroelectrics 162, 85 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Baranowski and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Plochocka, Advanced Energy Ma- terials 10, 1903659 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' 7 Supplementary Material for "Exciton spectrum in atomically thin monolayers: The role of hBN encapsulation" SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' THE MODIFIED RYTOVA-KELDYSH POTENTIAL FOR THE CASE OF ULTRATHIN TOP HBN FLAKE In our study, we modeled the top layer of hBN with its thickness L and the macroscopic in- (εhBN,∥) and out- of-plane (εhBN,⊥) dielectric constants of hBN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' in the limit of the continuous hBN medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' However, in the case of the ultrathin top hBN flake, where it consists of only a few layers of hBN monolayers, the use of such a limit is doubtful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' In order to find the limits of applica- bility of the continuous model, we derive the modified Rytova-Keldysh potential with a few layers of the hBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' To do it, we consider the modification of the system studied in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The bottom semi-infinite sub- strate remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Thus, it is characterized by in- (ε1,∥) and out-of-plane (ε1,⊥) dielectric constants and occupies the domain z ∈ (−∞, −δ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The S-TMD mono- layer is placed in the plane z = 0 with its in-plane po- larizability χTMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The first top hBN flake consists of N layer and has a thickness L = Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Here d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='33 nm is the distance between the hBN layers in a bulk hBN crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The jth layer of the hBN flake is placed in the zj = jd plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Each hBN layer is characterized by the in-plane polarizability χhBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The second top layer with the in- (ε3,∥) and out-of-plane (ε3,⊥) dielectric constants occupies the domain z ∈ [L + δ, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' In order to find the potential energy between two charges in S-TMD monolayer, we solve the following elec- trostatic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' We consider the point-like charge Q at the point (0, 0, z′), where 0 < z′ ̸= zj < L and calcu- late the potential in such a system following [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Simi- larly to the previous case, we consider 3 regions: bottom semi-infinite medium with potential Φ1(ρ, z), second top semi-infinite layer with the potential Φ3(ρ, z), and the space between them with the potential Φ2(ρ, z), where we introduced the in-plane vector ρ = (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Using the cylindrical symmetry of the problem, we present the po- tentials in the form Φj(ρ, z) = 1 (2π)2 � d2keikρΦj(k, z) (26) The Maxwell’s equation div D1,3 = 0 for 1-st and 3-rd regions (j = 1, 3) reads − εj,∥k2Φj(k, z) + εj,⊥ d2Φj(k, z) dz2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (27) The solutions to these equations are Φ1(k, z) = Aeκ1z, for z ∈ (−∞, −δ], (28) Φ3(k, z) = Be−κ3z, for z ∈ [L + δ, ∞), (29) where κj = |k| � εj,∥/εj,⊥ = k � εj,∥/εj,⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The equation for the potential Φ2(r, z) takes the form ∆∥Φ2(ρ, z)+ϵ⊥ d2Φ2(ρ, z) dz2 = − 4π[Qδ(ρ)δ(z − z′) − ϱind(ρ, z)], (30) where ∆∥ is 2D Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' We introduced the out- of-plane dielectric constant (ϵ⊥) of the hBN flake with N > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' In the case N = 1, one needs to put ϵ⊥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The first term on the right-hand side of the equation is the charge density of the charge Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The second term ϱind(ρ, z) represents the induced charge density due to the polarization of the hBN and S-TMD layer by charge Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [29], we present the induced charge in the form ϱind(ρ, z) = −χhBN N � j=1 δ(z − zj)∆∥Φ2(ρ, zj)− −χTMDδ(z)∆∥Φ2(ρ, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (31) Using the Fourier transform (26) of the potential Φ2(ρ, z) we present Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (30) as � k2 − ϵ⊥ d2 dz2 � Φ2(k, z) =4πQδ(z − z′) − 2r0k2δ(z)Φ2(k, 0) −2R0k2 N � j=1 δ(z − zj)Φ2(k, zj), (32) where we introduced the in-plane screening lengths r0 = 2πχTMD and R0 = 2πχhBN for S-TMD and hBN mono- layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Integrating this equation in the re- gions z ∈ [−ϵ, ϵ], z ∈ [zj −ϵ, zj +ϵ], and z ∈ [z′ −ϵ, z′ +ϵ] one obtains the following conditions in the limit ϵ → 0 ϵ⊥ dΦ2(k, z) dz ��� +0 −0 = 2r0k2Φ2(k, 0), (33) ϵ⊥ dΦ2(k, z) dz ��� zj+0 zj−0 = 2R0k2Φ2(k, zj), (34) ϵ⊥ dΦ2(k, z) dz ��� z′+0 z′−0 = −4πQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (35) At the points z ̸= 0 , zj , z′ the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (32) is simplified � k2 − ϵ⊥ d2 dz2 � Φ2(k, z) = 0 (36) Therefore, the general solution for Φ2(k, z) can be writ- ten in the form Φ2(k, z) =Ψ0e−K|z−z′| + N � j=1 Ψje−K|z−zj|+ +Ψe−K|z| + αeKz + βe−Kz, (37) 8 where Ψ0,Ψj,Ψ, α and β are unknown functions of K = k/√ϵ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The aforementioned boundary conditions to- gether with the equation give the following restrictions Ψ = − r0KΦ2(k, 0), Ψj = − R0KΦ2(k, zj), Ψ0 =2πQ/ϵ⊥K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (38) Considering the boundary conditions for the electrostatic potential and for the out-of-plane component of the dis- placement field at z = −δ Φ1(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' −δ) = Φ2(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' −δ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (39) ε1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='⊥ ϵ⊥ dΦ1(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' z) dz ��� z=−δ = dΦ2(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' z) dz ��� z=−δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (40) and at z = L + δ Φ3(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' L + δ) = Φ2(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' L + δ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (41) ε3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='⊥ ϵ⊥ dΦ3(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' z) dz ��� z=L+δ = dΦ2(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' z) dz ��� z=L+δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (42) we get the following equations Ae−κ1δ = Ψ0e−K(z′+δ) + N � j=1 Ψje−K(zj+δ) + Ψe−Kδ + αe−Kδ + βeKδ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (43) A ε1 √ϵ⊥ e−κ1δ = Ψ0e−K(z′+δ) + N � j=1 Ψje−K(zj+δ) + Ψe−Kδ + αe−Kδ − βeKδ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (44) Be−κ3(L+δ) = Ψ0e−K(L+δ−z′) + N � j=1 Ψje−K(L+δ−zj) + Ψe−K(L+δ) + αeK(L+δ) + βe−K(L+δ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (45) B ε3 √ϵ⊥ e−κ3(L+δ) = Ψ0e−K(L+δ−z′) + N � j=1 Ψje−K(L+δ−zj) + Ψe−K(L+δ) − αeK(L+δ) + βe−K(L+δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (46) where we introduced the short notation εj = √εj,⊥εj,∥ for j = 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Introducing a = ε1/√ϵ⊥ and b = ε3/√ϵ⊥, and considering the general case a, b ̸= 1 we remove the parameters A and B from the above equations and obtain Ψ0e−K(z′+δ) + N � j=1 Ψje−K(zj+δ) + Ψe−Kδ + αe−Kδ + �a + 1 a − 1 � βeKδ = 0, (47) Ψ0e−K(L+δ−z′) + N � j=1 Ψje−K(L+δ−zj) + Ψe−K(L+δ) + �b + 1 b − 1 � αeK(L+δ) + βe−K(L+δ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (48) Note that the solution of the aforementioned system of equations for the case a = 1 and/or b = 1 can be obtained from the general solution by taking the corresponding limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' We solve the general equation in a few steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' First, we write the system of equations in the following form e−Kδα + �a + 1 a − 1 � eKδβ =A, �b + 1 b − 1 � eK(L+δ)α + e−K(L+δ)β =B, (49) where we introduced the following notations A = −Ψ0e−K(z′+δ) − �N j=1 Ψje−K(zj+δ) − Ψe−Kδ, B = −Ψ0e−K(L+δ−z′) − �N j=1 Ψje−K(L+δ−zj) − Ψe−K(L+δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Solving the system of equations (49) one gets α = Ae−K(L+δ) − B � a+1 a−1 � eKδ D , (50) β = Be−Kδ − A � b+1 b−1 � eK(L+δ) D , (51) 9 with D = e−K(L+2δ) − � a+1 a−1 �� b+1 b−1 � eK(L+2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Substi- tuting these solutions into (37), we express Φ2(k, z) via N + 2 parameters Ψ0, Ψj and Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Then, using the latter expression, we calculate the expressions for the potential in z = 0, z = z′, and z = zl, l = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' N points Φ2(k, 0) =Ψ0e−Kz′ + N � j=1 Ψje−Kzj + Ψ + α + β, Φ2(k, z′) =Ψ0 + N � j=1 Ψje−K(zl−z′) + Ψe−Kz′ + αeKz′ + βe−Kz′, Φ2(k, zl) =Ψ0e−K(zl−z′) + N � j=1 Ψje−K|zl−zj| + Ψe−Kzl + αeKzl + βe−Kzl, (52) which together with (38) gives the complete set of equa- tions for the parameters Ψ0, Ψj, and Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' In the system of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (52), we considered the particular case 0 < z′ < z1 < z2 < · · · < zN = L for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Then we sim- plify our solutions by taking the following limits δ → 0 and z′ → 0, and obtain Φ2(k, z = 0) = (2πQ/k)v(k) which defines the electrostatic potential Φ(ρ) in the S- TMD plane, which reads Φ(ρ) = Q � ∞ 0 dkJ0(kρ)v(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (53) We derive the expressions for the case of monolayer (vm) and bilayer (vb) hBN flake to demonstrate the aforemen- tioned algorithm and then estimate the spectrum of ex- citons for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' First, we consider the case of monolayer hBN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' N = 1, ϵ⊥ = 1, and hence a = ε1, b = ε3 and K = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Following the procedure mentioned above, we obtain the result for the monolayer (m) vm(k) = 1 − � 2kR0+ε3−1 2kR0+ε3+1 � e−2dk (kr0 + ε1+1 2 ) − (kr0 + ε1−1 2 ) � 2kR0+ε3−1 2kR0+ε3+1 � e−2dk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (54) Note that this result looks similar to the general result in the main text, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The eigenvalue problem for this potential has a form � �b2 1 ξ � ξ d dξ � + 2�b � ∞ 0 dxJ0(xξ)vm(x) + ϵ � ψ(ξ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (55) Here vm(x = kr0) ≡ vm(k), ϵ = E/Ry and ξ = ρ/r0 are the dimensionless energy and coordinate, respec- tively, with Ry = µe4/2ℏ2 and �b = ℏ2/(µe2r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' To calculate the spectrum of excitons, we consider the par- ticular case ε1 = εhBN = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='5, ε3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Using for- mula (4) from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [31] and in-plane dielectric constant of the monolayer hBN εhBN,∥ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='93 from [34] R0 = (εhBN,∥ − 1)d/2, we obtain R0 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='78 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Note that this screening length is much smaller than, for example, in the WSe2 monolayer r0 = 45 Å [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Solving the eigen- value equation with �b ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='056 and taking into account that Ry = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='856 eV we obtain the following binding en- ergies: E1 ≈ −242 meV, E2 ≈ −74 meV, E3 ≈ −36 meV, E4 ≈ −21 meV, E5 ≈ −14 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Note that this result is close to the result of the homogeneous model proposed in the main text: E1 ≈ −231 meV, E2 ≈ −73 meV, E3 ≈ −35 meV, E4 ≈ −21 meV, E5 ≈ −14 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The main difference between the two calculated spectra is the binding energy of the 1s exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The smaller binding energy of the 1s state in the homogeneous model can be explained by the less effective screening of the Coulomb potential than in the model considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The potential for the case of the bilayer flake N = 2 Φ2(k, z = 0) = (2πQ/k√ϵ⊥)vb(k) with the correspond- ing out-of-plane dielectric constant ϵ⊥ is vb(k) = 1 − 2KR0(b+2KR0) (KR0+1)(b+2KR0+1)e−2dK + (KR0−1)(b+2KR0−1) (KR0+1)(b+2KR0+1)e−4dK a+1 2 + Kr0 − [(a+2Kr0)(b+2KR0)−1] (KR0+1)(b+2KR0+1) KR0e−2dK + (KR0−1)(b+2KR0−1) (KR0+1)(b+2KR0+1) � a−1 2 + Kr0 � e−4dK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (56) Here K = k/√ϵ⊥, a = ε1/√ϵ⊥, and b = ε3/√ϵ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The eigenvalue problem for the case of the bilayer reads � �b2ϵ⊥ 1 ξ � ξ d dξ � + 2�b � ∞ 0 dxJ0(xξ)vb(x) + ϵ � ψ(ξ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' (57) 10 Here vb(x = kr0/√ϵ⊥) ≡ vb(k), ϵ = E/Ry and ξ = ρ√ϵ⊥/r0 are the dimensionless energy and coordinate, respectively, with Ry = µe4/2ℏ2 and �b = ℏ2/(µe2r0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' For the numerical estimation of the exciton spectrum in this case, we use ε1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='5 and ε3 = 1 and the result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' [34] for ϵ⊥ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' The spectrum of the excitons then reads E1 ≈ −218 meV, E2 ≈ −69 meV, E3 ≈ −34 meV, E4 ≈ −20 meV, E5 ≈ −13 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' It is surprisingly very close to the result of the homogeneous model, described in the main text, E1 ≈ −217 meV, E2 ≈ −69 meV, E3 ≈ −34 meV, E4 ≈ −20 meV, E5 ≈ −13 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} +page_content=' Therefore, we conclude that the homogeneous model provides a very good method for the calculation of the exciton spectrum, even in the case of the extremely thin top hBN flake of about 1-2 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfhwTG/content/2301.03245v1.pdf'} diff --git a/tdE5T4oBgHgl3EQfmg_h/content/tmp_files/2301.05680v1.pdf.txt b/tdE5T4oBgHgl3EQfmg_h/content/tmp_files/2301.05680v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d2069a98cf31ba7ef3a699599dab66f4b4479a8a --- /dev/null +++ b/tdE5T4oBgHgl3EQfmg_h/content/tmp_files/2301.05680v1.pdf.txt @@ -0,0 +1,2226 @@ +Cumulative Memory Lower Bounds for Randomized and +Quantum Computation +Paul Beame* +Computer Science & Engineering +University of Washington +Niels Kornerup +Computer Science +University of Texas at Austin +January 16, 2023 +Abstract +Cumulative memory—the sum of space used over the steps of a computation—is a fine- +grained measure of time-space complexity that is a more accurate measure of cost for algorithms +with infrequent spikes in memory usage in the context of technologies such as cloud computing +that allow dynamic allocation and de-allocation of resources during their execution. We give +the first lower bounds on cumulative memory complexity that apply to general sequential +classical algorithms. We also prove the first such bounds for bounded-error quantum circuits. +Among many possible applications, we show that any classical sorting algorithm with suc- +cess probability at least 1/poly(n) requires cumulative memory ˜Ω(n2), any classical matrix +multiplication algorithm requires cumulative memory Ω(n6/T), any quantum sorting circuit +requires cumulative memory Ω(n3/T), and any quantum circuit that finds k disjoint collisions +in a random function requires cumulative memory Ω(k3n/T2). More generally, we present +theorems that can be used to convert a wide class of existing time-space tradeoff lower bounds +to matching lower bounds on cumulative memory complexity. +*Research supported by NSF grant CCF-2006359 +arXiv:2301.05680v1 [cs.CC] 13 Jan 2023 + +1 +Introduction +There are many problems where algorithms can use additional memory for faster running times +or expend additional time to reduce memory requirements. While there are many different kinds +of tradeoffs between time and space, the most common complexity metric for such algorithms is +the maximum time-space (TS) product. This metric is appropriate when a machine must allocate +an algorithm’s maximum space throughout its computation. However, recent technologies like +AWS Lambda [BBB+21], Flex [LL20], and CloudScale [SSGW11] suggest that in the context of +cloud computing, space can be allocated to a program only as it is needed. When using such +services, analyzing the average memory used per step leads to a more accurate picture than merely +measuring the maximum space used. +Cumulative memory (CM) - the sum over time of the space used per step of an algorithm - is +an alternative notion of time-space complexity that is more fair to algorithms that only require +rare spikes in memory. The term cumulative memory complexity was first coined by Alwen and +Serbinenko [AS15] who introduced it as a way to discuss time-space tradeoffs for "memory hard +functions" like password hashes. Since then, lower and upper bounds on the CM complexity +of problems in structured computational models using the black pebble game have been exten- +sively studied, beginning with the work of [AS15, AB16, RD16, ACP+17, ACK+16, ABP17]. These +structured models via pebble games are particularly natural in the context of the random oracle +assumptions that are common in cryptography. By carefully interweaving their memory intensive +steps, these authors devise algorithms for cracking passwords that compute many hashes in parallel +using only slightly more space than is necessary to compute a single hash. While such algorithms +can use parallelism to amortize costs and circumvent proven single instance TS complexity lower +bounds, their cumulative memory scales linearly with the number of computed hashes. Thus +cumulative memory complexity is a more robust metric than TS complexity. +Surprisingly there has been little research into CM complexity outside the setting of cryptogra- +phy. In [AdRNV17] the authors showed strong CM complexity results for the black-white pebble +game and used them to derive related results for resolution proof systems. Our work is the first to +explore CM complexity outside the regime of pebbling and the first to obtain results that apply to +general models of computation without cryptographic or black-box assumptions. +Our Results +In this work, we give generic methods that allow one to convert existing paradigms for obtaining +time-space tradeoff lower bounds involving worst-case space to new lower bounds that replace +the time-space product by cumulative space, immediately yielding a host of new lower bounds on +cumulative memory complexity. With these methods, we show how to extend almost all known +proofs for time-space tradeoffs to equivalent lower bounds on cumulative memory complexity. Our +results, like those of existing time-space tradeoffs, apply in models in which arbitrary sequential +computations may be performed between queries to a read-only input. Our lower bounds also +apply to randomized and quantum algorithms that are allowed to make errors. +Classical computation +We first focus on lower bound paradigms that apply to computations +of multi-output functions. We give general theorems showing how to translate the basic ideas +1 + +that yield virtually all time-space tradeoffs known for such functions to yield lower bounds on +cumulative memory complexity. As applications of our general methodology, we prove that the +cumulative memory required by any sorting algorithm is ˜Ω(n2) which generalizes [BC82, Bea91] +and the cumulative memory required for any matrix multiplication algorithm using time T is +Ω(n6/T), generalizing [Abr91]. +We also show how the paradigm can be extended to correspond to the best time-space tradeoff +lower bounds for single-output Boolean functions. In particular, we give examples of functions +for which algorithms running in time T require cumulative memory Ω((n2 log n)/2cT/n) for some +constant c > 0, generalizing [BSSV03]. This means, for example, that algorithms computing these +functions in time o(n log n) require cumulative memory Ω(n2 log n). +Quantum computation +We generalize the quantum time-space tradeoff for sorting proven in +[KŠdW07], which requires that the time order in which output values are produced must correspond +to the sorted order, to a matching cumulative memory complexity bound of Ω(n3/T) that works +for any fixed time-ordering of output production which yields a more general lower bound1. We +then show how our classical general theorems can be applied to known quantum time-space +tradeoffs and extend the quantum time-space tradeoff for k-collision pairs finding from [HM21] to +the matching cumulative memory complexity bound of Ω(k3n/T2). +Previous work +Memory hard functions and cumulative memory complexity +Memory hard functions (MHFs) +are functions designed to require large space to compute. In [AS15] Alwen and Serbinenko in- +troduced parallel cumulative (memory) complexity as a metric for analyzing the space footprint +required to compute these functions. Most MHFs are constructed using hashgraphs (see [DNW05]) +of DAGs whose output is a fixed length string and their proofs of security are based on pebbling ar- +guments on these DAGs while assuming access to truly random hash functions for their complexity +bounds [AS15, BCGS16, BDK16, RD16, ABP17, ACP+17, BZ17]. More recent MHF constructions +do not rely on random hash functions; however, they still require some cryptographic assumptions +[CT19, ABB21]. In general the major focus of these results has been on savings with parallel rather +than sequential computation. +Classical time-space tradeoffs +While early work focused on the kinds of restricted pebbling +models similar to those considered to date for cumulative memory complexity [Tom80, BFK+81], +the gold standard model for time-space tradeoff analysis is that of unrestricted branching programs, +which simultaneously capture both time and space for general sequential computation. This +analysis began with the work of [BC82] who proved lower bounds for sorting and introduced a +general methodology for multi-output functions that has been extended to many problems (e.g., +[Yes84, Abr87, Abr90, Bea91, MNT93]), including universal hashing and a wide array of problems +in linear algebra [Abr91]. A separate methodology for single-output functions, first introduced in +the context of restricted branching programs [BRS93, Oko93], was extended to general branching +programs in [BJS01], with further applications to other problems [Ajt02] including multi-precision +1For example, an algorithm may be able to determine the median output long before it determines the other outputs. +2 + +integer multiplication [SW03] and error-correcting codes [Juk09] as well as over Boolean input +domains [Ajt05, BSSV03]. +Both of these methodologies involve breaking the branching program into blocks. For multi- +output functions one needs to show that for any fixed node at the beginning of a block, the +probability over a random input that the program started at that node produces k correct output +values in that block decays exponentially in k. For single-output functions, one decomposes the +space of inputs based on the "trace" of nodes traversed at segment boundaries. Based on the traces, +one can determine the size and density properties of "embedded rectangles" of inputs on which the +function must be constant. Lower bounds require showing the given function does not have such +rectangles. +Quantum time-space tradeoffs +Though the basic notion of exponential decay in producing +correct outputs is similar to the classical multi-output bounds, the arguments are substantially more +subtle in the quantum setting. The quantum query model gives us access to an input X = x1, . . . , xn +via an oracle QX. Since the result of a single quantum query can change if we flip any bit of the +input, we need arguments that limit the sensitivity of a query to changes in the oracle. These +arguments generally follow one of two techniques: the adversary method [BBBV97, Amb02, ŠS05] +or the polynomial method [BBC+01]. +To obtain quantum time-space tradeoffs for multi-output functions, it is important to have +lemmas showing that query-bounded computations only yield a slight advantage over randomly +guessing outputs. Such lemmas often take the form of direct product theorems, which state that +if T queries are necessary to solve one instance of a problem with constant probability, then kT +queries are insufficient to solve k independent instances of that problem with probability 2−Ω(k). +While such results seems intuitive, Shaltiel proved that they are not true in general [Sha03]. The +polynomial method [Aar05, KŠdW07, She11] and the adversarial method [AŠdW09] have both +been extended to prove quantum direct product theorems. +In [KŠdW07] the authors use direct product theorems to prove a tight time-space tradeoff for +sorting and a time-space tradeoff for matrix multiplication in Boolean algebra. They also proved +somewhat weaker lower bounds for computing matrix-vector products for fixed matrices A; those +bounds were extended in [AŠdW09] to systems of linear inequalities. However, both of these latter +results apply to computations where the fixed matrix A defining the problem depends on the +space bound and, unlike the case of sorting or Boolean matrix multiplication, do not yield a fixed +problem for which the lower bound applies at all space bounds. More recently [HM21] extended +the recording query technique of Zhandry in [Zha19] to obtain time-space lower bounds for the +k-collision problem and match the aforementioned result for sorting. +Our methods +At the highest level, we employ part of the same paradigms previously used for time-space tradeoff +lower bounds. namely breaking up the computations into blocks of time and analyzing properties +of the branching programs or quantum circuits based on what happens at the boundaries between +those time blocks. However, for cumulative memory complexity, those boundaries cannot be at +fixed locations in time and their selection needs to depend on the space used in those time steps. +3 + +Further, in many cases, the time-space tradeoff lower bound needs to set the lengths of those +time blocks in a way that depends on the specific space bound. When extending the ideas to +bound cumulative memory usage, there is no single space bound that can be used throughout +the computation; this sets up a tricky interplay between the choices of boundaries between time +blocks and the lengths of the time blocks. Because the space usage within a block may grow and +shrink radically, even with optimal selection of block boundaries, the contribution of each time +block to the overall cumulative memory may be significantly lower than the time-space product +lower bound one would obtain for the individual block. +We show how to bound any loss in going from time-space tradeoff lower bounds to cumulative +memory lower bounds in a way that depends solely on the bound on the lengths of blocks as +a function h0 of the target space bound. For many classes of bounding functions we are able to +bound the loss by a constant factor, and we are able show that it is always at most an O(log n) +factor loss. Once we have this, if this bounding function h0 is non-constant, there is still a matter of +bounding the optimum way for the algorithm to allocate its space budget for producing the require +outputs throughout its computation. This optimization again depends on the bounding function +h0. This involves minimizing a convex function based on h0 subject to a mix of convex and concave +constraints which is not generally tractable. However, assuming that h0 is nicely behaved, we +are able to apply specialized convexity arguments to yield lower bounds on cumulative memory +complexity that in many instances match those of previous time-space tradeoffs up to a constant +factor. +Road map +We give the overall definitions in Section 2, including a review of the standard +definitions of the work space used by quantum circuits. We then give the very simple version of our +methods that is needed to prove results on the cumulative memory complexity of classical sorting +algorithms in Section 3. In Section 4, we give our lower bound for quantum sorting algorithms. +This example shows something of the complexity required for our general arguments; in this case, +the bounding function is simple enough that we can apply an alternative direct argument to show +only constant loss in the choices of boundaries for time blocks, but it still requires some of the +complexity of the general optimization of space allocation. This section also includes the additional +ideas that allow us to analyze circuits that produce sorted outputs in arbitrary sequential time +steps. +We give the full general theorems that let us convert classical time-space tradeoffs for multi- +output functions to cumulative memory lower bounds, even for randomized algorithms, in Sec- +tion 5. In Section 6 we apply these general theorems to a few other problems, particularly those +in linear algebra, to give an indication of how they can be used. Next, we show how to convert +time-space tradeoff lower bounds for single-output functions to cumulative memory lower bounds +in Section 7. Finally, in Section 8 and Section 9 we show how to extend our generic method to +quantum circuits and discuss its application to other existing quantum time-space tradeoffs. The +appendices contain some of the technical arguments that allow us to bound the loss functions and +to give bounds on the optimum allocations of cumulative space budgets to time steps. +4 + +2 +Preliminaries +Cumulative memory is an abstract notion of time-space complexity that can be applied to any +model of computation with a natural notion of space. +Definition 2.1. The cumulative memory of a discrete time computation A that uses |At| space during +the t-th step and runs in time T is: +CM(A) = +T +∑ +t=1 +|At| +The cumulative memory complexity of a function f with respect to a computational model M is: +CMC( f ) = +min +A∈M computes f CM(A). +In this paper we consider both branching programs and quantum circuits. +Branching Programs +Branching programs with input {x1, . . . , xn} ∈ Dn are known as D-way +branching programs and are defined using a rooted DAG in which each non-sink vertex is labeled +with an i ∈ [n] and has |D| outgoing edges that correspond to possible values of xi. Each edge is +optionally labeled by some number of output statements expressed as pairs (j, oj) where j ∈ [m] +is an output index and oj ∈ R (if outputs are to be ordered) or simply oj ∈ R (if outputs are to +be unordered). Evaluation is performed by starting at the root v0 and following the appropriate +labels of the respective xi. We consider branching programs P that contain T + 1 layers where +the outgoing edges from nodes in each layer t are all in layer t + 1. We impose no restriction +on the query pattern of the branching program or when it can produce parts of the output. We +define the following complexity measures for such a branching program P. The time of the +branching program is T(P) = T. The space of the branching program is S(P) = maxt log2 |Lt| +where Lt is the set of nodes in layer t. Observe that in the absence of any limit on its space, a +branching program could equally well be a decision tree; hence the minimum time for branching +programs to compute a function f is its decision tree complexity. The time-space (product) used by the +branching program is TS(P) = T(P)S(P). The cumulative memory used by the branching program +is CM(P) = ∑t log2 |Lt|. +Branching programs are very general and simultaneously model time and space for sequential +computation. In particular they model time and space for random-access off-line multitape Turing +machines and random-access machines (RAMs) when time is unit-cost, space is log-cost, and the +input and output are read-only and write-only respectively2. Branching programs are much more +flexible than these models since they can make arbitrary changes to their storage in a single step. +2In prior work, branching program space has often been defined to be the logarithm of the total number of nodes +(e.g., [BC82, Abr91]) rather than the logarithm of the width (maximum number of nodes per layer), though the latter has +been used (e.g., [CFL83]). The natural conversion from an arbitrary space-bounded machine to a branching program +produces one that is not leveled (i.e., nodes are not segregated by time step). After leveling the branching program, the +space of the original machine becomes the logarithm of the width (cf. [Pip79]). The width-based definition is also the +only natural one by which to measure cumulative memory complexity and, in any case, the two definitions differ by at +most an additive log2 T amount, with lower bounds on width implying lower bounds on size. +5 + +Quantum Circuits +We also consider quantum circuits C classical read-only input X = x1, . . . , xn +that can be queried using an XOR query oracle as shown in Figure 1. As is normal in circuit models, +each output wire is associated with a fixed position in the output sequence, independent of the +input. As shown in Figure 2 following [KŠdW07], we abstract an arbitrary quantum circuit C into +layers C = {L1, . . . , LT} where layer Lt starts with the t-th query Q to the input and ends with +the start of the next layer. During each layer, an arbitrary unitary transformation V gets applied +which can express an arbitrary sub-circuit involving input-independent computation. The sub- +circuit/transformation V outputs St qubits for use in the next layer in addition to some qubits that +are immediately measured in the standard basis, some of which are treated as classical write-only +output. The time of C is lower bounded by the number of layers T and we say that the space of +layer Lt is St. Observe that to compute a function f, T must be at least the quantum query complexity +of f since that measure corresponds the above circuit model when the space is unbounded. Note +that the cumulative memory of a circuit is lower-bounded by the sum of the St. For convenience +we define S0, the space of the circuit before its first query, to be zero. Thus we only consider the +space after the input is queried. +|x⟩ +f +|x⟩ +|b⟩ +|b ⊕ f (x)⟩ +Figure 1: The XOR oracle for a function f : D → R where D ⊆ {0, 1}n and R ⊆ {0, 1}m is the linear +operator that, for all x ∈ D and b ∈ {0, 1}m, maps the basis state |x⟩ |b⟩ to |x⟩ |b ⊕ f (x)⟩. When +x ̸∈ D, the XOR oracle acts like the identity. The query oracle QX (where X = x1, . . . , xn) is the XOR +oracle for the function f (i) = xi. +Figure 2: The abstraction of a quantum circuit into layers. +Simulating a quantum query from modified input +Without making any additional assumptions +on our query oracle, it is possible to simulate a query for a modified (possibly larger) input using at +most two queries to the original input. Let QX be an XOR query oracle for some input X = x1, . . . xn +where xi ∈ {0, 1}m and let +ˆxi = +� +xi +i ∈ [n] +0 +otherwise +By definition, this makes QX the permutation that maps any basis state |i, j, k⟩ where i ∈ {0, 1}⌈log2 n⌉ +and j ∈ {0, 1}m to the basis state |i, j ⊕ ˆxi⟩. We want to use QX to simulate queries to some modified +6 + +input X′ = {x′ +1, . . . , x′ +n+t} where x′ +i ∈ {0, 1}ℓ is defined by x′ +i = g(i, ˆxi). Let G be the XOR query +oracle for g(i, j) and Pi>n be the XOR query oracle for the predicate function p(i) = 1i>n. Then the +circuit in Figure 3 simulates an XOR query on modified input X′. +Note that G and Pi>n both compute classical functions and therefore can be computed using a +network of Toffoli gates. Since G is independent of X, this circuit simulates a query to X′ using at +most two queries to X. The second query of QX is necessary to uncompute the m qubit ancillary +register. +Figure 3: The above circuit uses two calls to a query oracle QX in order to simulate one query to the +modified input X′. +3 +Cumulative memory complexity of classical sorting algorithms +For a natural number N, the standard version of sorting is a function Sortn,N : [N]n → [N]n that +on input x ∈ [N]n produces an output y ∈ [N]n in non-decreasing order where y is a permutation +of x; that is, there is some permutation π such that yi = xπ(i) for all i ∈ [n]. A related problem is +the ranking problem Rankn,N : [N]n → [n]n which on input x ∈ [N]n produces a permutation π +represented as the vector (π(1), . . . , π(n)) such that Sortn,N(x) = (xπ(1), . . . , xπ(n)) and whenever +xi = xj for i < j we have π(i) < π(j). +Proposition 3.1 ([BC82]). +(a) If there is an [nN]-way branching program P computing Sortn,nN then +there is an [N]-way branching program P′ computing Rankn,N with T(P′) ≤ T(P), S(P′) ≤ S(P), +and CM(P′) ≤ CM(P). +(b) If there is an [N]-way branching program P′′ computing Rankn,N then there is an [N]-way branch- +ing program P′′′ computing Sortn,N with T(P′′′) ≤ 2T(P′′), S(P′′′) ≤ S(P′′) + log2 N, and +CM(P′′′) ≤ 2CM(P′′) + T(P′′′) log2 N. +Proof. For part (a), the program P′ is exactly P except that when P queries xi ∈ [Nn], P′ reads +x′ +i ∈ [N] and branches on value xi = (x′ +i, i) and when P outputs (i, yi) = (i, xπ(i)) on an edge +for xπ(i) = (x′ +π(i), π(i)), P′ outputs (i, π(i)). For part (b), the program P′′′ is exactly P′′ except +that whenever P′′ outputs (i, π(i)) on an edge, P′′′ queries xπ(i) and outputs (i, xπ(o)). One layer +becomes two layers and the number of nodes per layer of P′′′ is at most N times that of P′′. +Following [BC82], we focus on inputs where the xi are distinct. In this case, the tie-breaking we +enforced in defining Rankn,N when there are equal elements is irrelevant. +7 + +Proposition 3.2 ([BC82]). There is an α > 0 such that the following holds. Let n be sufficiently large and +µ be the uniform distribution over lists of n distinct integers from [n2]. Then for any branching program B +of height h ≤ αn and for all integers k ≤ 2αn, the probability for x ∼ µ that B produces at least k correct +output values of Rankn,n2 on input x is at most 2−k/⌈log2 n⌉. +Theorem 3.3. Let P be a branching program computing Sortn,n3 with probability at least n−O(1) and +T = T(P). Then T is Ω(n2/ log2 n) or CM(P) is Ω(n2/ log n). Further, any random access machine +computing Sortn,n3 with n−O(1) probability requires cumulative memory of Ω(n2/ log n) bits. +Proof. We prove the same bounds for branching programs P computing Rankn,n2 which, by Propo- +sition 3.1, implies the bounds for computing Sortn,n3. +For simplicity we first assume that P is determistic and is always correct. Let α be the constant +and µ be the probability distributuon on [n2]n from Proposition 3.2, and let H = +� α +2n +� +. We partition +P into ℓ = ⌈T/H⌉ intervals {I1, . . . , Iℓ}, all of length H except for the first, which may be shorter +than the rest. Let t1 = 0, tℓ+1 = T, and for i ∈ [2, ℓ], ti be the time-step in Ii with the fewest number +of nodes. We define Si = log2(|Lti|) where Lj is the set of nodes of P in layer j. The i-th time block +Bi will contain all layers from ti to ti+1. We observe: +CM(P) ≥ +ℓ +∑ +i=2 +Si H = H +ℓ +∑ +i=1 +Si +(1) +since S1 = 0. Define ki = ⌈⌈log2 n⌉ (Si + log2(2T))⌉, which will be our target number of outputs +for block Bi. By our choice of Bi we know its length is at most αn and it starts at a layer with 2Si +nodes. So, by Proposition 3.2, combined with a union bound, the probability for x ∼ µ that Bi +produces at least ki correct output values of Rankn,n2 on input x ∼ µ is at most 1/(2T). Thus the +probability over µ that at least one block Bi produces at least ki correct output values is at most 1/2 +and the probability that the total number of outputs produced is at most ∑ℓ +i=1(ki − 1) is at least 1/2. +Since P must always produce n correct outputs, we must have: +ℓ +∑ +i=1 +(ki − 1) ≥ n. +Inserting the definition of ki we get: +ℓ +∑ +i=1 +(⌈log2 n⌉ (Si + log2(2T))) ≥ n. +Using Equation (1) to express this in terms of CM(P) gives us: +CM(P)/H + ℓ log2(2T) ≥ +n +⌈log2 n⌉ +or +CM(P) + T log2(2T) ≥ n +� α +2n +� +⌈log2 n⌉ ≥ +αn2 +3 log2 n. +Thus at least one of T log2(2T) or CM(P) is at least αn2/(6 log2 n), as required, since log T is +O(log n) wlog. The bound for random-access machines comes from observing that such a machine +requires at least one memory cell of Ω(log T) bits at every time step. +8 + +To prove the bound for algorithms with success probability n−c, we multiply log2(2T) in the +above argument by (c + 1). Since any sorting algorithm must have T ≥ n, on randomly chosen +inputs the probability that it produces at least ∑ℓ +i=1(ki − 1) correct outputs becomes +1 +2nc < 1 +nc and +hence the above bounds (reduced by the constant factor c + 1) apply to deterministic algorithms +with success probability 1/nc. By Yao’s lemma this implies the same lower bound for randomized +algorithms with success probability n−c. +Theorem 3.3 applies to cumulative working memory of any algorithm that produces its sorted +output in a write-only output vector and can compute those values in arbitrary time order. If the al- +gorithm is constrained to produce its sorted output in the natural time order then, following [Bea91], +one can obtain a slightly stronger bound. +Theorem 3.4. Any branching program P computing the outputs of Sortn,n in order in time T and probability +at least 4/5 requires T to be Ω(n2/ log n) or CM(P) to be Ω(n2). Further, any random access machine +computng Sortn,n in order with probability at least 4/5 requires cumulative memory Ω(n2). +Proof Sketch. Any such algorithm can easily determine all the elements of the input that occur +uniquely and the lower bounds follow from the bounds on Unique Elements that we prove +in Section 6. +4 +Quantum cumulative memory complexity of sorting +We now show with a similar argument that the quantum cumulative memory complexity of sorting +is Ω(n3/T), matching the ST complexity bounds given in [KŠdW07, HM21]. This involves the +quantum circuit model which, as we have noted, produces each output position at a predetermined +input-independent layer. We restrict our attention to circuits that output all elements in the input +according to their sorted order with a constant total success probability. While our proof is inspired +by the time-space lower bound of [KŠdW07], it can be easily adapted to follow the proof in [HM21] +instead. +Definition 4.1. In the k-threshold problem we receive an input X = x1, . . . , xn where xi ∈ {0, 1}. We +want to accept iff there are at least k distinct values for i where xi = 1. +We say that a quantum circuit C that computes a boolean function f : {0, 1}n → {0, 1} has +completeness a and soundness b on inputs in domain D iff for all x ∈ D, Pr[C(x) = 1] ≥ a when +f (x) = 1 and Pr[C(x) = 1] ≤ b when f (x) = 0. We say that a circuit has perfect completeness +(soundness) iff a = 1 (respectively, b = 0). +Proposition 4.2 (Theorem 13 in [KŠdW07]). For every γ > 0 there is an α > 0 such that any quantum +k-threshold circuit with at most T ≤ α +√ +kn queries and with perfect soundness must have completeness +σ ≤ e−γk on inputs with hamming weight k. +Using the above theorem, we present a generalization of a lemma first proven in [KŠdW07]. +Lemma 4.3. Choose any constant γ > 0. Let n be sufficiently large and C(X) be a quantum circuit with +input X = x1, . . . , xn. There exists a constant β that depends only on γ such that for all k ≤ β2n and +9 + +R ⊆ {n/2 + 1, . . . , n} where |R| = k, if C(X) makes at most β +√ +kn queries, then the probability that C(X) +can correctly output all k pairs (xi, rj) where rj ∈ R and xi is the rj’th smallest element of X is at most +e(1−γ)k−1. If R is a contiguous set of integers, then the probability is at most e−γk. +The version of Lemma 4.3 proved in [KŠdW07] had the additional assumption that the set of +output ranks R is a contiguous set of integers; this was sufficient to show that any quantum circuit +that produces its sorted output in sorted time order requires that T2S is Ω(n3). The authors stated +that their proof can be generalized to any fixed rank ordering, but the generalization is not obvious. +We generalize their lemma to non-contiguous R, which is sufficient to obtain an Ω(n3/T) lower +bound on the cumulative complexity of sorting independent of the time order in which the sorted +output is produced. +Proof of Lemma 4.3. Choose α as the constant for γ in Proposition 4.2 and let β = +√ +2α/6. Let C be +a circuit with at most β +√ +kn layers that outputs the k correct pairs (xi, rj) with probability p. Let +R = {r1, . . . rk} where r1 < r2 < . . . < rk. We describe our construction of a circuit C′(X) solving +the k-threshold problem on inputs X = x1, . . . , xn/2 with exactly k ones in terms of a function +f : [n/2] → R. Given f, we re-interpret the input as follows: we replace each xi with x′ +i = f (i)xi, +add k dummy values of 0, and add one dummy value of j for each j ∈ {n/2 + 1, . . . , n} \ R. Doing +this gives us an input X′ = x′ +1, . . . , x′ +n that has n/2 zeroes. If we assume that f is 1-1 on the k ones +of X, then the image of the ones of X will be R and there will be precisely one element of X′ for +each j ∈ {n/2 + 1, . . . , n}. Therefore the element of rank j > n/2 in X′ will have value j, and hence +the rank r1, . . . , rk elements of X′ will be the images of precisely those elements of X with xi = 1. +To obtain perfect soundness, we cannot rely on the output of C(X′) and must be able to check +that each of the output ranks was truly mapped to by a distinct one of X. For each element xi of +X we simply append its index i as log2 n low order bits to its image x′ +i and append an all-zero +bit-vector of length log2 n to each dummy value to obtain input X′′. Doing so will not change the +ranks of the elements in X′, but will allow recovery of the k indices that should be the ones in X. +In particular, circuit C′(X) will run C(X′′) and then for each output x′′ +j with low order bits i, C′(X) +will query xi, accepting if and only if all of those xi = 1. More precisely, since the mapping from +each xi to the corresponding x′′ +i is only a function of f, xi, and i, as long as C′(X) has an explicit +representation of f, it can simulate each query of C(X′′) with two oracle queries to X (see Section 2 +for details). Since C′ has at most +2β +√ +kn + k ≤ 3β +√ +kn ≤ α +√ +kn/2 +layers, by Proposition 4.2, it can only accept with probability at most e−γk when the input has k +ones. +We now observe that for each fixed X with exactly k ones, for a randomly chosen function +f : [n/2] → R, the probability that f is 1-1 on the ones of X′ is exactly k!/kk ≥ e1−k. Therefore C′(X) +will give the indices of the k ones in X with probability3 at least p · e1−k. However, this probability +must be at most e−γk, so we can conclude that p ≤ e(1−γ)k−1. In the event that R is a contiguous +set of integers, observe that any choice for the function f will make X′′ have the ones of X become +ranks r1, . . . , rk. So the probability of finding the ones is at least p ≤ e−γk. +3Note that though this is exponentially small in k it is still sufficiently large compared to the completeness required in +the lower bound for the k-threshold problem. +10 + +By setting k and γ appropriately, Lemma 4.3 gives a useful upper bound on the number of +fixed ranks successfully output by any β +√ +Sn query quantum circuit that has access to S qubits of +input dependent initial state. To handle input-dependent initial state, we will need the following +proposition. +Proposition 4.4 ([Aar05]). Let C be a quantum circuit, ρ be any S qubit (possibly mixed) state, and I be +the S qubit maximally mixed state. If C with initial state ρ produces some output O with probability p, then +C with initial state I produces O with probability at least p/22S. +This allows us to bound the overall progress made by any short quantum circuit. +Lemma 4.5. Let γ = 1 + ln(4) and β be the constant from Lemma 4.3 that depending on γ. Then for any +fixed set of S ≤ β2n ranks that are greater than n/2, the probability that any quantum circuit C with at most +β +√ +Sn queries and S qubits of input-dependent initial state correctly produces the outputs for these S ranks +is at most 1/e. +Proof. Applying Proposition 4.4 to the bound in Lemma 4.3 gives us that a quantum circuit with S +qubits of input-dependent state can produce a fixed set of k ≤ β2n outputs larger than median with +a probability at most 22Se(1−γ)k−1. Since γ = 1 + ln(4) setting k = S yields a probability bound on +of at most 1/e on the event in question. +Theorem 4.6. When n is sufficiently large, any quantum circuit C for sorting a list of length n with success +probability at least 1/e and at most T layers that produces its sorted outputs in any fixed time order requires +cumulative memory that is Ω(n3/T). +Proof. We partition C into blocks with large cumulative memory that can only produce a small +number of outputs. We achieve this by starting at last unpartitioned layer and finding a suitably +low space layer before it so that we can apply Lemma 4.5 to upper bound the number of correct +outputs that can be produced in that block with a success probability of at least 1/e. Let β be +the constant from Lemma 4.5 and k∗(t) be the least non-negative integer value of k such that the +interval: +I(k, t) = +� +t − β +2 (2k+1 − 1)√ +n, t − β +2 (2k − 1)√ +n +� +contains some t′ such that St′ ≤ 4k − 1. We recursively define our blocks as follows. Let ℓ be the +number of blocks generated by this method. The final block Cℓ starts with the first layer tℓ−1 ∈ +I(k∗(T), T) where Stℓ−1 ≤ 4k∗(T) − 1 and ends with layer tℓ = T. Let ti be the first layer of block Ci+1. +Then the block Ci starts with the first layer ti−1 ∈ I(k∗(ti), ti) where Sti−1 ≤ 4k∗(ti) − 1 and ends with +ti. See Figure 4 for an illustration of our partitioning. Since S0 = 0 we know that k∗(t) ≤ log(T). +Likewise since St > 0 when t > 0, for all t > β +2 +√n we know that 0 < k∗(t) ≤ log(T). +By construction, block Ci starts with less than 4k∗(ti) qubits of initial state and has length at most +β2k∗(ti)√n; so by Lemma 4.5, if 4k∗(ti) ≤ β2n, the block Ci can output at most 4k∗(ti) inputs with +failure probability at most 1/e. Additionally Ci has at least β +22k∗(ti)−1√n layers that each have at +least 4k∗(ti)−1 qubits,4 so the cumulative memory of Ci is at least β +223k∗(ti)−3√n. +4This may not hold for C1 with length less than β +2 +√ +N, but Lemma 4.3 with Appendix C give us that this number of +layers is insufficient to find a fixed rank input with probability at least 1/e. Thus we can omit such a block from our +analysis. +11 + +Figure 4: How we define the block Ci that ends at layer Lti. The grey layers are the ones used to +lower bound the cumulative memory complexity of Ci, as each of these layers uses at least 4k∗(ti)−1 +qubits and the length of this interval is β +22k∗(ti)−1√n. +We now have two possibilities. If we have some i such that 4k∗(ti) > β2n, the cumulative memory +of Ci alone is at least β4n2/16 which is Ω(n2) and hence C has cumulatively memory Ω(n3/T) since +T ≥ n. Otherwise, since we require that the algorithm is correct with probability at least 1/e, each +block Ci can produce at most 4k∗(ti) outputs. Since our circuit must output all n/2 elements larger +than the median, we know ∑ℓ +i=1 4k∗(ti) ≥ n/2. For convenience we define wi = 2k∗(ti) and get the +following bound on the sum of the w2 +i : +ℓ +∑ +i=1 +w2 +i ≥ n/2 +We obtain the following lower bound on the cumulative memory: +CM(C) ≥ +ℓ +∑ +i=1 +β +2 23k∗(ti)−3√ +n = β +16 +√ +n +ℓ +∑ +i=1 +w3 +i +(2) +To lower bound the cumulative complexity, this gives us the non-convex optimization problem in +Figure 5. In Appendix A we prove that if ∑ xi ≤ ∑ x2 +i , then ∑ x2 +i ≤ ∑ x3 +i . This gives us that: +ℓ +∑ +i=1 +x3 +i ≥ ξ +Reversing the variable substitution gives us: +ℓ +∑ +i=1 +w3 +i ≥ βn5/2 +16T +Then applying Equation (2) gives us the bound: +CM(C) ≥ β2n3 +256T +12 + +≤ 4*(t) - 1(a) +CM(C) ≥ min β +16 +√ +n +ℓ +∑ +i=1 +w3 +i +s.t. +ℓ +∑ +i=1 +w2 +i ≥ n/2 +β +4 +√ +n +ℓ +∑ +i=1 +wi ≤ T +(b) +min +ℓ +∑ +i=1 +x3 +i +s.t. +ℓ +∑ +i=1 +x2 +i ≥ ξ +ℓ +∑ +i=1 +xi ≤ ξ +Figure 5: The non-convex optimization problem that bounds the cumulative memory for quantum +sorting. The objective function of system (a) is a lower bound on the cumulative memory complexity +and system (b) is the same system after scaling the objective function and applying the variable +substitutions wi = βn3/2 +8T xi and ξ = 32T2 +β2n2 . +And therefore the cumulative memory of C is Ω(n3/T). +To extend our results to arbitrary success probability at most 1 − δ, it is important to know how +α and γ are related in Proposition 4.2. In Appendix C we show that we can have α that is Ω(e−γ/2) +and get a probability of at most e−γk. Thus for any S, k, and δ ∈ (0, 1), we can choose +γ = ln(22S/(1 − δ)) − 1 +k ++ 1 +to get a probability of at most 1 − δ for circuits with +Ω +�� +22S +e(1 − δ) +�1/2k √ +kn +� +layers and S qubits of advice to produce k outputs. When S = k and δ = 1 − 1/e, this is exactly the +bound from Lemma 4.5. If we repeat the proof of Theorem 4.6 for failure probability at most δ, we +can set β to a value that is Ω(1/ +√ +1 − δ) to obtain a lower bound on the cumulative memory that is +Ω(n3/((1 − δ)T)). +5 +A general method for proving cumulative memory complexity lower +bounds +Our method involves adapting techniques previously used to prove tradeoff lower bounds on +worst-case time and worst-case space. We show that the same properties that yield lower bounds +on the product of time and space in the worst case can also be used to produce nearly identical +lower bounds on cumulative memory. To do so, we first revisit the standard approach to such +time-space tradeoff lower bounds. +13 + +The standard method for time-space tradeoff lower bounds for multi-output functions +Consider a multi-output function f on Dn where the output f (x) is either unordered (the output +is simply a set of elements from R) or ordered (the output is a vector of elements from R). Then +| f (x)| is either the size of the set or the length of the vector of elements. The standard method +for obtaining an ordinary time-space tradeoff lower bounds for multi-output functions on D-way +branching programs is the following: +The part that depends on f: +Choose a suitable probability distribution µ on Dn, often simply the +uniform distribution on Dn and then: +(A) Prove that Prx∼µ[| f (x)| ≥ m] ≥ α. +(B) Prove that for all k ≤ m′ and any branching program B of height ≤ h′(k, n), the probability for +x ∼ µ that B produces at least k correct output values of f on input x is at most C · |R|−k/r(n) +for some m′, h′, r and constant C independent of n. +Observe that under any distribution µ, a branching program with ordered outputs that makes no +queries can produce k outputs that are all correct with probability at least |R|−k, so the bound in +(B) shows that, roughly, up to the power 1/r(n) there is not much gained by using a branching +program of height h. +The generic completion: +In the following outline we omit integer rounding for readability. +• Suppose that +S ≤ log2 |R| +r(n) +· m′ − log2(2C/α). +(3) +• Let k = [S + log2(2C/α)] · r(n)/ log2 |R|, which is at most m′ by hypothesis on S, and define +h(S, n) = h′(k, n). +• Divide time T into ℓ = T/h blocks of length h = h(S, n). +• The original branching program can be split into at most 2S sub-branching programs of height +≤ h, each beginning at a boundary node between layers. By property (B) and a union bound, +for x ∼ µ the probability that at least one of these ≤ 2S sub-branching programs of height at +most h produces k correct outputs on input x is at most +2S · C · |R|−k/r(n) ≤ α/2 +by our choice of k. +• Under distribution µ, by (A), with probability at least α, an input x ∼ µ has some block of +time during which at least m/ℓ = m · h(S, n)/T outputs of f must be produced on input x. +• If m · h(S, n)/T ≤ k, this can occur for at most an α/2 fraction of inputs under µ. Therefore +we have +m · h(S, n)/T > k = [S + log2(2C/α)] · r(n)/ log2 |R| +14 + +and hence, combining with Equation (3), we have +T · S ≥ min +� +m · h(S, n), m′ · n′� · log2 |R| +r(n) +− log2(C/α) · T +where n′ ≤ n is the decision tree complexity of f and hence a lower bound on T. +Remark 5.1. Though it will not impact our argument, for many instances of the above outline, the +proof of property (B) is shown for a decision tree of the same height by proving an analog for the +conditional probability along each path in the decision tree separately; this will apply to the tree as +a whole since the paths are followed by disjoint inputs, so property (B) follows from the alternative +property below: +(B’) For any partial assignment τ of k ≤ m′ output values over R and any restriction (i.e., partial +assignment) π of h′(k, n) coordinates within Dn, +Pr +x∼µ[ f (x) is consistent with τ | x is consistent with π] ≤ C · |R|−k/r(n). +Remark 5.2. The above method still gives lower bounds for many multi-output functions g : DN → +RM that have individual output values that are easy to compute or large portions of the input space +on which they are easy to compute. The bounds follow by applying the method to some subfunction +f of g given by f (x) = ΠO(g(x, π)) where π is a partial assignment to the input coordinates and +ΠO is a projection onto a subset O of output coordinates. In the subsequent discussions we ignore +this issue, but the idea can be applied to all of our lower bound methods. +A general extension to cumulative memory bounds +To give a feel for the basic ideas of the method, we first show this for a simple case. Observe that, +other than the separate bound on time, the lower bound on cumulative memory usage we prove in +this case is asymptotically identical to the bound achieved for the product of time and worst-case +space using the standard outline. +Theorem 5.3. Let c > 0. Suppose that properties (A) and (B) apply for h′(k, n) = h(n), m′ = m, and +α = C = 1. If +T log2 T ≤ m · h(n) · log2 |R| +6(c + 1)r(n) +then the cumulative memory used in computing f : Dn → Rm in time T with success probability at least +T−c is at least +m · h(n) · log2 |R| +6r(n) +. +Proof. Fix a deterministic branching program P of length T computing f. Rather than choosing +fixed blocks of height h = h(n), layers of nodes at a fixed distance from each other, and a fixed +target of k outputs per block, we choose the block boundaries depending on the properties of P +and the target k depending on the property of the boundary layer chosen. +15 + +Let H = ⌊h(n)/2⌋. We break P into ℓ = ⌈T/H⌉ time segments of length H working backwards +from step T so that the first segment may be shorter than the rest. We let t1 = 0 and for 1 < i ≤ ℓ +we let +ti = arg min{ |Lt| : T − (ℓ − i + 1) · H ≤ t < T − (ℓ − i) · H } +be the time step with the fewest nodes among all time steps t ∈ [T − (ℓ − i + 1) · H, T − (ℓ − i) · H]. +The i-th time block of P will be between times ti and ti+1. Observe that by construction +|ti+1 − ti| ≤ h(n) so each block has length at most h(n). Set Si = log2 |Lti| so that Lti has at 2Si +nodes. By definition of each ti, the cumulative memory used by P, +CM(P) ≥ +ℓ +∑ +i=1 +Si · H. +(4) +(Note that since S1 = 0, it does not matter that the first segment is shorter than the rest5.) +We now define the target ki for the number of output values produced in each time block to be +the smallest integer such that |R|−ki/r(n) ≤ 2−Si/Tc+1. That is, +ki = ⌈r(n) · (Si + (c + 1) log2 T)/ log2 |R|⌉. +For x ∼ µ, for each i ∈ [ℓ] and each sub-branching program B rooted at some node in Lti and +extending until time ti+1, by our choice of ki and property (B), if ki ≤ m, the probability that B +produces at least ki correct outputs on input x is at most 2−Si/Tc+1. Therefore, by a union bound, +for x ∼ µ the probability that P produces at least ki correct outputs in the i-th time block on input x +is at most +|Lti| · 2−Si/Tc+1 = 1/Tc+1. +Therefore, if each ki ≤ m, the probability for x ∼ µ that there is some i such that P produces at least +ki correct outputs on input x during the i-th block is at most ℓ/Tc+1 < Tc. Therefore, if each ki ≤ m, +the probability for x ∼ µ that P produces at most ∑ℓ +i=1(ki − 1) correct outputs in total on input x is +> 1 − 1/Tc. +If each ki ≤ m, since P must produce m correct outputs on x ∈ Dn with probability at least +1/Tc, we must have ∑ℓ +i=1(ki − 1) ≥ m. On the other hand, if some ki > m we have the same bound. +Using our definition of ki we have +ℓ +∑ +i=1 +[r(n) · (Si + (c + 1) log2 T)]/ log2 |R|)] ≥ m +or +ℓ +∑ +i=1 +(Si + (c + 1) log2 T) ≥ m · log2 |R| +r(n) +. +In particular, plugging in the bound (4) on the cumulative memory and the value of ℓ, it implies +that +CM(P)/H + (c + 1)⌈T/H⌉ · log2 T ≥ m · log2 |R| +r(n) +5This simplifies some calculations and is the prime reason for starting the time segment boundaries at T rather than +at 0. +16 + +or that +CM(P) + (c + 1)T log2 T ≥ m · h(n) · log2 |R| +3 · r(n) +, +where the 3 on the right rather than a 2 allows us to remove the integer rounding. Therefore either +T log2 T > m · h(n) · log2 |R| +6(c + 1) · r(n) +or +CM(P) ≥ m · h(n) · log2 |R| +6r(n) +, +which is what we wanted to show. +In the general version of our theorem there are a number of additional complications, most +especially because the branching program height limit h(k, n) in property (B) can depend on k, the +target for the number of outputs produced. This forces the lengths of the blocks and the space +used at the boundaries between blocks to depend on each other in a quite delicate way. In order +to discuss the impact of that dependence and state our general theorem, we need the following +definition. +Definition 5.4. Given a non-decreasing function p : R → R with p(1) = 1, we define p−1 : R → +R ∪ {∞} by p−1(R) = min{j | p(j) ≥ k}. We also define the loss, Lp, of p by +Lp(n) = +min +1≤k≤p(n) +∑k +j=1 p−1(j) +k · p−1(k) . +Lemma 5.5. The following hold for every non-decreasing function p : R → R with p(1) = 1: +(a) 1/p(n) ≤ Lp(n) ≤ 1. +(b) If p is a polynomial function p(s) = s1/c then Lp(n) > 1/2c+1. +(c) For any c > 1, Lp(n) ≥ min +1≤s≤n +p(s) − p(s/c) +cp(s) +. +(d) We say that p is nice if it is differentiable and there is an integer c > 1 such that for all x, p′(cx) ≥ +p′(x)/c. If p is nice then Lp(n) is Ω(1/ log2 n). This is tight for p with p(s) = 1 + log2 s. +We prove these technical statements in Appendix B. The following is our full general theorem. +Theorem 5.6. Let c > 0. Suppose that function f defined on Dn has properties (A) and (B) with α that +is 1/nO(1) and m′ that is ω(log2 n). For s > 0, define h(s, n) to be h′(k, n) for k = s · r(n)/ log2 |R|. +Suppose that h(s, n) = h0(s) h1(n) with h0(1) = 1 and h0 a differentiable function such that s/h0(s) is +increasing and concave. Define S∗ = S∗(T, n) by +S∗ +h0(S∗) = m · h1(n) · log2 |R| +6r(n)T +. +17 + +(a) Either +T log2(2CTc+1/α) > m · h1(n) · log2 |R| +6r(n) +, +which implies that T is Ω( m·h1(n)·log |R| +r(n) log n +), or the cumulative memory used by a randomized branching +program in computing f in time T with error ε ≤ α(1 − 1/(2Tc)) is at least +Lh0(n log2 |D|) · min +� +m · h(S∗(T, n), n), 3m′ · h′(m′/2, n) +� · log2 |R| +6r(n) . +(b) Further any randomized random-access machine computing f in time T with error ε ≤ α(1 − +1/(2Tc)) requires cumulative memory +Ω +� +Lh0(n log2 |D|) · min +� +m · h(S∗(T, n), n), m′ · h′(m′/2, n) +� · log2 |R| +r(n) +� +. +Before we give the proof of the theorem, we note that by Lemma 5.5, in the case that h0 is +constant or a polynomial function of its input, which together account for all existing applications +we are aware of, the function Lh0 is lower bounded by a constant. Further, the value S∗ in the +statement of this theorem is at least a constant factor times the value of S used in the generic +time-space tradeoff lower bound methodology. Therefore, for example, the cumulative memory +lower bound derived for random-access machines via Theorem 5.6 is close to the lower bound on +the product of time and worst-case space given by standard methods. +Proof of Theorem 5.6. We prove both (a) and (b) directly for branching programs, which can model +random-access machines, and will describe the small variation that occurs in the case that the +branching program in question comes from a random-access machine. To prove these properties +for randomized branching programs, by Yao’s Lemma [Yao77] it suffices to prove the properties +for deterministic branching programs that have error at most ε under distribution µ. Fix a (deter- +ministic) branching program P of length T computing f with error at most ε under distribution µ. +Without loss of generality, P has maximum space usage at most Smax = n log2 |D| space since there +are at most |Dn| inputs. +Let H = ⌊h1(n)/2⌋. We break P into ℓ = ⌈T/H⌉ time segments of length H working backwards +from step T so that the first segment may be shorter than the rest. We then choose a sequence +of candidates for the time steps in which to begin new blocks, as follows: We let τ1 = 0 and for +1 < i ≤ ℓ we let +τi = arg min{ |Lt| : T − (ℓ − i + 1) · H ≤ t < T − (ℓ − i) · H } +be the time step with the fewest nodes among all time steps t ∈ [T − (ℓ − i + 1) · H, T − (ℓ − i) · H]. +Set σi = log2 |Lτi| so that Lτi has at 2σi nodes. This segment contributes at least σi · H to the +cumulative memory bound of P. +To choose the beginning ti∗ of the last time block6. we find the smallest k such that h0(σℓ−k+1) < +k. Such a k must exist since h0 is a non-decreasing non-negative function, h0(1) = 1 and σ1 = 0 < 1. +6Since we are working backwards from the end of the branching program and we do not know how many segments +are included in each block, we don’t actually know this index until things stop with t1 = 0 +18 + +We now observe that the length of the last block is at most k · H which by choice of k is less than +h(σℓ−k+1, n) and hence we have satisfied the requirements for property (B) to apply at each starting +node of the last time block. +By our choice of each τi, the total cumulative memory used in the last k segments is at least +k +∑ +j=1 +σℓ+1−j · H. +Further, since k was chosen as smallest with the above property, we know that for every j ∈ [k − 1] +we have +h0(σℓ−j+1) ≥ j +Hence we have σℓ−j+1 ≥ h−1 +0 (j) and we get a cumulative memory bound for the last k segments of +at least +(σℓ−k+1 + +k−1 +∑ +j=1 +h−1 +0 (j)) · H. +(5) +CLAIM: σℓ−k+1 + ∑k−1 +j=1 h−1 +0 (j) ≥ Lh0(Smax) · σℓ−k+1 · k. +Proof of Claim. Observe that it suffices to prove the claim when we replace σℓ−k+1, which appears +on both sides, by a larger quantity. In particular, we show how to prove the claim with h−1 +0 (k) +instead, which is larger since h0(σℓ−k+1) < k. But this follows immediately since by definition +Lh0(Smax) ≤ ∑k +j=1 h−1 +0 (j) +k · h−1 +0 (k) +, +which is equivalent to what we want to prove. +Write Si∗ = σℓ−k+1. By the claim, the cumulative memory contribution associated with the last +block beginning at ti∗ is at least +Lh0(Smax) · Si∗ · h0(Si∗)H. +We repeat this in turn to find the time step for the beginning of the next block from the end, +ti∗−1. One small difference now is that there is a last partial segment of height at most H from +the beginning of segment containing ti∗ to layer ti∗. However, this only adds at most h1(n)/2 +to the length of the segment which still remains well within the height bound of h(Si∗−1, n) = +h0(Si∗−1)h1(n) for property (B) to apply. +Repeating this back to the beginning of the branching program we obtain a decomposition +of the branching program into some number i∗ of blocks, the i-th block beginning at time step +ti with 2Si nodes, height between h0(Si)H and h0(Si)H + H ≤ 2h0(Si)H, and with an associated +cumulative memory contribution in the i-th block of at least +Lh0(Smax) · Si · h0(Si)H. +19 + +(This is correct even for the partial block starting at time t1 = 0 since S1 = 0.) Since we know that +i∗ ≤ ℓ, for convenience, we also define Si = 0 for i∗ + 1 ≤ i ≤ ℓ. Then, by definition we have +CM(P) ≥ Lh0(Smax) · +� +i∗ +∑ +i=1 +Si · h0(Si) +� +· H = Lh0(Smax) · +� +ℓ +∑ +i=1 +Si · h0(Si) +� +. +(6) +and +ℓ +∑ +i=1 +h0(Si) ≤ T/H. +(7) +As in the previous argument for the simple case, for i ≤ i∗, we define the target ki for the number +of output values produced in each time block to be the smallest integer such that C˙|R|−ki/r(n) ≤ +2−Siα/(2Tc+1). That is, +ki = ⌈r(n) · (Si + log2(2CTc+1/α))/ log2 |R|⌉. +If ki > m′ for some i, then Si ≥ m′ · log2 |R|/r(n) − log2(2CTc+1/α) ≥ m′ · log2 |R|/(2r(n)) +since m′ is ω(log n) and 1/α and T are nO(1). Therefore h0(Si) ≥ h′(m′/2, n) and hence +CM(P) ≥ Lh0(Smax) · m′ · h′(m′/2, n) · log2 |R| +2r(n) +Suppose instead that ki ≤ m′ for all i ≤ i∗. Then, for x ∼ µ, for each i ∈ [i∗] and each sub- +branching program B rooted at some node in Lti and extending until time ti+1, by our choice of ki +and property (B), the probability that B produces at least ki correct outputs on input x is at most +α · 2−Si/(2Tc+1). +Therefore, by a union bound, for x ∼ µ the probability that P produces at least ki correct outputs in +the i-th time block on input x is at most +|Lti| · α · 2−Si/(2Tc+1) = α/(2Tc+1) +and hence the probability for x ∼ µ that there is some i such that P produces at least ki correct +outputs on input x during the i-th block is at most ℓ · α/(2Tc+1) < α/(2Tc). Therefore, the +probability for x ∼ µ that P produces at most ∑ℓ +i=1(ki − 1) correct outputs in total on input x is +> 1 − α/(2Tc). +Since, by property (A) and the maximum error it allows, P must produce at least m correct +outputs with probability at least α − ϵ ≥ α − α(1 − 1/(2Tc)) = α/(2Tc) for x ∼ µ, we must have +∑i∗ +i=1(ki − 1) ≥ m. Using our definition of ki we have +i∗ +∑ +i=1 +[r(n) · (Si + log2(2CTc+1/α))]/ log2 |R|)] ≥ m +or +i∗ +∑ +i=1 +(Si + log2(2CTc+1/α)) ≥ m · log2 |R| +r(n) +. +20 + +This is the one place in the proof where there is a distinction between an arbitrary branching +program and one that comes from a random access machine. +We first start with the case of arbitrary branching programs: Note that i∗ ≤ ℓ = ⌈T/H⌉ = +⌈T/⌊h1(n)/2⌋⌉. Suppose that T log2(2CTc+1/α) ≤ m·h1(n)·log2 |R| +6r(n) +. Then, even with rounding, we +obtain ∑i∗ +i=1 Si ≥ m·log2 |R| +2r(n) +. +Unlike an arbitrary branching program that may do non-trivial computation with sub- +logarithmic Si, a random-access machine with even one register requires at least log2 n bits of +memory (just to index the input for example) and hence Si + log2(2CTc+1/α) will be O(Si), since +T is at most polynomial in n without loss of generality and 1/α is at most polynomial in n by +assumption. Therefore we obtain that ∑i∗ +i=1 Si is Ω( m·log2 |R| +r(n) +) without the assumption on T. +In the remainder we continue the argument for the case of arbitrary branching programs and +track the constants involved. The same argument obviously applies for programs coming from +random-access machines with slightly different constants that we will not track. In particular, since +Si = 0 for i > i∗ we have +ℓ +∑ +i=1 +Si ≥ m · log2 |R| +2r(n) +. +(8) +From this point we need to do something different from the argument in the simple case because +the lower bound on the total cumulative memory contribution is given by Equation (6) and is not +simply ∑ℓ +i=1 Si · H. Instead, we combine Equation (8) and Equation (7) using the following technical +lemma that we prove in Appendix A. +Lemma 5.7. Let p : R≥0 → R≥0 be a differentiable function such that q(x) = x/p(x) is a concave +increasing function of x. For x1, x2, . . . ∈ R≥0, if ∑i xi ≥ K and ∑i p(xi) ≤ L then ∑i xip(xi) ≥ +q−1(K/L) · L. +In our application of the lemma p = h0, K = m·log2 |R| +2r(n) +, and L = T/H. Let S∗ be the solution to +S∗ +h0(S∗) = K/L = m · H · log2 |R| +2r(n)T +≥ m · h1(n) log2 |R| +6r(n)T +. +Then Lemma 5.7 implies that +ℓ +∑ +i=1 +Si · h0(Si) ≥ S∗ · T/H = m · h0(S∗) · log2 |R| +2r(n) +. +and hence +CM(P) ≥ Lh0(Smax) · m · h0(S∗) · H · log2 |R| +2r(n) +≥ Lh0(Smax) · m · h(S∗, n) · log2 |R| +6 · r(n) +since H = ⌊h1(n)/2⌋ and h(S∗, n) = h0(S∗) · h1(n). +By Lemma 5.5, in the case that h0 is a polynomial function of its input, the function Lh0 is +lower bounded by a constant and the bound in Theorem 5.6 only loses a constant factor in moving +from the product of worst-case space and time to cumulative memory complexity. In that special +case (and indeed for any nice function h0), there is an alternative variant of the above in which +21 + +one breaks up time into exponentially growing segments starting with time step T. We use that +alternative approach to obtain lower bounds on the cumulative memory complexity of sorting by +quantum algorithms in Section 4. +Remark 5.8. If we restrict our attention to o( m′ log |R| +r(n) +)-space bounded computation, then each ki ≤ m′ +and the cumulative memory bound for a branching program in Theorem 5.6 becomes +Lh0(n log2 |D|) · m · h(S∗(T, n), n) · log2 |R| +6r(n) . +And the bound for RAM cumulative memory becomes +Ω +� +Lh0(n log2 |D|) · m · h(S∗(T, n), n) · log2 |R| +r(n) +� +. +6 +Sample applications to cumulative complexity of classical algorithms +Theorems 5.3 and 5.6 are powerful tools that can convert most existing time-space lower bounds +into asymptotically equivalent lower bounds on the required cumulative memory. We give a few +examples to indicate how our general theorems can be used. +Unique elements +Define Uniquen,N : [N]n → P([N]) by Uniquen,N(x) = { xi | xj ̸= xi for all j ̸= i }. +Proposition 6.1 (Lemmas 2 and 3 in [Bea91]). For the uniform distribution µ on [N]n with N ≥ n, +(A) Prx∼µ[|Uniquen,N(x)| ≥ n/(2e)] ≥ 1/(2e − 1) +(B’) For any partial assignment τ of k ≤ n/4 output values over [N] and any restriction π of n/4 +coordinates in [n]n, Prx∼µ[Uniquen,N(x) is consistent with τ | x is consistent with π] ≤ e−k/2. +The above lemma is sufficient to prove that TS is Ω(n2) for the unique elements problem, and +can be easily extended to a cumulative complexity bound using Theorem 5.6. +Theorem 6.2. For n ≥ N, any branching program computing Uniquen,N in time T and probability at least +4/5 requires T to be Ω(n2/ log n) or CM(P) to be Ω(n2). Further, any random access machine computing +Uniquen,N with probability at least 4/5 requires cumulative memory Ω(n2) +Proof. By Proposition 6.1, Uniquen,N satisfies conditions (A) and (B) of Section 5 with h′(k, n) = n/4, +m′ = n/4, m = n/(2e), C = 1, r(n) = 2 ln N and α = 1/(2e − 1) ≥ 0.2254. Since h′(k, n) is +independent of k, the function h0 defined in Theorem 5.6 is the constant function 1 and h1(n) = n/4 +so Lh0 ≡ 1. We then apply Theorem 5.6 to obtain the claimed lower bounds. +The above theorem is tight for N = n using the algorithm in [Bea91]. +22 + +Linear Algebra +We consider linear algebra over some finite field F. Let D be a subset of F with d elements. +Definition 6.3. An m × n matrix is (g, h, c)-rigid iff every k × w submatrix where k ≤ g and +w ≥ n − h has rank at least ck. We call (g, h, 1)-rigid matrices (g, h)-rigid. +Matrix rigidity is a robust notion of rank and is an important property for proving time-space +and cumulative complexity lower bounds for linear algebra. Fortunately, Abrahamson proved that +there are always rigid square matrices. +Proposition 6.4 (Lemma 4.3 in [Abr91]). There is a constant γ ∈ (0, 1 +2) where at least a 1 − d−1(2/3)γn +fraction of the matrices over Dn×n are (γn, γn)-rigid. +Abrahamson shows in [Abr91] that for any constant c ∈ (0, 1 +2) and m × n matrix A that is +(cm, cn, c)-rigid, any D-way branching program that computes the function f (x) = Ax with +expected time T ≥ n and expected space7 S has TS = Ω(nm log d) where d = |D|. We restate the +key property used in that proof. +Proposition 6.5 (Theorem 4.6 in [Abr91]). Let c ∈ (0, 1 +2], A be any m × n matrix that is (g, h, c)-rigid +and f be the function f (x) = Ax over F. Let µ be the uniform distribution on Dn for D ⊆ F with +|D| = d. For any restriction π of h coordinates to values in D and any partial assignment τ of k ≤ g output +coordinates over Fm, +Pr +x∼µ[ f (x) is consistent with τ | x is consistent with π] ≤ d−ck +Theorem 6.6. Let c ∈ (0, 1 +2]. Let A be an m × n matrix over D, with |D| = d that is (g(m), h(n), c)-rigid. +Then, for any D-way branching program P computing f (x) = Ax in T steps with probability at least +n−O(1), either T is Ω(g(m)h(n) logn d) or CM(P) is Ω(g(m)h(n) log d). Further, computing f on a +random access machine requires cumulative memory Ω(g(m)h(n) log d) unconditionally. +Proof. We invoke Theorem 5.3 using Proposition 6.5 to obtain condition (B’). Condition (A) is trivial +since | f (x)| = m. +By Proposition 6.4 we know that for some constant γ, a random matrix has a good chance of +being (γm, γn)-rigid. This means that computing f (x) = Ax for a random matrix A in time at +most T is likely to require either the cumulative memory or T log T to be Ω(mn log d). Since Yesha +[Yes84] proved that the n × n DFT matrix is (n/4, n/4, 1/2)-rigid, the DFT is a concrete example +where the cumulative memory or T log T is Ω(n2 log d); other examples include generalized Fourier +transform matrices over finite fields [BJS01, Lemma 28]. +Corollary 6.7. If A is an n × n generalized Fourier transform matrix over field F with characteristic +relatively prime to n then any random-access machine computing f (x) = Ax for x ∈ Dn where D ⊆ F has +|D| = d with probability at least n−O(1) requires cumulative memory that is Ω(n2 log d). +It is easy to see that our lower bound is asymptotically optimal in these cases. +7[Abr91] defines expected space as the expected value of the log2 of the largest number of a branching program node +that is visited during a computation under best case node numbering. +23 + +Proposition 6.8 (Theorem 7.1 in [Abr91]). Let f : D2n2 → Fn2 for D ⊆ F and d = |D| be the matrix +multiplication function, γ be the constant from Proposition 6.4, and µ be the uniform distribution over +(γm, γn)-rigid matrices. Choose any integers h and k such that 2(h/γn)2 ≤ k. If γn ≥ 1 then for any +D-way branching program B of height ≤ h the probability that B produces at least k correct output values of +f is at most d2−γk/4. +Theorem 6.9. Multiplying two random matrices in Dn2 with D ⊆ F and d = |D| with probability at +least n−O(1) requires time T that is Ω((n3� +log d)/ log n) or cumulative memory Ω((n6 log d)/T). On +random access machines, the cumulative memory bound is unconditional. +Proof. Proposition 6.8 lets us apply Theorem 5.6 with m = n2, h′(k, n) = γn√ +k/2, C = d2, α = 1, +|R| = |F|, and r(n) = (4 logd |F|)/γ. This gives us that h(s, n) = n +� +2γs/ log2 d, so h0(s) = √s. +Then we get that +√ +S∗ = +mn√ +2γ/ log2 d·log2 |F| +6r(n)T +and hence +S∗ is Ω +�n6 log d +T2 +� +. +Therefore we get that either +T is Ω +� +n3 log1/2 d +log n +� +or, since the loss function for h0 is a constant, the cumulative memory is +Ω +� +min +� +(n6 log d)/T, n5 log1/2 d +�� +. +Since the decision tree complexity of matrix multiplication is Ω(n2), this is Ω((n6 log d)/T). For +random access machines, the same cumulative memory bound applies without the condition on +T. +7 +Cumulative memory complexity of single-output functions +The time-space tradeoff lower bounds known for classical algorithms computing single-output +functions are quite a bit weaker than those for multi-output functions, but the bounds we can +obtain on cumulative memory for slightly super-linear time bounds are nearly as strong as those +for multi-output functions. +For simplicity we focus on branching programs with Boolean output, in which case, we can +simply assume that the output is determined by which of two nodes the branching program reaches +at time step T. +The general method for bounds for single output functions is based on the notion of the trace +of a branching program computation. We fix a branching program P computing f : Dn → { 0, 1 }. +As in the case of the simple bounds for multi-output functions, we break up P into a sequence of +blocks, say ℓ of them, that are separated by time steps 0 = t1, . . . , tℓ, tℓ+1 = T. A trace τ in P is a +sequence of ℓ nodes of P, one node in the set of nodes Lti at time step ti for each i = 1, . . . , ℓ. The +set of all traces T = Lt1 × · · · × Ltℓ. +24 + +A key object under consideration is the notion of an embedded rectangle, which is a subset of +R ⊆ Dn with associated disjoint subsets A ⊂ [n] and B ⊂ [n] with |A| = |B| = m(R) = m and +assignment σ ∈ D[n]−A−B such that R = RA × RB × σ. We write α(R) = min(|RA|, |RB|)/|D|m. +Proposition 7.1 (Implicit in Corollary 5.2 of [BSSV03]). Let P be a branching program of length T +computing a function f : Dn → {0, 1}. Suppose that T ≤ kn for k ≥ 4 and n ≥ ℓ ≥ k22k+6. If +0 = t1 < t2 < · · · < tℓ+1 = T are time steps with ti+1 − ti ≤ n/(k2k+6), then there is an embedded +rectangle R ⊆ f −1(1) with m(R) = m ≥ n/2k+1 and α(R) ≥ 2−12(k+1)m−2 · |T |−1 · | f −1(1)|/|D|n +where T is the set of traces of P associated with time steps t1, . . . , tℓ. +Corollary 7.2. Let P be a D-way branching program of length T computing a function f : Dn → {0, 1}. If +T ≤ kn for k ≥ 4 and n ≥ k22k+8, then there is an embedded rectangle R ⊆ f −1(1) with m(R) = m ≥ +n/2k+1 and α(R) ≥ 2−12(k+2)m−k·2k+9·CM(P)/n−2 · | f −1(1)|/|Dn|. +Proof. Fix a branching program P of length T ≤ kn computing f. We can extend P to length exactly +kn by adding a chain of nodes to the root. This does not impact the cumulative memory bound of +P – a single node per level is 0 space – so we assume that T = kn without loss of generality. Let +ℓ = k22k+8. We apply the same basic idea for the choice of time steps 0 = t1, t1, . . . , tℓ+1 = T used +in the simple general method for multi-output functions: Namely, we break P into ℓ time segments +of length either h = ⌊kn/ℓ⌋ or ⌈kn/ℓ⌉. We define t1 = 0 and define ti for 1 < i ≤ ℓ to be the time +step during the next segment at which the set |Lti| is minimized. Write Si = log2 |Lti|. Then the +cumulative memory complexity used by P satisfies +CM(P) ≥ +ℓ +∑ +i=1 +Si · h = h · log2 |T |, +since |T | = ∏t +i=1 |Lti|. +Clearly each ti+1 − ti is at most 2⌈kn/ℓ⌉ ≤ n/(k2k+6) by definition, since their difference is at +most the length of two consecutive time segments. Therefore, the conditions of Proposition 7.1 +apply and we obtain that there is an embedded rectangle R ⊆ f −1(1) with m(R) ≥ n/2k+1 and +α(R) ≥ 2−12(k+2)m−2 · |T |−1 · | f −1(1)|/|Dn| +≥ 2−12(k+2)m−2−CM(P)/h · | f −1(1)|/|Dn| +≥ 2−12(k+2)m−k·2k+9·CM(P)/n−2 · | f −1(1)|/|Dn|. +An example of a natural problem that we can apply this to is the Hamming Closeness problem +HAM1/8,n,N : [N]n → {0, 1} which outputs 1 iff there is a pair of input coordinates xi, xj ∈ [N] such +that the Hamming distance between the binary representations of xi and xj is at most 1 +8 log2 N. +Proposition 7.3 ([BSSV03]). For f (x) = 1 − HAM1/8,n,N(x), and N ≥ n4.39 we have +• (Proposition 6.15) | f −1(1)| ≥ Nn/2, and +• (Lemma 6.17) there is a constant β > 0 such that any embedded rectangle R ⊆ f −1(1) has α(R) ≤ +N−βm(R). +25 + +[BSSV03] apply the above to prove that any [N]-way branching program computing HAM1/8,n,N +for N ≥ n4.39 in time T and space S requires T that is Ω(n log +� +n log n +S +� +). +Theorem 7.4. For N ≥ n4.39 any [N]-way branching program computing HAM1/8,n,N in time T that is +o(n log n) requires cumulative memory (n2 log n)/2O(T/n) which is n2−o(1). +Proof. Let P be an [N]-way branching program computing HAM1/8,n,N in time T that is o(n log n). +We can swap the sink nodes to obtain a branching program P′ computing f = 1 − HAM1/8,n,N. +Write k = T/n and assume wlog that k ≥ 4. Therefore k is o(log n) and hence k22k+8 is no(1) and +hence ≤ n. Therefore by Corollary 7.2, there is an embedded rectangle R ⊆ f −1(1) such that +m(R) = m ≥ n/2k+1 and +α(R) ≥ 2−12(k+2)m−k·2k+9·CM(P′)/n−2 · | f −1(1)|/Nn. +Therefore by Proposition 7.3, for some constant β > 0 we have +N−βm ≥ α(R) ≥ 2−12(k+2)m−k·2k+9·CM(P′)/n−3. +Since CM(P) = CM(P′), solving we obtain +k · 2k+9 · CM(P) ≥ βnm log2 N − 12(k + 2)mn − 3n. +Since k + 2 is o(log N) we obtain that k · 2k+9 · CM(P) ≥ δnm log2 N for some constant δ > 0. +Therefore, plugging in the value of T/n for k, we see that CM(P) is (n2 log n)/2O(T/n). This is +n2−o(1) by the bound on T. +Similar bounds can also be shown by related means for various problems involving computation +of quadratic forms, parity-check matrices of codes and others. For some problems the following +stronger lower bound method is required. +Proposition 7.5 (Implicit in Corollary 5.4 of [BSSV03]). Let P be a D-way branching program of length +T computing a function f : Dn → {0, 1}. Suppose that T ≤ (k − 2)n for k ≥ 8 and n ≥ ℓ ≥ 2q5k2 for q ≥ +240k8. If 0 = t1 < t2 < · · · < tℓ+1 = T are time steps with ti+1 − ti ≤ kn/q5k2, then there is an embedded +rectangle R ⊆ f −1(1) with m(R) = m ≥ q−2k2n/2 and α(R) ≥ 2−q−1/2m · |T |−1 · | f −1(1)|/|D|n where +T is the set of traces of P associated with time steps t1, . . . , tℓ. +Corollary 7.6. Let P be a branching program of length T computing a function f : Dn → {0, 1}. If +T ≤ (k − 2)n for k ≥ 8 and n ≥ 2q5k2 for q = 240k8, then there is an embedded rectangle R ⊆ f −1(1) with +m(R) = m ≥ q−2k2n/2 and α(R) ≥ 2−q−1/2m−q5k2CM(P)/n · | f −1(1)|/|Dn|. +Proof Sketch. The proof is the analog of that of Corollary 7.2 using Proposition 7.5 in place of +Proposition 7.1. +Define the Element Distinctness function EDn,N on [N]n to be the Boolean function that is 1 iff +all values in the input are distinct. +Proposition 7.7 ([BSSV03]). For N ≥ n2, +26 + +• (Proposition 6.11) |ED−1 +n,N(1)| ≥ Nn/e, and +• (Lemma 6.12) Every embedded rectangle R in ED−1 +n,N(1) has α(R) ≤ 2−m(R). +[BSSV03] used this to prove that the time T and space S for computing EDn,n2 must satisfy T = +Ω(n +� +log(n/S)/ log log(n/S)). We strengthen this to the following theorem using Corollary 7.6. +Theorem 7.8. Any [n2]-way branching program computing EDn,n2 in time T that is o(n +� +log n/ log log n) +requires cumulative memory n2/(T/n)O(T2/n2) which is n2−o(1). +Proof. Let P compute EDn,n2 in time T that is o(n +� +log n/ log log n). Write k = T/n + 2 so that +T ≤ (k − 2)/n and assume wlog that k ≥ 8. Write q = 240k8. Since T is o(n +� +log n/ log log n), k +is o( +� +log n/ log log n) and 2q5k2 which is kO(k2) and hence no(1) and therefore ≤ n. We can then +apply Corollary 7.6 to say that there is a rectangle R ⊆ ED−1 +n,n2(1) with m(R) = m ≥ q−2k2n/2 and +α(R) ≥ 2−q−1/2m−q5k2CM(P)/n · |ED−1 +n,n2(1)|/|Dn|. By Proposition 7.7, we have +2−m ≥ α(R) ≥ 2−q−1/2m−q5k2CM(P)/n/e. +Solving, we obtain that +q5k2CM(P) ≥ n · m(1 − 1/q1/2) − 2n. +Therefore, since m ≥ q−2k2n/2, we have constant c such that CM(P) ≥ n2/qck2. As noted above, +qck2 is no(1). More precisely, the bound we obtain is +CM(P) ≥ n2/(T/n)O(T2/n2). +8 +Extending the general method to quantum lower bounds +Quantum circuit time-space lower bounds have the same general structure as their classical branch- +ing program counterparts. +The standard method for quantum time-space tradeoff lower bounds +Let f : Dn → Rm be a multi-output function. For simplicity, we will assume that the output of f +is always m elements in R. To obtain a time-space tradeoff lower bound on f, we must prove a +lemma of the following form for some m′, h(k, n), µ, r(n) and constant C: +Lemma 8.1 (Quantum generic property). For all k ≤ m and any quantum circuit C with at most h(k, n) +layers, there exists a distribution µ such that when x ∼ µ, the probability that C produces at least k correct +output values of f (x) is at most C · |R|−k/r(n). +Such lemmas have historically been proving using direct product theorems [KŠdW07, AŠdW09] +and, more recently, using the recording query technique [HM21]. This is the quantum version of +condition (B) for the classical general method. In the classical setting, the lemma could be extended +27 + +to account for the 2S boundary nodes between layers by using a union bound over 2S possible +branching programs. However in the quantum setting it is not as obvious how to use a lemma that +does not account for initial state. Aaronson showed in [Aar05] how to do exactly this using the +following proposition, which we previously used in Section 4. +Proposition 4.4 ([Aar05]). Let C be a quantum circuit, ρ be any S qubit (possibly mixed) state, and I be +the S qubit maximally mixed state. If C with initial state ρ produces some output O with probability p, then +C with initial state I produces O with probability at least p/22S. +Thus for any problem where we can prove something similar to Lemma 8.1, we can bound the +probability of circuits with S qubits of input-dependent state producing k correct outputs as being +at most 22S · C · |R|−k/r(n). This idea has been applied in [KŠdW07, AŠdW09, HM21] to bound +the probability that blocks produce correct outputs, even when they are given initial state from +previous blocks. +From here we take any circuit C with T layers and S qubits and split it into sub-circuits C1, . . . , Cℓ +with h = h(k, n) layers each. This makes ℓ = ⌈T/h⌉. While Lemma 8.1 gives us that C1 produces at +least k correct outputs with probability at most |R|−k/r(n), sub-circuits C2, . . . , Cℓ start with some +initial state ρ2, . . . , ρℓ that can depend on the input. Since ρi has at most S qubits, the probability +that Ci produces at least k correct outputs is at most +22S · C · |R|−k/r(n) +Assume that: +2S ≤ log2 |R| +r(n) +· m′ − log2(2C) +(9) +Then we can set k = [2S + log2(2C)] · r(n)/ log2 |R| and get that the probability of producing K +correct outputs is at most 1/2. There must be some block that produces at least m · h(S, n)/T correct +outputs, so we must have that +m · h(S, n)/T > k = [2S + log2(2C)] · r(N)/ log2 |R| +This gives us that +TS is Ω +�mh(S, n) log |R| +r(n) +� +. +In the event that (9) is not satisfied, we can instead use the bounded-error quantum query complex- +ity of f (denoted Q( f )) instead of the decision tree complexity to obtain that TS is +Ω +�Q( f ) · m′ log2 |R| +r(n) +� +. +Generic quantum cumulative complexity +Quantum sorting +In Section 4 we are able to exploit some specific structure that leads to a cleaner +proof than we can obtain in general. Specifically for quantum sorting we have h0(s) = √s. Since h0 +is not a constant function, we cannot apply arguments like Theorem 3.3 or Theorem 5.3. However +h0 is a polynomial function. This means that a block with at least s/4 qubits per layer for at least +28 + +h(s/4, n)/2 layers has a constant fraction of the cumulative complexity of a block with h(s, n) layers +that each have s qubits. This means that we can use Lemma 4.3 to upper bound the number of +outputs for a block with h(4s, n) layers and 4s initial qubits while obtaining a lower bound on the +cumulative complexity of such a block that is within a constant factor of the TS complexity of that +block. To obtain such a bound on the cumulative complexity, we can start with a segment of length +h(s, n) when s = 1 that ends at the start of the next block and then repeatedly multiply s by four +until we find a block of length h(s, n) where one of the first h(s, n)/2 layers has less than s space. +Since this is the first such segment, we know that there must be h(s/4, n)/2 layers that each have +s/4 qubits, which gives us the asymptotically tight cumulative complexity lower bound for the +block. The argument we used in our quantum sorting proof can be applied to other classical and +quantum time-space tradeoffs where h0(s) is a polynomial function. +The generic completion +In general, h0(s) may not be a polynomial function. When this is not the +case, we can observe that Theorem 5.6 does not exploit any structure of branching programs that +cannot be applied to quantum circuits. It depends only on the existence of a lemma that bounds +the number of outputs for short computation and a way to apply that lemma to computation with +input dependent initial state, which are given by our generic Lemma 8.1 and Proposition 4.4. We +state the quantum versions of Theorem 5.6 and Remark 5.8 when α = 1 here for completeness. +Note that since Proposition 4.4 gives us a bound of p/22S rather than p/2S, the cumulative memory +bounds we obtain in the quantum setting are half of those from Theorem 5.6. +Corollary 8.2. Let c > 0. Suppose that function f defined on Dn satisfies generic Lemma 8.1 with m′ that +is ω(log2 n). For s > 0, let h(s, n) = h′(s · r(n)/ log2 |R|, n). Let h(s, n) = h0(s)h1(n) where h0(1) = 1 +and h0 is a differentiable function where s/h0(s) is increasing and concave. Let S∗ be defined by: +S∗ +h0(S∗) = m · h1(n) · log2 |R| +12r(n)T +Then either +T log2(2CTc+1) > m · h1(n) · log2 |R| +12r(n) +Which implies that T is Ω( m·h1(n)·log |R| +r(n) log n +) or the cumulative memory used by a quantum circuit that computes +f in time T with error ε ≤ (1 − 1/(2Tc)) is at least +Lh0(n log2 |D|) · min +� +m · h(S∗, n), 3m′ · h′(m′/2, n) +� · log2 |R| +12r(n) . +Additionally if the quantum circuit uses o( m′ log |R| +r(n) +) qubits, then the cumulative memory bound instead is +Lh0(n log2 |D|) · m · h(S∗, n) · log2 |R| +12r(n) . +29 + +9 +Quantum applications of the generic method +Disjoint Collision Pairs Finding +In [HM21] the authors considered the problem of finding k disjoint collisions in a random function +f : [m] → [n], and were able to prove a time-space tradeoff that T3S is Ω(k3n) for circuits that +solve the problem with success probability 2/3. Specifically, they consider circuits that must output +triples (xj2i, xj2i+1, yji) where f (xj2i) = f (xj2i+1) = yji. To obtain this result, they prove the following +theorem using the recording query technique: +Proposition 9.1 (Theorem 4.6 in [HM21]). For all 1 ≤ k ≤ n/8 and any quantum circuit C with at most +T quantum queries to a random function f : [m] → [n], the probability that C produces at least k disjoint +collisions in f is at most O(T3/(k2n))k/2 + 2−k. +The above theorem can be extended to a lemma matching Lemma 8.1 by choosing a sufficiently +small constant α and setting T = αk2n to obtain a probability of at most 2S+1−k. This is sufficient to +obtain a matching lower bound on the cumulative memory complexity using Corollary 8.2. +Theorem 9.2. Finding ω(log2 n) ≤ k ≤ n/8 disjoint collisions in a random function f : [m] → [n] with +probability at least 2/3 requires time T is Ω(kn1/3/ log n) or cumulative memory Ω(k3n/T2). +Proof. Proposition 9.1 lets us apply Corollary 8.2 with m = m′ = k, h′(k, n) = αk2/3n1/3, |R| = +m2n − mn, and r(n) = log2 |R|. Thus we have h(s, n) = h′(s, n) and h0 is a differentiable function +where s/h0(s) is an increasing and concave function. With these parameters, we have: +S∗ is Ω +�k3n +T3 +� +By Corollary 8.2 with the observation that the loss is constant we get that: +T is Ω +�kn1/3 +log n +� +or the quantum cumulative memory is: +Ω +� +min +�k3n +T2 , k5/3n1/3 +�� +. +By Proposition 9.1 we know that any quantum circuit with at most T′ = αk2/3n1/3 layers can +produce k disjoint collisions with probability at most 21−k. Thus we know that T > T′ and our +cumulative memory bound becomes Ω(k3n/T2). +On Tradeoffs for Linear Inequalities and Boolean Linear Algebra +In this section we consider problems in Boolean linear algebra where we write A • x for Boolean +(i.e. and-or) matrix-vector product and A • B for Boolean matrix multiplication. In [KŠdW07] the +authors prove the following time-space tradeoff for Boolean matrix vector products: +30 + +Proposition 9.3 (Theorem 23 in [KŠdW07]). For every S in o(n/ log n), there is an n × n Boolean +matrix AS such that every bounded-error quantum circuit with space at most S that computes Boolean +matrix vector product AS • x in T queries requires that T is Ω( +√ +n3/S). +This result is weaker than a standard time-space tradeoff since the function involved is not +independent of the circuits that might compute it. In particular, [KŠdW07] does not find a single +function that is hard for all space bounds, as the matrix A that they use changes depending on +the value of S. For example, a circuit using space S′ ≫ S could potentially compute AS • x +using o(n3/2/(S′)1/2) queries. This means that an extension of their bound to cumulative memory +complexity does not follow from our Corollary 8.2, as blocks with distinct numbers of initial +qubits would be computing outputs for different functions. In [AŠdW09] the authors use the same +space-dependent matrices to prove a result for systems of linear inequalities. +Proposition 9.4 (Theorem 19 in [AŠdW09]). Let S be in min(O(n/t), o(n/ log n)) and⃗t be the all-t +vector. There is an n × n Boolean matrix AS such that every bounded error quantum circuit using space S +for evaluating the system ASx ≥⃗t using T queries requires T that is Ω( +� +(tn3/S)). +Again this result is not a general time-space tradeoff and hence is not compatible with obtaining +a true cumulative memory bound8. While neither of the above results is a time-space tradeoff for +a fixed function, [KŠdW07] leverages the ideas for Proposition 9.3 to compute a true time-space +tradeoff lower bound for computing Boolean matrix multiplication. +Proposition 9.5 (Theorem 25 in [KŠdW07]). If a quantum circuit computes the Boolean matrix product +A • B with bounded error using T queries and S space, then TS is Ω(n5/T). +In Proposition 9.5, unlike in Proposition 9.3 and Proposition 9.4, both A and B are inputs to the +problem. This allows the lower bound argument to use the properties of the circuit to find matrices +A and B for which the circuit will be particularly challenged. More precisely, to prove the above +result, the authors use a lemma matching the form of Lemma 8.1 that we extract from their lower +bound argument. +Proposition 9.6 (from Theorem 25 in [KŠdW07]). Let R ⊆ [n] × [n] be any fixed set of k ∈ o(n) outputs +to the function f (A, B) = A • B. Then there are constants α, γ > 0 such that for any quantum circuit C +with at most α +√ +kn layers, there is a distribution µC over pairs of matrices such that when (A, B) ∼ µC, the +probability that C produces the correct values for R is at most 2−γk. +Note that, though there are Ω(n2) total output values, Proposition 9.6 only works when k — +the number of output values in a block — is sublinear in n. This is not a problem in the time-space +tradeoff lower bound. Proposition 9.6 upper bounds the value of k for a block as O(S). Since the +time T must be Ω(n2) simply to read the input, the bound T2S = Ω(n5) trivially holds when S is +Ω(n). Thus the time-space tradeoff proof only needs to apply Proposition 9.6 when S (and therefore +k) is sublinear in n. +We cannot apply such an argument when considering cumulative memory complexity, as a +circuit can use Ω(n) qubits for a small number of layers without having an asymptotic effect on the +8The analogous cumulative complexity result would require the matrix A to depend extensively on the structural +properties of the circuit, including the number of qubits after each layer and the locations of each fixed output gate. It is +unclear whether the TS results also may need the matrix AS to depend on the locations of the output gates. +31 + +cumulative memory complexity. However, if we consider o(n) space bounded computation, we +can get a matching bound on the cumulative memory complexity. +Theorem 9.7. Any quantum circuit that computes the boolean matrix product A • B requires Ω(n) ancilla +qubits, time T that is Ω(n5/2/ log n), or cumulative memory that is Ω(n5/T). +Proof. Proposition 9.6 lets us apply Corollary 8.2 with m′ being o(n), m = n2, h′(k, n) = α +√ +kn, |R| = +2, and r(n) = 1/γ. Thus we have h(s, n) = h′(s/γ, n) = α +� +sn/γ and h0(s) = √s. Therefore we +define S∗ to be +S∗ = γα2n5 +36T2 +Thus by Corollary 8.2 we get that T is Ω(n5/2/ log n) or since the space bound is o(n) and the loss +function is constant, the cumulative memory is Ω(n5/T). +Though this is somewhat limited in its range of applicability, it still yields a strict generalization +of the time-space tradeoff lower bound of Proposition 9.5 when S is o(n) and T is o(n5/2/ log n). +10 +Acknowledgements +Many thanks to David Soloveichik for his guidance and contributions to our initial results. Thanks +also to Scott Aaronson for encouraging us to consider cumulative memory complexity in the context +of quantum computation. +References +[Aar05] +Scott Aaronson. Limitations of quantum advice and one-way communication. 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In Alexandra Boldyreva and Daniele Micciancio, editors, Advances +in Cryptology – CRYPTO 2019, pages 239–268, Cham, 2019. Springer International +Publishing. +36 + +A +Optimizations +In this section we prove general optimization lemmas that allow us to derive worst-case properties +of the allocation of branching program layers into blocks. +The first special case is relevant for our analysis of quantum sorting algorithms. +Lemma A.1. If ∑i xi ≤ ∑i x2 +i then ∑i x3 +i ≥ ∑i x2 +i . +Proof. Without loss generality we remove all xi that are 0 or 1 since they contribute the same amount +to each of ∑i xi, ∑i x2 +i , and ∑i x3 +i . Therefore every xi satisfies 0 < xi < 1 or it satisfies xi > 1. For +simplicity we rename those xi with 0 < xi < 1 by yi and those xi with xi > 1 by zj. +Then ∑i xi ≤ ∑i x2 +i can be rewritten as +∑ +i +yi(1 − yi) ≤ ∑ +j +zj(zj − 1), +and both quantities are positive. Let y∗ be the largest value < 1 and z∗ be the smallest value > 1. +Therefore we have +∑ +i +(y2 +i − y3 +i ) = ∑ +i +y2 +i (1 − yi) +≤ ∑ +i +y∗yi(1 − yi) += y∗ ∑ +i +yi(1 − yi) +≤ y∗ ∑ +j +zj(zj − 1) +< z∗ ∑ +j +zj(zj − 1) += ∑ +j +z∗zj(zj − 1) +≤ ∑ +j +z2 +j (zj − 1) += ∑ +j +(z3 +j − z2 +j ). +Rewriting we have ∑i y2 +i + ∑j z2 +j < ∑i y3 +i + ∑j z3 +j , or equivalently ∑i x3 +i > ∑i x2 +i , as required. +The following is a generalization of the above to all differentiable functions p : R≥0 → R≥0 +such that s/p(s) is a concave increasing function of s. +Lemma 5.7. Let p : R≥0 → R≥0 be a differentiable function such that q(x) = x/p(x) is a concave +increasing function of x. For x1, x2, . . . ∈ R≥0, if ∑i xi ≥ K and ∑i p(xi) ≤ L then ∑i xip(xi) ≥ +q−1(K/L) · L. +Proof. By hypothesis, +∑ +i +(xi − Kp(xi)/L) ≥ 0. +(10) +37 + +Observe that s − Kp(s)/L is an increasing function of s since s/p(s) is an increasing function +of s that is 0 precisely when s = q−1(K/L). Since all xi with xi = q−1(K/L) evaluate to 0 in +Equation (10), we can rewrite it as +∑ +xi>q−1(K/L) +(xi − Kp(xi)/L) ≥ +∑ +xi q−1(K/L) both the numerator and denominator are positive and for +xi < q−1(K/L) both the numerator and denominator are negative. Hence f (xi) is non-negative for +every xi ̸= q−1(K/L). +CLAIM: If q is a convex differentiable function, we can complete f to a (non-decreasing) +continuous function of x with f ′(x) ≥ 0 for all x with 0 < x ̸= q−1(K/L). +Proof of Claim. Write a = q−1(K/L). Then since p(x) > 0 and q(a) > 0, we have +f (x) = x · p(x) − x · a/q(a) +x − q(a) · p(x) += x − (x/p(x)) · a/q(a) +x/p(x) − q(a) += x − q(x) · a/q(a) +q(x) − q(a) += +1 +q(a) · q(a) · x − a · q(x) +q(x) − q(a) +. +Therefore +f ′(x) = +1 +q(a) · (q(a) − a · q′(x))(q(x) − q(a)) − (q(a) · x − a · q(x)) · q′(x) +(q(x) − q(a))2 += q(x) − q(a) + (a − x) · q′(x) +(q(x) − q(a))2 +. +Since the denominator is a square and q is increasing, to prove that f ′(x) ≥ 0 for x ̸= a it suffices to +prove that the numerator is non-negative. +Suppose first that x < a, Then a − x > 0 and the numerator q(x) − q(a) + (a − x) · q′(x) ≥ 0 if +and only if q′(x) ≥ q(a)−q(x) +a−x +, which is equivalent to the slope of the tangent to q at x being at least +that of the chord from x to a. This is certainly true since q is a concave function. +Suppose now that x > a. Then a − x < 0 and the numerator q(x) − q(a) + (a − x) · q′(x) ≥ 0 if +and only if q′(x) ≤ q(x)−q(a) +x−a +. Again, this is true since q is a concave function. +It remains to show that we can complete f to a continuous function by giving it a finite value at +a = q−1(K/L). By l’Hôpital’s rule, the limit of q(a) · f (x) as x approaches a is +q(a) − a · q′(a) +q′(a) +if the denominator is non-zero, which it is, since q is an increasing differentiable function at a. +38 + +We now have the tools we need. Let x∗ +− be the largest xi < q−1(K/L) and x∗ ++ be the smallest +xi > q−1(K/L). Then we have f (x∗ ++) ≥ f (x∗ +−) and +∑ +xi>q−1(K/L) +� +xi p(xi) − q−1(K/L) · L/K · xi +� += +∑ +xi>q−1(K/L) +xi · p(xi) − q−1(K/L) · L/K +xi − Kp(xi)/L +· (xi − Kp(xi)/L) += +∑ +xi>q−1(K/L) +f (xi) · (xi − Kp(xi)/L) +≥ +∑ +xi>q−1(K/L) +f (x∗ ++) · (xi − Kp(xi)/L) += f (x∗ ++) +∑ +xi>q−1(K/L) +(xi − Kp(xi)/L) +≥ f (x∗ +−) +∑ +xi>q−1(K/L) +(xi − Kp(xi)/L) +≥ f (x∗ +−) +∑ +xi 1/2c+1. +(c) For any c > 1, Lp(n) ≥ min +1≤s≤n +p(s) − p(s/c) +cp(s) +. +(d) We say that p is nice if it is differentiable and there is an integer c > 1 such that for all x, p′(cx) ≥ +p′(x)/c. If p is nice then Lp(n) is Ω(1/ log2 n). This is tight for p with p(s) = 1 + log2 s. +Proof. Since p is non-decreasing, 1 ≤ p−1(j) ≤ p−1(k) for 1 ≤ j ≤ k and hence +1 +k ≤ ∑k +j=1 p−1(j) +k · p−1(k) +≤ 1 +(12) +since p−1(k) is included in the numerator. Lp(n) is the minimum over all integers k ∈ [1, p(n)] of +∑k +j=1 p−1(j) +k·p−1(k) +and p is non-decreasing so we have 1/p(n) ≤ Lp(n) ≤ 1, which proves part (a) +When p(s) = s1/c we have +k +∑ +j=1 +p−1(j) ≥ +k +∑ +j=⌈(k+1)/2⌉ +j c > ⌈k/2⌉(k/2)c ≥ (k/2)c+1 = k · p−1(k)/2c+1 +so each term in the definition of Lp(n) is larger than 1/2c+1 which proves part (b). (More precise +bounds can be shown but we are not focused on the specific constant.) +Let 1 ≤ k ≤ p(n) be an integer. Then 1 ≤ s = p−1(k) ≤ n. Observe that there are at least +p(s) − p(s/c) integers j ≤ k with p−1(j) ≥ s/c. Therefore +∑k +j=1 p−1(j) +k · p−1(k) +≥ (p(s) − p(s/c)) · s/c +kp−1(k) += p(s) − p(s/c) +ck += p(s) − p(s/c) +cp(s) +. +(13) +The minimum over all k ∈ [1, p(n)] is equivalent to the minimum over all s ∈ [1, n], which proves +part (c). +Now suppose that p is nice. Since p is differentiable, for any s, +p(cs) − p(s) = +� cs +s +p′(y) dy += +� c +s/c p′(cx)c dx +by substitution y = cx +≥ +� c +s/c p′(x) dx +since p is nice += p(s) − p(s/c). +40 + +Then by induction we have that for every positive integer i ≤ logc s, p(s) − p(s/c) ≥ p(s/ci−1) − +p(s/ci). Write ℓ = ⌊logc s⌋. Then s/cℓ < c and +p(s) − p(s/cℓ) = +ℓ +∑ +i=1 +[p(s/ci−1) − p(s/ci)] ≤ ℓ · [p(s) − p(s/c)], +or equivalently that p(s) − p(s/c) ≥ (p(s) − p(s/cℓ)/ℓ and hence +p(s) − p(s/c) ≥ (p(s) − p(c))/ logc s +since p is a non-decreasing function. Applying the lower bound from Equation (12) when k = +p(s) < 2p(c) and the lower bound from Equation (13) when p(s) ≥ 2p(c) we obtain +Lp(n) ≥ min +� +1 +2p(c), +min +1≤s≤n:p(s)≥2p(c)(1 − p(c)/p(s))/(c logc s) +� +. +Since c is a constant, we obtain that Lp(n) is Ω(1/ log n). +Observe that p given by p(s) = 1 + log2 s is nice for every constant c > 0 since p′(cx) = +(ln 2)−1/(cx) = p′(x)/c. In this case we have p−1(j) = 2j−1 and Lp(n) < 2/p(n) < 2/ log2 n since +the largest term p−1(k) in each numerator is (a little) more than the sum of all smaller terms put +together. Together with the lower bound, this proves part (d). +C +The dependence of α on γ in Proposition 4.2 +The following lemma is sufficient to prove Theorem 13 in [KŠdW07] (our Proposition 4.2). Although +the authors prove a more general version of this proposition, the statement below captures what is +necessary in our proof. +Proposition C.1 (Special case of Lemma 12 in [KŠdW07]). Let p be a degree 2α +√ +kn univariate polyno- +mial such that: +• p(i) = 0 when i ∈ {0, . . . , k − 1} +• p(k) = σ +• p(i) ∈ [0, 1] when i ∈ {k + 1, . . . , n} +Then there exists universal positive constants a and b such that for any γ > 0 where keγ+1 ≤ n − k: +σ ≤ a · exp +� +b(2α +√ +kn − k)2 + 4eγ/2+1/2k +√ +n − k(2α√n − +√ +k) +n − k(eγ+1 + 1) +− k − γk +� +. +The σ in this bound gives the completeness bound on the k-threshold problem. So we need a +choice of α such that σ ≤ e−γk. We now prove that this is possible when α ∈ Ω(e−γ/2). +Lemma C.2. If √ +k/n ≤ α ≤ min(1/(16 +√ +eγ+1 + 1), 1/(2 +√ +2b)) then σ ≤ a · e−γk. +41 + +Proof. We show this with a chain of inequalities from Proposition C.1 using our bounds on α: +σ ≤ a · exp +� +b(2α +√ +kn − k)2 + 4eγ/2+1/2k +√ +n − k(2α√n − +√ +k) +n − k(eγ+1 + 1) +− k − γk +� += a · exp +� +b(4α2kn − 4αk +√ +kn + k2) + 4eγ/2+1/2k +√ +n − k(2α√n − +√ +k) +n − k(eγ+1 + 1) +− k − γk +� += a · exp +� +kbα2(4n − 4 +√ +kn/α + k/α2) + 4eγ/2+1/2k +√ +n − k(2α√n − +√ +k) +n − k(eγ+1 + 1) +− k − γk +� +≤ a · exp +� +kbα2(4n − 3k/α2) + 4eγ/2+1/2k +√ +n − k(2α√n − +√ +k) +n − k(eγ+1 + 1) +− k − γk +� +≤ a · exp +� +kbα2(4n − 4k(eγ+1 + 1)) + 4eγ/2+1/2k +√ +n − k(2α√n − +√ +k) +n − k(eγ+1 + 1) +− k − γk +� += a · exp +� +4eγ/2+1/2k +√ +n − k(2α√n − +√ +k) +n − k(eγ+1 + 1) ++ k(4bα2 − 1 − γ) +� +≤ a · exp +� +4eγ/2+1/2k(2αn − +√ +nk) +n − k(eγ+1 + 1) ++ k(4bα2 − 1 − γ) +� +≤ a · exp +� +4eγ/2+1/2kα(2n − +√ +nk/α) +n − k(eγ+1 + 1) ++ k(4bα2 − 1 − γ) +� +≤ a · exp +�4eγ/2+1/2kα(2n − k/α2) +n − k(eγ+1 + 1) ++ k(4bα2 − 1 − γ) +� +≤ a · exp +�4eγ/2+1/2kα(2n − 2k(eγ+1 + 1)) +n − k(eγ+1 + 1) ++ k(4bα2 − 1 − γ) +� += a · exp +� +k(4bα2 + 8eγ/2+1/2α − 1 − γ) +� +≤ a · exp +� +k(4bα2 − 1/2 − γ) +� +≤ a · exp (−γk) . +To go from this lemma to σ ≤ e−γk, we set γ′ = γ + ln(a) in Lemma C.2 and observe that we +get the same asymptotic lower bound on the corresponding α′. +42 + diff --git a/tdE5T4oBgHgl3EQfmg_h/content/tmp_files/load_file.txt b/tdE5T4oBgHgl3EQfmg_h/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c1fd34792980b7020fa0f4a5593bf98027af54f --- /dev/null +++ b/tdE5T4oBgHgl3EQfmg_h/content/tmp_files/load_file.txt @@ -0,0 +1,1374 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf,len=1373 +page_content='Cumulative Memory Lower Bounds for Randomized and Quantum Computation Paul Beame* Computer Science & Engineering University of Washington Niels Kornerup Computer Science University of Texas at Austin January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 2023 Abstract Cumulative memory—the sum of space used over the steps of a computation—is a fine- grained measure of time-space complexity that is a more accurate measure of cost for algorithms with infrequent spikes in memory usage in the context of technologies such as cloud computing that allow dynamic allocation and de-allocation of resources during their execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We give the first lower bounds on cumulative memory complexity that apply to general sequential classical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We also prove the first such bounds for bounded-error quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Among many possible applications, we show that any classical sorting algorithm with suc- cess probability at least 1/poly(n) requires cumulative memory ˜Ω(n2), any classical matrix multiplication algorithm requires cumulative memory Ω(n6/T), any quantum sorting circuit requires cumulative memory Ω(n3/T), and any quantum circuit that finds k disjoint collisions in a random function requires cumulative memory Ω(k3n/T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' More generally, we present theorems that can be used to convert a wide class of existing time-space tradeoff lower bounds to matching lower bounds on cumulative memory complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Research supported by NSF grant CCF-2006359 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='05680v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='CC] 13 Jan 2023 1 Introduction There are many problems where algorithms can use additional memory for faster running times or expend additional time to reduce memory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' While there are many different kinds of tradeoffs between time and space, the most common complexity metric for such algorithms is the maximum time-space (TS) product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This metric is appropriate when a machine must allocate an algorithm’s maximum space throughout its computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' However, recent technologies like AWS Lambda [BBB+21], Flex [LL20], and CloudScale [SSGW11] suggest that in the context of cloud computing, space can be allocated to a program only as it is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' When using such services, analyzing the average memory used per step leads to a more accurate picture than merely measuring the maximum space used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Cumulative memory (CM) - the sum over time of the space used per step of an algorithm - is an alternative notion of time-space complexity that is more fair to algorithms that only require rare spikes in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The term cumulative memory complexity was first coined by Alwen and Serbinenko [AS15] who introduced it as a way to discuss time-space tradeoffs for "memory hard functions" like password hashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since then, lower and upper bounds on the CM complexity of problems in structured computational models using the black pebble game have been exten- sively studied, beginning with the work of [AS15, AB16, RD16, ACP+17, ACK+16, ABP17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' These structured models via pebble games are particularly natural in the context of the random oracle assumptions that are common in cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' By carefully interweaving their memory intensive steps, these authors devise algorithms for cracking passwords that compute many hashes in parallel using only slightly more space than is necessary to compute a single hash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' While such algorithms can use parallelism to amortize costs and circumvent proven single instance TS complexity lower bounds, their cumulative memory scales linearly with the number of computed hashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Thus cumulative memory complexity is a more robust metric than TS complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Surprisingly there has been little research into CM complexity outside the setting of cryptogra- phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In [AdRNV17] the authors showed strong CM complexity results for the black-white pebble game and used them to derive related results for resolution proof systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Our work is the first to explore CM complexity outside the regime of pebbling and the first to obtain results that apply to general models of computation without cryptographic or black-box assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Our Results In this work, we give generic methods that allow one to convert existing paradigms for obtaining time-space tradeoff lower bounds involving worst-case space to new lower bounds that replace the time-space product by cumulative space, immediately yielding a host of new lower bounds on cumulative memory complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' With these methods, we show how to extend almost all known proofs for time-space tradeoffs to equivalent lower bounds on cumulative memory complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Our results, like those of existing time-space tradeoffs, apply in models in which arbitrary sequential computations may be performed between queries to a read-only input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Our lower bounds also apply to randomized and quantum algorithms that are allowed to make errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Classical computation We first focus on lower bound paradigms that apply to computations of multi-output functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We give general theorems showing how to translate the basic ideas 1 that yield virtually all time-space tradeoffs known for such functions to yield lower bounds on cumulative memory complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' As applications of our general methodology, we prove that the cumulative memory required by any sorting algorithm is ˜Ω(n2) which generalizes [BC82, Bea91] and the cumulative memory required for any matrix multiplication algorithm using time T is Ω(n6/T), generalizing [Abr91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We also show how the paradigm can be extended to correspond to the best time-space tradeoff lower bounds for single-output Boolean functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In particular, we give examples of functions for which algorithms running in time T require cumulative memory Ω((n2 log n)/2cT/n) for some constant c > 0, generalizing [BSSV03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This means, for example, that algorithms computing these functions in time o(n log n) require cumulative memory Ω(n2 log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Quantum computation We generalize the quantum time-space tradeoff for sorting proven in [KŠdW07], which requires that the time order in which output values are produced must correspond to the sorted order, to a matching cumulative memory complexity bound of Ω(n3/T) that works for any fixed time-ordering of output production which yields a more general lower bound1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We then show how our classical general theorems can be applied to known quantum time-space tradeoffs and extend the quantum time-space tradeoff for k-collision pairs finding from [HM21] to the matching cumulative memory complexity bound of Ω(k3n/T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Previous work Memory hard functions and cumulative memory complexity Memory hard functions (MHFs) are functions designed to require large space to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In [AS15] Alwen and Serbinenko in- troduced parallel cumulative (memory) complexity as a metric for analyzing the space footprint required to compute these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Most MHFs are constructed using hashgraphs (see [DNW05]) of DAGs whose output is a fixed length string and their proofs of security are based on pebbling ar- guments on these DAGs while assuming access to truly random hash functions for their complexity bounds [AS15, BCGS16, BDK16, RD16, ABP17, ACP+17, BZ17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' More recent MHF constructions do not rely on random hash functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' however, they still require some cryptographic assumptions [CT19, ABB21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In general the major focus of these results has been on savings with parallel rather than sequential computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Classical time-space tradeoffs While early work focused on the kinds of restricted pebbling models similar to those considered to date for cumulative memory complexity [Tom80, BFK+81], the gold standard model for time-space tradeoff analysis is that of unrestricted branching programs, which simultaneously capture both time and space for general sequential computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This analysis began with the work of [BC82] who proved lower bounds for sorting and introduced a general methodology for multi-output functions that has been extended to many problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=', [Yes84, Abr87, Abr90, Bea91, MNT93]), including universal hashing and a wide array of problems in linear algebra [Abr91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' A separate methodology for single-output functions, first introduced in the context of restricted branching programs [BRS93, Oko93], was extended to general branching programs in [BJS01], with further applications to other problems [Ajt02] including multi-precision 1For example, an algorithm may be able to determine the median output long before it determines the other outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 2 integer multiplication [SW03] and error-correcting codes [Juk09] as well as over Boolean input domains [Ajt05, BSSV03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Both of these methodologies involve breaking the branching program into blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For multi- output functions one needs to show that for any fixed node at the beginning of a block, the probability over a random input that the program started at that node produces k correct output values in that block decays exponentially in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For single-output functions, one decomposes the space of inputs based on the "trace" of nodes traversed at segment boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Based on the traces, one can determine the size and density properties of "embedded rectangles" of inputs on which the function must be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Lower bounds require showing the given function does not have such rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Quantum time-space tradeoffs Though the basic notion of exponential decay in producing correct outputs is similar to the classical multi-output bounds, the arguments are substantially more subtle in the quantum setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The quantum query model gives us access to an input X = x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , xn via an oracle QX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since the result of a single quantum query can change if we flip any bit of the input, we need arguments that limit the sensitivity of a query to changes in the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' These arguments generally follow one of two techniques: the adversary method [BBBV97, Amb02, ŠS05] or the polynomial method [BBC+01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' To obtain quantum time-space tradeoffs for multi-output functions, it is important to have lemmas showing that query-bounded computations only yield a slight advantage over randomly guessing outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Such lemmas often take the form of direct product theorems, which state that if T queries are necessary to solve one instance of a problem with constant probability, then kT queries are insufficient to solve k independent instances of that problem with probability 2−Ω(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' While such results seems intuitive, Shaltiel proved that they are not true in general [Sha03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The polynomial method [Aar05, KŠdW07, She11] and the adversarial method [AŠdW09] have both been extended to prove quantum direct product theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In [KŠdW07] the authors use direct product theorems to prove a tight time-space tradeoff for sorting and a time-space tradeoff for matrix multiplication in Boolean algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' They also proved somewhat weaker lower bounds for computing matrix-vector products for fixed matrices A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' those bounds were extended in [AŠdW09] to systems of linear inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' However, both of these latter results apply to computations where the fixed matrix A defining the problem depends on the space bound and, unlike the case of sorting or Boolean matrix multiplication, do not yield a fixed problem for which the lower bound applies at all space bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' More recently [HM21] extended the recording query technique of Zhandry in [Zha19] to obtain time-space lower bounds for the k-collision problem and match the aforementioned result for sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Our methods At the highest level, we employ part of the same paradigms previously used for time-space tradeoff lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' namely breaking up the computations into blocks of time and analyzing properties of the branching programs or quantum circuits based on what happens at the boundaries between those time blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' However, for cumulative memory complexity, those boundaries cannot be at fixed locations in time and their selection needs to depend on the space used in those time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 3 Further, in many cases, the time-space tradeoff lower bound needs to set the lengths of those time blocks in a way that depends on the specific space bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' When extending the ideas to bound cumulative memory usage, there is no single space bound that can be used throughout the computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' this sets up a tricky interplay between the choices of boundaries between time blocks and the lengths of the time blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Because the space usage within a block may grow and shrink radically, even with optimal selection of block boundaries, the contribution of each time block to the overall cumulative memory may be significantly lower than the time-space product lower bound one would obtain for the individual block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We show how to bound any loss in going from time-space tradeoff lower bounds to cumulative memory lower bounds in a way that depends solely on the bound on the lengths of blocks as a function h0 of the target space bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For many classes of bounding functions we are able to bound the loss by a constant factor, and we are able show that it is always at most an O(log n) factor loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Once we have this, if this bounding function h0 is non-constant, there is still a matter of bounding the optimum way for the algorithm to allocate its space budget for producing the require outputs throughout its computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This optimization again depends on the bounding function h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This involves minimizing a convex function based on h0 subject to a mix of convex and concave constraints which is not generally tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' However, assuming that h0 is nicely behaved, we are able to apply specialized convexity arguments to yield lower bounds on cumulative memory complexity that in many instances match those of previous time-space tradeoffs up to a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Road map We give the overall definitions in Section 2, including a review of the standard definitions of the work space used by quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We then give the very simple version of our methods that is needed to prove results on the cumulative memory complexity of classical sorting algorithms in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In Section 4, we give our lower bound for quantum sorting algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This example shows something of the complexity required for our general arguments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' in this case, the bounding function is simple enough that we can apply an alternative direct argument to show only constant loss in the choices of boundaries for time blocks, but it still requires some of the complexity of the general optimization of space allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This section also includes the additional ideas that allow us to analyze circuits that produce sorted outputs in arbitrary sequential time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We give the full general theorems that let us convert classical time-space tradeoffs for multi- output functions to cumulative memory lower bounds, even for randomized algorithms, in Sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In Section 6 we apply these general theorems to a few other problems, particularly those in linear algebra, to give an indication of how they can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Next, we show how to convert time-space tradeoff lower bounds for single-output functions to cumulative memory lower bounds in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Finally, in Section 8 and Section 9 we show how to extend our generic method to quantum circuits and discuss its application to other existing quantum time-space tradeoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The appendices contain some of the technical arguments that allow us to bound the loss functions and to give bounds on the optimum allocations of cumulative space budgets to time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 4 2 Preliminaries Cumulative memory is an abstract notion of time-space complexity that can be applied to any model of computation with a natural notion of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The cumulative memory of a discrete time computation A that uses |At| space during the t-th step and runs in time T is: CM(A) = T ∑ t=1 |At| The cumulative memory complexity of a function f with respect to a computational model M is: CMC( f ) = min A∈M computes f CM(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In this paper we consider both branching programs and quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Branching Programs Branching programs with input {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , xn} ∈ Dn are known as D-way branching programs and are defined using a rooted DAG in which each non-sink vertex is labeled with an i ∈ [n] and has |D| outgoing edges that correspond to possible values of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Each edge is optionally labeled by some number of output statements expressed as pairs (j, oj) where j ∈ [m] is an output index and oj ∈ R (if outputs are to be ordered) or simply oj ∈ R (if outputs are to be unordered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Evaluation is performed by starting at the root v0 and following the appropriate labels of the respective xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We consider branching programs P that contain T + 1 layers where the outgoing edges from nodes in each layer t are all in layer t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We impose no restriction on the query pattern of the branching program or when it can produce parts of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We define the following complexity measures for such a branching program P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The time of the branching program is T(P) = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The space of the branching program is S(P) = maxt log2 |Lt| where Lt is the set of nodes in layer t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Observe that in the absence of any limit on its space, a branching program could equally well be a decision tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' hence the minimum time for branching programs to compute a function f is its decision tree complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The time-space (product) used by the branching program is TS(P) = T(P)S(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The cumulative memory used by the branching program is CM(P) = ∑t log2 |Lt|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Branching programs are very general and simultaneously model time and space for sequential computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In particular they model time and space for random-access off-line multitape Turing machines and random-access machines (RAMs) when time is unit-cost, space is log-cost, and the input and output are read-only and write-only respectively2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Branching programs are much more flexible than these models since they can make arbitrary changes to their storage in a single step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 2In prior work, branching program space has often been defined to be the logarithm of the total number of nodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=', [BC82, Abr91]) rather than the logarithm of the width (maximum number of nodes per layer), though the latter has been used (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=', [CFL83]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The natural conversion from an arbitrary space-bounded machine to a branching program produces one that is not leveled (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=', nodes are not segregated by time step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' After leveling the branching program, the space of the original machine becomes the logarithm of the width (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' [Pip79]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The width-based definition is also the only natural one by which to measure cumulative memory complexity and, in any case, the two definitions differ by at most an additive log2 T amount, with lower bounds on width implying lower bounds on size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 5 Quantum Circuits We also consider quantum circuits C classical read-only input X = x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , xn that can be queried using an XOR query oracle as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' As is normal in circuit models, each output wire is associated with a fixed position in the output sequence, independent of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' As shown in Figure 2 following [KŠdW07], we abstract an arbitrary quantum circuit C into layers C = {L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , LT} where layer Lt starts with the t-th query Q to the input and ends with the start of the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' During each layer, an arbitrary unitary transformation V gets applied which can express an arbitrary sub-circuit involving input-independent computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The sub- circuit/transformation V outputs St qubits for use in the next layer in addition to some qubits that are immediately measured in the standard basis, some of which are treated as classical write-only output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The time of C is lower bounded by the number of layers T and we say that the space of layer Lt is St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Observe that to compute a function f, T must be at least the quantum query complexity of f since that measure corresponds the above circuit model when the space is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Note that the cumulative memory of a circuit is lower-bounded by the sum of the St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For convenience we define S0, the space of the circuit before its first query, to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Thus we only consider the space after the input is queried.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' |x⟩ f |x⟩ |b⟩ |b ⊕ f (x)⟩ Figure 1: The XOR oracle for a function f : D → R where D ⊆ {0, 1}n and R ⊆ {0, 1}m is the linear operator that, for all x ∈ D and b ∈ {0, 1}m, maps the basis state |x⟩ |b⟩ to |x⟩ |b ⊕ f (x)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' When x ̸∈ D, the XOR oracle acts like the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The query oracle QX (where X = x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , xn) is the XOR oracle for the function f (i) = xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Figure 2: The abstraction of a quantum circuit into layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Simulating a quantum query from modified input Without making any additional assumptions on our query oracle, it is possible to simulate a query for a modified (possibly larger) input using at most two queries to the original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let QX be an XOR query oracle for some input X = x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' xn where xi ∈ {0, 1}m and let ˆxi = � xi i ∈ [n] 0 otherwise By definition, this makes QX the permutation that maps any basis state |i, j, k⟩ where i ∈ {0, 1}⌈log2 n⌉ and j ∈ {0, 1}m to the basis state |i, j ⊕ ˆxi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We want to use QX to simulate queries to some modified 6 input X′ = {x′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , x′ n+t} where x′ i ∈ {0, 1}ℓ is defined by x′ i = g(i, ˆxi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let G be the XOR query oracle for g(i, j) and Pi>n be the XOR query oracle for the predicate function p(i) = 1i>n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then the circuit in Figure 3 simulates an XOR query on modified input X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Note that G and Pi>n both compute classical functions and therefore can be computed using a network of Toffoli gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since G is independent of X, this circuit simulates a query to X′ using at most two queries to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The second query of QX is necessary to uncompute the m qubit ancillary register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Figure 3: The above circuit uses two calls to a query oracle QX in order to simulate one query to the modified input X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 3 Cumulative memory complexity of classical sorting algorithms For a natural number N, the standard version of sorting is a function Sortn,N : [N]n → [N]n that on input x ∈ [N]n produces an output y ∈ [N]n in non-decreasing order where y is a permutation of x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' that is, there is some permutation π such that yi = xπ(i) for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' A related problem is the ranking problem Rankn,N : [N]n → [n]n which on input x ∈ [N]n produces a permutation π represented as the vector (π(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , π(n)) such that Sortn,N(x) = (xπ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , xπ(n)) and whenever xi = xj for i < j we have π(i) < π(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1 ([BC82]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' (a) If there is an [nN]-way branching program P computing Sortn,nN then there is an [N]-way branching program P′ computing Rankn,N with T(P′) ≤ T(P), S(P′) ≤ S(P), and CM(P′) ≤ CM(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' (b) If there is an [N]-way branching program P′′ computing Rankn,N then there is an [N]-way branch- ing program P′′′ computing Sortn,N with T(P′′′) ≤ 2T(P′′), S(P′′′) ≤ S(P′′) + log2 N, and CM(P′′′) ≤ 2CM(P′′) + T(P′′′) log2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For part (a), the program P′ is exactly P except that when P queries xi ∈ [Nn], P′ reads x′ i ∈ [N] and branches on value xi = (x′ i, i) and when P outputs (i, yi) = (i, xπ(i)) on an edge for xπ(i) = (x′ π(i), π(i)), P′ outputs (i, π(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For part (b), the program P′′′ is exactly P′′ except that whenever P′′ outputs (i, π(i)) on an edge, P′′′ queries xπ(i) and outputs (i, xπ(o)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' One layer becomes two layers and the number of nodes per layer of P′′′ is at most N times that of P′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Following [BC82], we focus on inputs where the xi are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In this case, the tie-breaking we enforced in defining Rankn,N when there are equal elements is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 7 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2 ([BC82]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' There is an α > 0 such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let n be sufficiently large and µ be the uniform distribution over lists of n distinct integers from [n2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then for any branching program B of height h ≤ αn and for all integers k ≤ 2αn, the probability for x ∼ µ that B produces at least k correct output values of Rankn,n2 on input x is at most 2−k/⌈log2 n⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let P be a branching program computing Sortn,n3 with probability at least n−O(1) and T = T(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then T is Ω(n2/ log2 n) or CM(P) is Ω(n2/ log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Further, any random access machine computing Sortn,n3 with n−O(1) probability requires cumulative memory of Ω(n2/ log n) bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We prove the same bounds for branching programs P computing Rankn,n2 which, by Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1, implies the bounds for computing Sortn,n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For simplicity we first assume that P is determistic and is always correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let α be the constant and µ be the probability distributuon on [n2]n from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2, and let H = � α 2n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We partition P into ℓ = ⌈T/H⌉ intervals {I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , Iℓ}, all of length H except for the first, which may be shorter than the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let t1 = 0, tℓ+1 = T, and for i ∈ [2, ℓ], ti be the time-step in Ii with the fewest number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We define Si = log2(|Lti|) where Lj is the set of nodes of P in layer j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The i-th time block Bi will contain all layers from ti to ti+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We observe: CM(P) ≥ ℓ ∑ i=2 Si H = H ℓ ∑ i=1 Si (1) since S1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Define ki = ⌈⌈log2 n⌉ (Si + log2(2T))⌉, which will be our target number of outputs for block Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' By our choice of Bi we know its length is at most αn and it starts at a layer with 2Si nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' So, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2, combined with a union bound, the probability for x ∼ µ that Bi produces at least ki correct output values of Rankn,n2 on input x ∼ µ is at most 1/(2T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Thus the probability over µ that at least one block Bi produces at least ki correct output values is at most 1/2 and the probability that the total number of outputs produced is at most ∑ℓ i=1(ki − 1) is at least 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since P must always produce n correct outputs, we must have: ℓ ∑ i=1 (ki − 1) ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Inserting the definition of ki we get: ℓ ∑ i=1 (⌈log2 n⌉ (Si + log2(2T))) ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Using Equation (1) to express this in terms of CM(P) gives us: CM(P)/H + ℓ log2(2T) ≥ n ⌈log2 n⌉ or CM(P) + T log2(2T) ≥ n � α 2n � ⌈log2 n⌉ ≥ αn2 3 log2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Thus at least one of T log2(2T) or CM(P) is at least αn2/(6 log2 n), as required, since log T is O(log n) wlog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The bound for random-access machines comes from observing that such a machine requires at least one memory cell of Ω(log T) bits at every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 8 To prove the bound for algorithms with success probability n−c, we multiply log2(2T) in the above argument by (c + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since any sorting algorithm must have T ≥ n, on randomly chosen inputs the probability that it produces at least ∑ℓ i=1(ki − 1) correct outputs becomes 1 2nc < 1 nc and hence the above bounds (reduced by the constant factor c + 1) apply to deterministic algorithms with success probability 1/nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' By Yao’s lemma this implies the same lower bound for randomized algorithms with success probability n−c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 applies to cumulative working memory of any algorithm that produces its sorted output in a write-only output vector and can compute those values in arbitrary time order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If the al- gorithm is constrained to produce its sorted output in the natural time order then, following [Bea91], one can obtain a slightly stronger bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Any branching program P computing the outputs of Sortn,n in order in time T and probability at least 4/5 requires T to be Ω(n2/ log n) or CM(P) to be Ω(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Further, any random access machine computng Sortn,n in order with probability at least 4/5 requires cumulative memory Ω(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof Sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Any such algorithm can easily determine all the elements of the input that occur uniquely and the lower bounds follow from the bounds on Unique Elements that we prove in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 4 Quantum cumulative memory complexity of sorting We now show with a similar argument that the quantum cumulative memory complexity of sorting is Ω(n3/T), matching the ST complexity bounds given in [KŠdW07, HM21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This involves the quantum circuit model which, as we have noted, produces each output position at a predetermined input-independent layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We restrict our attention to circuits that output all elements in the input according to their sorted order with a constant total success probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' While our proof is inspired by the time-space lower bound of [KŠdW07], it can be easily adapted to follow the proof in [HM21] instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In the k-threshold problem we receive an input X = x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , xn where xi ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We want to accept iff there are at least k distinct values for i where xi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We say that a quantum circuit C that computes a boolean function f : {0, 1}n → {0, 1} has completeness a and soundness b on inputs in domain D iff for all x ∈ D, Pr[C(x) = 1] ≥ a when f (x) = 1 and Pr[C(x) = 1] ≤ b when f (x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We say that a circuit has perfect completeness (soundness) iff a = 1 (respectively, b = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2 (Theorem 13 in [KŠdW07]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For every γ > 0 there is an α > 0 such that any quantum k-threshold circuit with at most T ≤ α √ kn queries and with perfect soundness must have completeness σ ≤ e−γk on inputs with hamming weight k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Using the above theorem, we present a generalization of a lemma first proven in [KŠdW07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Choose any constant γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let n be sufficiently large and C(X) be a quantum circuit with input X = x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' There exists a constant β that depends only on γ such that for all k ≤ β2n and 9 R ⊆ {n/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , n} where |R| = k, if C(X) makes at most β √ kn queries, then the probability that C(X) can correctly output all k pairs (xi, rj) where rj ∈ R and xi is the rj’th smallest element of X is at most e(1−γ)k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If R is a contiguous set of integers, then the probability is at most e−γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The version of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 proved in [KŠdW07] had the additional assumption that the set of output ranks R is a contiguous set of integers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' this was sufficient to show that any quantum circuit that produces its sorted output in sorted time order requires that T2S is Ω(n3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The authors stated that their proof can be generalized to any fixed rank ordering, but the generalization is not obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We generalize their lemma to non-contiguous R, which is sufficient to obtain an Ω(n3/T) lower bound on the cumulative complexity of sorting independent of the time order in which the sorted output is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Choose α as the constant for γ in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2 and let β = √ 2α/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let C be a circuit with at most β √ kn layers that outputs the k correct pairs (xi, rj) with probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let R = {r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' rk} where r1 < r2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' < rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We describe our construction of a circuit C′(X) solving the k-threshold problem on inputs X = x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , xn/2 with exactly k ones in terms of a function f : [n/2] → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Given f, we re-interpret the input as follows: we replace each xi with x′ i = f (i)xi, add k dummy values of 0, and add one dummy value of j for each j ∈ {n/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , n} \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Doing this gives us an input X′ = x′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , x′ n that has n/2 zeroes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If we assume that f is 1-1 on the k ones of X, then the image of the ones of X will be R and there will be precisely one element of X′ for each j ∈ {n/2 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore the element of rank j > n/2 in X′ will have value j, and hence the rank r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , rk elements of X′ will be the images of precisely those elements of X with xi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' To obtain perfect soundness, we cannot rely on the output of C(X′) and must be able to check that each of the output ranks was truly mapped to by a distinct one of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For each element xi of X we simply append its index i as log2 n low order bits to its image x′ i and append an all-zero bit-vector of length log2 n to each dummy value to obtain input X′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Doing so will not change the ranks of the elements in X′, but will allow recovery of the k indices that should be the ones in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In particular, circuit C′(X) will run C(X′′) and then for each output x′′ j with low order bits i, C′(X) will query xi, accepting if and only if all of those xi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' More precisely, since the mapping from each xi to the corresponding x′′ i is only a function of f, xi, and i, as long as C′(X) has an explicit representation of f, it can simulate each query of C(X′′) with two oracle queries to X (see Section 2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since C′ has at most 2β √ kn + k ≤ 3β √ kn ≤ α √ kn/2 layers, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2, it can only accept with probability at most e−γk when the input has k ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We now observe that for each fixed X with exactly k ones, for a randomly chosen function f : [n/2] → R, the probability that f is 1-1 on the ones of X′ is exactly k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='/kk ≥ e1−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore C′(X) will give the indices of the k ones in X with probability3 at least p · e1−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' However, this probability must be at most e−γk, so we can conclude that p ≤ e(1−γ)k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In the event that R is a contiguous set of integers, observe that any choice for the function f will make X′′ have the ones of X become ranks r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' So the probability of finding the ones is at least p ≤ e−γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 3Note that though this is exponentially small in k it is still sufficiently large compared to the completeness required in the lower bound for the k-threshold problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 10 By setting k and γ appropriately, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 gives a useful upper bound on the number of fixed ranks successfully output by any β √ Sn query quantum circuit that has access to S qubits of input dependent initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' To handle input-dependent initial state, we will need the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='4 ([Aar05]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let C be a quantum circuit, ρ be any S qubit (possibly mixed) state, and I be the S qubit maximally mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If C with initial state ρ produces some output O with probability p, then C with initial state I produces O with probability at least p/22S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This allows us to bound the overall progress made by any short quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let γ = 1 + ln(4) and β be the constant from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 that depending on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then for any fixed set of S ≤ β2n ranks that are greater than n/2, the probability that any quantum circuit C with at most β √ Sn queries and S qubits of input-dependent initial state correctly produces the outputs for these S ranks is at most 1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Applying Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='4 to the bound in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 gives us that a quantum circuit with S qubits of input-dependent state can produce a fixed set of k ≤ β2n outputs larger than median with a probability at most 22Se(1−γ)k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since γ = 1 + ln(4) setting k = S yields a probability bound on of at most 1/e on the event in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' When n is sufficiently large, any quantum circuit C for sorting a list of length n with success probability at least 1/e and at most T layers that produces its sorted outputs in any fixed time order requires cumulative memory that is Ω(n3/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We partition C into blocks with large cumulative memory that can only produce a small number of outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We achieve this by starting at last unpartitioned layer and finding a suitably low space layer before it so that we can apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5 to upper bound the number of correct outputs that can be produced in that block with a success probability of at least 1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let β be the constant from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5 and k∗(t) be the least non-negative integer value of k such that the interval: I(k, t) = � t − β 2 (2k+1 − 1)√ n, t − β 2 (2k − 1)√ n � contains some t′ such that St′ ≤ 4k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We recursively define our blocks as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let ℓ be the number of blocks generated by this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The final block Cℓ starts with the first layer tℓ−1 ∈ I(k∗(T), T) where Stℓ−1 ≤ 4k∗(T) − 1 and ends with layer tℓ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let ti be the first layer of block Ci+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then the block Ci starts with the first layer ti−1 ∈ I(k∗(ti), ti) where Sti−1 ≤ 4k∗(ti) − 1 and ends with ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' See Figure 4 for an illustration of our partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since S0 = 0 we know that k∗(t) ≤ log(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Likewise since St > 0 when t > 0, for all t > β 2 √n we know that 0 < k∗(t) ≤ log(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' By construction, block Ci starts with less than 4k∗(ti) qubits of initial state and has length at most β2k∗(ti)√n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' so by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5, if 4k∗(ti) ≤ β2n, the block Ci can output at most 4k∗(ti) inputs with failure probability at most 1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Additionally Ci has at least β 22k∗(ti)−1√n layers that each have at least 4k∗(ti)−1 qubits,4 so the cumulative memory of Ci is at least β 223k∗(ti)−3√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 4This may not hold for C1 with length less than β 2 √ N, but Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 with Appendix C give us that this number of layers is insufficient to find a fixed rank input with probability at least 1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Thus we can omit such a block from our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 11 Figure 4: How we define the block Ci that ends at layer Lti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The grey layers are the ones used to lower bound the cumulative memory complexity of Ci, as each of these layers uses at least 4k∗(ti)−1 qubits and the length of this interval is β 22k∗(ti)−1√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We now have two possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If we have some i such that 4k∗(ti) > β2n, the cumulative memory of Ci alone is at least β4n2/16 which is Ω(n2) and hence C has cumulatively memory Ω(n3/T) since T ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Otherwise, since we require that the algorithm is correct with probability at least 1/e, each block Ci can produce at most 4k∗(ti) outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since our circuit must output all n/2 elements larger than the median, we know ∑ℓ i=1 4k∗(ti) ≥ n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For convenience we define wi = 2k∗(ti) and get the following bound on the sum of the w2 i : ℓ ∑ i=1 w2 i ≥ n/2 We obtain the following lower bound on the cumulative memory: CM(C) ≥ ℓ ∑ i=1 β 2 23k∗(ti)−3√ n = β 16 √ n ℓ ∑ i=1 w3 i (2) To lower bound the cumulative complexity, this gives us the non-convex optimization problem in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In Appendix A we prove that if ∑ xi ≤ ∑ x2 i , then ∑ x2 i ≤ ∑ x3 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This gives us that: ℓ ∑ i=1 x3 i ≥ ξ Reversing the variable substitution gives us: ℓ ∑ i=1 w3 i ≥ βn5/2 16T Then applying Equation (2) gives us the bound: CM(C) ≥ β2n3 256T 12 ≤ 4*(t) - 1(a) CM(C) ≥ min β 16 √ n ℓ ∑ i=1 w3 i s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' ℓ ∑ i=1 w2 i ≥ n/2 β 4 √ n ℓ ∑ i=1 wi ≤ T (b) min ℓ ∑ i=1 x3 i s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' ℓ ∑ i=1 x2 i ≥ ξ ℓ ∑ i=1 xi ≤ ξ Figure 5: The non-convex optimization problem that bounds the cumulative memory for quantum sorting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The objective function of system (a) is a lower bound on the cumulative memory complexity and system (b) is the same system after scaling the objective function and applying the variable substitutions wi = βn3/2 8T xi and ξ = 32T2 β2n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' And therefore the cumulative memory of C is Ω(n3/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' To extend our results to arbitrary success probability at most 1 − δ, it is important to know how α and γ are related in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In Appendix C we show that we can have α that is Ω(e−γ/2) and get a probability of at most e−γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Thus for any S, k, and δ ∈ (0, 1), we can choose γ = ln(22S/(1 − δ)) − 1 k + 1 to get a probability of at most 1 − δ for circuits with Ω �� 22S e(1 − δ) �1/2k √ kn � layers and S qubits of advice to produce k outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' When S = k and δ = 1 − 1/e, this is exactly the bound from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If we repeat the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 for failure probability at most δ, we can set β to a value that is Ω(1/ √ 1 − δ) to obtain a lower bound on the cumulative memory that is Ω(n3/((1 − δ)T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 5 A general method for proving cumulative memory complexity lower bounds Our method involves adapting techniques previously used to prove tradeoff lower bounds on worst-case time and worst-case space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We show that the same properties that yield lower bounds on the product of time and space in the worst case can also be used to produce nearly identical lower bounds on cumulative memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' To do so, we first revisit the standard approach to such time-space tradeoff lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 13 The standard method for time-space tradeoff lower bounds for multi-output functions Consider a multi-output function f on Dn where the output f (x) is either unordered (the output is simply a set of elements from R) or ordered (the output is a vector of elements from R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then | f (x)| is either the size of the set or the length of the vector of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The standard method for obtaining an ordinary time-space tradeoff lower bounds for multi-output functions on D-way branching programs is the following: The part that depends on f: Choose a suitable probability distribution µ on Dn, often simply the uniform distribution on Dn and then: (A) Prove that Prx∼µ[| f (x)| ≥ m] ≥ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' (B) Prove that for all k ≤ m′ and any branching program B of height ≤ h′(k, n), the probability for x ∼ µ that B produces at least k correct output values of f on input x is at most C · |R|−k/r(n) for some m′, h′, r and constant C independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Observe that under any distribution µ, a branching program with ordered outputs that makes no queries can produce k outputs that are all correct with probability at least |R|−k, so the bound in (B) shows that, roughly, up to the power 1/r(n) there is not much gained by using a branching program of height h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The generic completion: In the following outline we omit integer rounding for readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Suppose that S ≤ log2 |R| r(n) m′ − log2(2C/α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' (3) Let k = [S + log2(2C/α)] · r(n)/ log2 |R|, which is at most m′ by hypothesis on S, and define h(S, n) = h′(k, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Divide time T into ℓ = T/h blocks of length h = h(S, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The original branching program can be split into at most 2S sub-branching programs of height ≤ h, each beginning at a boundary node between layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' By property (B) and a union bound, for x ∼ µ the probability that at least one of these ≤ 2S sub-branching programs of height at most h produces k correct outputs on input x is at most 2S · C · |R|−k/r(n) ≤ α/2 by our choice of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Under distribution µ, by (A), with probability at least α, an input x ∼ µ has some block of time during which at least m/ℓ = m · h(S, n)/T outputs of f must be produced on input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If m · h(S, n)/T ≤ k, this can occur for at most an α/2 fraction of inputs under µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore we have m · h(S, n)/T > k = [S + log2(2C/α)] · r(n)/ log2 |R| 14 and hence, combining with Equation (3), we have T · S ≥ min � m · h(S, n), m′ · n′� · log2 |R| r(n) − log2(C/α) · T where n′ ≤ n is the decision tree complexity of f and hence a lower bound on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Though it will not impact our argument, for many instances of the above outline, the proof of property (B) is shown for a decision tree of the same height by proving an analog for the conditional probability along each path in the decision tree separately;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' this will apply to the tree as a whole since the paths are followed by disjoint inputs, so property (B) follows from the alternative property below: (B’) For any partial assignment τ of k ≤ m′ output values over R and any restriction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=', partial assignment) π of h′(k, n) coordinates within Dn, Pr x∼µ[ f (x) is consistent with τ | x is consistent with π] ≤ C · |R|−k/r(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The above method still gives lower bounds for many multi-output functions g : DN → RM that have individual output values that are easy to compute or large portions of the input space on which they are easy to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The bounds follow by applying the method to some subfunction f of g given by f (x) = ΠO(g(x, π)) where π is a partial assignment to the input coordinates and ΠO is a projection onto a subset O of output coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In the subsequent discussions we ignore this issue, but the idea can be applied to all of our lower bound methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' A general extension to cumulative memory bounds To give a feel for the basic ideas of the method, we first show this for a simple case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Observe that, other than the separate bound on time, the lower bound on cumulative memory usage we prove in this case is asymptotically identical to the bound achieved for the product of time and worst-case space using the standard outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Suppose that properties (A) and (B) apply for h′(k, n) = h(n), m′ = m, and α = C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If T log2 T ≤ m · h(n) · log2 |R| 6(c + 1)r(n) then the cumulative memory used in computing f : Dn → Rm in time T with success probability at least T−c is at least m · h(n) · log2 |R| 6r(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Fix a deterministic branching program P of length T computing f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Rather than choosing fixed blocks of height h = h(n), layers of nodes at a fixed distance from each other, and a fixed target of k outputs per block, we choose the block boundaries depending on the properties of P and the target k depending on the property of the boundary layer chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 15 Let H = ⌊h(n)/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We break P into ℓ = ⌈T/H⌉ time segments of length H working backwards from step T so that the first segment may be shorter than the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We let t1 = 0 and for 1 < i ≤ ℓ we let ti = arg min{ |Lt| : T − (ℓ − i + 1) · H ≤ t < T − (ℓ − i) · H } be the time step with the fewest nodes among all time steps t ∈ [T − (ℓ − i + 1) · H, T − (ℓ − i) · H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The i-th time block of P will be between times ti and ti+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Observe that by construction |ti+1 − ti| ≤ h(n) so each block has length at most h(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Set Si = log2 |Lti| so that Lti has at 2Si nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' By definition of each ti, the cumulative memory used by P, CM(P) ≥ ℓ ∑ i=1 Si · H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' (4) (Note that since S1 = 0, it does not matter that the first segment is shorter than the rest5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=') We now define the target ki for the number of output values produced in each time block to be the smallest integer such that |R|−ki/r(n) ≤ 2−Si/Tc+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' That is, ki = ⌈r(n) · (Si + (c + 1) log2 T)/ log2 |R|⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For x ∼ µ, for each i ∈ [ℓ] and each sub-branching program B rooted at some node in Lti and extending until time ti+1, by our choice of ki and property (B), if ki ≤ m, the probability that B produces at least ki correct outputs on input x is at most 2−Si/Tc+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore, by a union bound, for x ∼ µ the probability that P produces at least ki correct outputs in the i-th time block on input x is at most |Lti| · 2−Si/Tc+1 = 1/Tc+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore, if each ki ≤ m, the probability for x ∼ µ that there is some i such that P produces at least ki correct outputs on input x during the i-th block is at most ℓ/Tc+1 < Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore, if each ki ≤ m, the probability for x ∼ µ that P produces at most ∑ℓ i=1(ki − 1) correct outputs in total on input x is > 1 − 1/Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If each ki ≤ m, since P must produce m correct outputs on x ∈ Dn with probability at least 1/Tc, we must have ∑ℓ i=1(ki − 1) ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' On the other hand, if some ki > m we have the same bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Using our definition of ki we have ℓ ∑ i=1 [r(n) · (Si + (c + 1) log2 T)]/ log2 |R|)] ≥ m or ℓ ∑ i=1 (Si + (c + 1) log2 T) ≥ m · log2 |R| r(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In particular, plugging in the bound (4) on the cumulative memory and the value of ℓ, it implies that CM(P)/H + (c + 1)⌈T/H⌉ · log2 T ≥ m · log2 |R| r(n) 5This simplifies some calculations and is the prime reason for starting the time segment boundaries at T rather than at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 16 or that CM(P) + (c + 1)T log2 T ≥ m · h(n) · log2 |R| 3 · r(n) , where the 3 on the right rather than a 2 allows us to remove the integer rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore either T log2 T > m · h(n) · log2 |R| 6(c + 1) · r(n) or CM(P) ≥ m · h(n) · log2 |R| 6r(n) , which is what we wanted to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In the general version of our theorem there are a number of additional complications, most especially because the branching program height limit h(k, n) in property (B) can depend on k, the target for the number of outputs produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This forces the lengths of the blocks and the space used at the boundaries between blocks to depend on each other in a quite delicate way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In order to discuss the impact of that dependence and state our general theorem, we need the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Given a non-decreasing function p : R → R with p(1) = 1, we define p−1 : R → R ∪ {∞} by p−1(R) = min{j | p(j) ≥ k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We also define the loss, Lp, of p by Lp(n) = min 1≤k≤p(n) ∑k j=1 p−1(j) k · p−1(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The following hold for every non-decreasing function p : R → R with p(1) = 1: (a) 1/p(n) ≤ Lp(n) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' (b) If p is a polynomial function p(s) = s1/c then Lp(n) > 1/2c+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' (c) For any c > 1, Lp(n) ≥ min 1≤s≤n p(s) − p(s/c) cp(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' (d) We say that p is nice if it is differentiable and there is an integer c > 1 such that for all x, p′(cx) ≥ p′(x)/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If p is nice then Lp(n) is Ω(1/ log2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This is tight for p with p(s) = 1 + log2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We prove these technical statements in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The following is our full general theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Suppose that function f defined on Dn has properties (A) and (B) with α that is 1/nO(1) and m′ that is ω(log2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For s > 0, define h(s, n) to be h′(k, n) for k = s · r(n)/ log2 |R|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Suppose that h(s, n) = h0(s) h1(n) with h0(1) = 1 and h0 a differentiable function such that s/h0(s) is increasing and concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Define S∗ = S∗(T, n) by S∗ h0(S∗) = m · h1(n) · log2 |R| 6r(n)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 17 (a) Either T log2(2CTc+1/α) > m · h1(n) · log2 |R| 6r(n) , which implies that T is Ω( m·h1(n)·log |R| r(n) log n ), or the cumulative memory used by a randomized branching program in computing f in time T with error ε ≤ α(1 − 1/(2Tc)) is at least Lh0(n log2 |D|) · min � m · h(S∗(T, n), n), 3m′ · h′(m′/2, n) � · log2 |R| 6r(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' (b) Further any randomized random-access machine computing f in time T with error ε ≤ α(1 − 1/(2Tc)) requires cumulative memory Ω � Lh0(n log2 |D|) · min � m · h(S∗(T, n), n), m′ · h′(m′/2, n) � · log2 |R| r(n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Before we give the proof of the theorem, we note that by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5, in the case that h0 is constant or a polynomial function of its input, which together account for all existing applications we are aware of, the function Lh0 is lower bounded by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Further, the value S∗ in the statement of this theorem is at least a constant factor times the value of S used in the generic time-space tradeoff lower bound methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore, for example, the cumulative memory lower bound derived for random-access machines via Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 is close to the lower bound on the product of time and worst-case space given by standard methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We prove both (a) and (b) directly for branching programs, which can model random-access machines, and will describe the small variation that occurs in the case that the branching program in question comes from a random-access machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' To prove these properties for randomized branching programs, by Yao’s Lemma [Yao77] it suffices to prove the properties for deterministic branching programs that have error at most ε under distribution µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Fix a (deter- ministic) branching program P of length T computing f with error at most ε under distribution µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Without loss of generality, P has maximum space usage at most Smax = n log2 |D| space since there are at most |Dn| inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let H = ⌊h1(n)/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We break P into ℓ = ⌈T/H⌉ time segments of length H working backwards from step T so that the first segment may be shorter than the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We then choose a sequence of candidates for the time steps in which to begin new blocks, as follows: We let τ1 = 0 and for 1 < i ≤ ℓ we let τi = arg min{ |Lt| : T − (ℓ − i + 1) · H ≤ t < T − (ℓ − i) · H } be the time step with the fewest nodes among all time steps t ∈ [T − (ℓ − i + 1) · H, T − (ℓ − i) · H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Set σi = log2 |Lτi| so that Lτi has at 2σi nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This segment contributes at least σi · H to the cumulative memory bound of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' To choose the beginning ti∗ of the last time block6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' we find the smallest k such that h0(σℓ−k+1) < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Such a k must exist since h0 is a non-decreasing non-negative function, h0(1) = 1 and σ1 = 0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 6Since we are working backwards from the end of the branching program and we do not know how many segments are included in each block, we don’t actually know this index until things stop with t1 = 0 18 We now observe that the length of the last block is at most k · H which by choice of k is less than h(σℓ−k+1, n) and hence we have satisfied the requirements for property (B) to apply at each starting node of the last time block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' By our choice of each τi, the total cumulative memory used in the last k segments is at least k ∑ j=1 σℓ+1−j · H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Further, since k was chosen as smallest with the above property, we know that for every j ∈ [k − 1] we have h0(σℓ−j+1) ≥ j Hence we have σℓ−j+1 ≥ h−1 0 (j) and we get a cumulative memory bound for the last k segments of at least (σℓ−k+1 + k−1 ∑ j=1 h−1 0 (j)) · H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' (5) CLAIM: σℓ−k+1 + ∑k−1 j=1 h−1 0 (j) ≥ Lh0(Smax) · σℓ−k+1 · k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof of Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Observe that it suffices to prove the claim when we replace σℓ−k+1, which appears on both sides, by a larger quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In particular, we show how to prove the claim with h−1 0 (k) instead, which is larger since h0(σℓ−k+1) < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' But this follows immediately since by definition Lh0(Smax) ≤ ∑k j=1 h−1 0 (j) k · h−1 0 (k) , which is equivalent to what we want to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Write Si∗ = σℓ−k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' By the claim, the cumulative memory contribution associated with the last block beginning at ti∗ is at least Lh0(Smax) · Si∗ · h0(Si∗)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We repeat this in turn to find the time step for the beginning of the next block from the end, ti∗−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' One small difference now is that there is a last partial segment of height at most H from the beginning of segment containing ti∗ to layer ti∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' However, this only adds at most h1(n)/2 to the length of the segment which still remains well within the height bound of h(Si∗−1, n) = h0(Si∗−1)h1(n) for property (B) to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Repeating this back to the beginning of the branching program we obtain a decomposition of the branching program into some number i∗ of blocks, the i-th block beginning at time step ti with 2Si nodes, height between h0(Si)H and h0(Si)H + H ≤ 2h0(Si)H, and with an associated cumulative memory contribution in the i-th block of at least Lh0(Smax) · Si · h0(Si)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 19 (This is correct even for the partial block starting at time t1 = 0 since S1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=') Since we know that i∗ ≤ ℓ, for convenience, we also define Si = 0 for i∗ + 1 ≤ i ≤ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then, by definition we have CM(P) ≥ Lh0(Smax) · � i∗ ∑ i=1 Si · h0(Si) � H = Lh0(Smax) · � ℓ ∑ i=1 Si · h0(Si) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' (6) and ℓ ∑ i=1 h0(Si) ≤ T/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' (7) As in the previous argument for the simple case, for i ≤ i∗, we define the target ki for the number of output values produced in each time block to be the smallest integer such that C˙|R|−ki/r(n) ≤ 2−Siα/(2Tc+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' That is, ki = ⌈r(n) · (Si + log2(2CTc+1/α))/ log2 |R|⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If ki > m′ for some i, then Si ≥ m′ · log2 |R|/r(n) − log2(2CTc+1/α) ≥ m′ · log2 |R|/(2r(n)) since m′ is ω(log n) and 1/α and T are nO(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore h0(Si) ≥ h′(m′/2, n) and hence CM(P) ≥ Lh0(Smax) · m′ · h′(m′/2, n) · log2 |R| 2r(n) Suppose instead that ki ≤ m′ for all i ≤ i∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then, for x ∼ µ, for each i ∈ [i∗] and each sub- branching program B rooted at some node in Lti and extending until time ti+1, by our choice of ki and property (B), the probability that B produces at least ki correct outputs on input x is at most α · 2−Si/(2Tc+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore, by a union bound, for x ∼ µ the probability that P produces at least ki correct outputs in the i-th time block on input x is at most |Lti| · α · 2−Si/(2Tc+1) = α/(2Tc+1) and hence the probability for x ∼ µ that there is some i such that P produces at least ki correct outputs on input x during the i-th block is at most ℓ · α/(2Tc+1) < α/(2Tc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore, the probability for x ∼ µ that P produces at most ∑ℓ i=1(ki − 1) correct outputs in total on input x is > 1 − α/(2Tc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since, by property (A) and the maximum error it allows, P must produce at least m correct outputs with probability at least α − ϵ ≥ α − α(1 − 1/(2Tc)) = α/(2Tc) for x ∼ µ, we must have ∑i∗ i=1(ki − 1) ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Using our definition of ki we have i∗ ∑ i=1 [r(n) · (Si + log2(2CTc+1/α))]/ log2 |R|)] ≥ m or i∗ ∑ i=1 (Si + log2(2CTc+1/α)) ≥ m · log2 |R| r(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 20 This is the one place in the proof where there is a distinction between an arbitrary branching program and one that comes from a random access machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We first start with the case of arbitrary branching programs: Note that i∗ ≤ ℓ = ⌈T/H⌉ = ⌈T/⌊h1(n)/2⌋⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Suppose that T log2(2CTc+1/α) ≤ m·h1(n)·log2 |R| 6r(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then, even with rounding, we obtain ∑i∗ i=1 Si ≥ m·log2 |R| 2r(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Unlike an arbitrary branching program that may do non-trivial computation with sub- logarithmic Si, a random-access machine with even one register requires at least log2 n bits of memory (just to index the input for example) and hence Si + log2(2CTc+1/α) will be O(Si), since T is at most polynomial in n without loss of generality and 1/α is at most polynomial in n by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore we obtain that ∑i∗ i=1 Si is Ω( m·log2 |R| r(n) ) without the assumption on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In the remainder we continue the argument for the case of arbitrary branching programs and track the constants involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The same argument obviously applies for programs coming from random-access machines with slightly different constants that we will not track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In particular, since Si = 0 for i > i∗ we have ℓ ∑ i=1 Si ≥ m · log2 |R| 2r(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' (8) From this point we need to do something different from the argument in the simple case because the lower bound on the total cumulative memory contribution is given by Equation (6) and is not simply ∑ℓ i=1 Si · H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Instead, we combine Equation (8) and Equation (7) using the following technical lemma that we prove in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let p : R≥0 → R≥0 be a differentiable function such that q(x) = x/p(x) is a concave increasing function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' ∈ R≥0, if ∑i xi ≥ K and ∑i p(xi) ≤ L then ∑i xip(xi) ≥ q−1(K/L) · L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In our application of the lemma p = h0, K = m·log2 |R| 2r(n) , and L = T/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let S∗ be the solution to S∗ h0(S∗) = K/L = m · H · log2 |R| 2r(n)T ≥ m · h1(n) log2 |R| 6r(n)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='7 implies that ℓ ∑ i=1 Si · h0(Si) ≥ S∗ · T/H = m · h0(S∗) · log2 |R| 2r(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' and hence CM(P) ≥ Lh0(Smax) · m · h0(S∗) · H · log2 |R| 2r(n) ≥ Lh0(Smax) · m · h(S∗, n) · log2 |R| 6 · r(n) since H = ⌊h1(n)/2⌋ and h(S∗, n) = h0(S∗) · h1(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5, in the case that h0 is a polynomial function of its input, the function Lh0 is lower bounded by a constant and the bound in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 only loses a constant factor in moving from the product of worst-case space and time to cumulative memory complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In that special case (and indeed for any nice function h0), there is an alternative variant of the above in which 21 one breaks up time into exponentially growing segments starting with time step T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We use that alternative approach to obtain lower bounds on the cumulative memory complexity of sorting by quantum algorithms in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If we restrict our attention to o( m′ log |R| r(n) )-space bounded computation, then each ki ≤ m′ and the cumulative memory bound for a branching program in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 becomes Lh0(n log2 |D|) · m · h(S∗(T, n), n) · log2 |R| 6r(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' And the bound for RAM cumulative memory becomes Ω � Lh0(n log2 |D|) · m · h(S∗(T, n), n) · log2 |R| r(n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 6 Sample applications to cumulative complexity of classical algorithms Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 are powerful tools that can convert most existing time-space lower bounds into asymptotically equivalent lower bounds on the required cumulative memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We give a few examples to indicate how our general theorems can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Unique elements Define Uniquen,N : [N]n → P([N]) by Uniquen,N(x) = { xi | xj ̸= xi for all j ̸= i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1 (Lemmas 2 and 3 in [Bea91]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For the uniform distribution µ on [N]n with N ≥ n, (A) Prx∼µ[|Uniquen,N(x)| ≥ n/(2e)] ≥ 1/(2e − 1) (B’) For any partial assignment τ of k ≤ n/4 output values over [N] and any restriction π of n/4 coordinates in [n]n, Prx∼µ[Uniquen,N(x) is consistent with τ | x is consistent with π] ≤ e−k/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The above lemma is sufficient to prove that TS is Ω(n2) for the unique elements problem, and can be easily extended to a cumulative complexity bound using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For n ≥ N, any branching program computing Uniquen,N in time T and probability at least 4/5 requires T to be Ω(n2/ log n) or CM(P) to be Ω(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Further, any random access machine computing Uniquen,N with probability at least 4/5 requires cumulative memory Ω(n2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1, Uniquen,N satisfies conditions (A) and (B) of Section 5 with h′(k, n) = n/4, m′ = n/4, m = n/(2e), C = 1, r(n) = 2 ln N and α = 1/(2e − 1) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since h′(k, n) is independent of k, the function h0 defined in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 is the constant function 1 and h1(n) = n/4 so Lh0 ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We then apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 to obtain the claimed lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The above theorem is tight for N = n using the algorithm in [Bea91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 22 Linear Algebra We consider linear algebra over some finite field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let D be a subset of F with d elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' An m × n matrix is (g, h, c)-rigid iff every k × w submatrix where k ≤ g and w ≥ n − h has rank at least ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We call (g, h, 1)-rigid matrices (g, h)-rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Matrix rigidity is a robust notion of rank and is an important property for proving time-space and cumulative complexity lower bounds for linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Fortunately, Abrahamson proved that there are always rigid square matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='4 (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 in [Abr91]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' There is a constant γ ∈ (0, 1 2) where at least a 1 − d−1(2/3)γn fraction of the matrices over Dn×n are (γn, γn)-rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Abrahamson shows in [Abr91] that for any constant c ∈ (0, 1 2) and m × n matrix A that is (cm, cn, c)-rigid, any D-way branching program that computes the function f (x) = Ax with expected time T ≥ n and expected space7 S has TS = Ω(nm log d) where d = |D|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We restate the key property used in that proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 in [Abr91]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let c ∈ (0, 1 2], A be any m × n matrix that is (g, h, c)-rigid and f be the function f (x) = Ax over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let µ be the uniform distribution on Dn for D ⊆ F with |D| = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For any restriction π of h coordinates to values in D and any partial assignment τ of k ≤ g output coordinates over Fm, Pr x∼µ[ f (x) is consistent with τ | x is consistent with π] ≤ d−ck Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let c ∈ (0, 1 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let A be an m × n matrix over D, with |D| = d that is (g(m), h(n), c)-rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then, for any D-way branching program P computing f (x) = Ax in T steps with probability at least n−O(1), either T is Ω(g(m)h(n) logn d) or CM(P) is Ω(g(m)h(n) log d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Further, computing f on a random access machine requires cumulative memory Ω(g(m)h(n) log d) unconditionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We invoke Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 using Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5 to obtain condition (B’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Condition (A) is trivial since | f (x)| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='4 we know that for some constant γ, a random matrix has a good chance of being (γm, γn)-rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This means that computing f (x) = Ax for a random matrix A in time at most T is likely to require either the cumulative memory or T log T to be Ω(mn log d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since Yesha [Yes84] proved that the n × n DFT matrix is (n/4, n/4, 1/2)-rigid, the DFT is a concrete example where the cumulative memory or T log T is Ω(n2 log d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' other examples include generalized Fourier transform matrices over finite fields [BJS01, Lemma 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If A is an n × n generalized Fourier transform matrix over field F with characteristic relatively prime to n then any random-access machine computing f (x) = Ax for x ∈ Dn where D ⊆ F has |D| = d with probability at least n−O(1) requires cumulative memory that is Ω(n2 log d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' It is easy to see that our lower bound is asymptotically optimal in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 7[Abr91] defines expected space as the expected value of the log2 of the largest number of a branching program node that is visited during a computation under best case node numbering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 23 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='8 (Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1 in [Abr91]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let f : D2n2 → Fn2 for D ⊆ F and d = |D| be the matrix multiplication function, γ be the constant from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='4, and µ be the uniform distribution over (γm, γn)-rigid matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Choose any integers h and k such that 2(h/γn)2 ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If γn ≥ 1 then for any D-way branching program B of height ≤ h the probability that B produces at least k correct output values of f is at most d2−γk/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Multiplying two random matrices in Dn2 with D ⊆ F and d = |D| with probability at least n−O(1) requires time T that is Ω((n3� log d)/ log n) or cumulative memory Ω((n6 log d)/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' On random access machines, the cumulative memory bound is unconditional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='8 lets us apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 with m = n2, h′(k, n) = γn√ k/2, C = d2, α = 1, |R| = |F|, and r(n) = (4 logd |F|)/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This gives us that h(s, n) = n � 2γs/ log2 d, so h0(s) = √s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then we get that √ S∗ = mn√ 2γ/ log2 d·log2 |F| 6r(n)T and hence S∗ is Ω �n6 log d T2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore we get that either T is Ω � n3 log1/2 d log n � or, since the loss function for h0 is a constant, the cumulative memory is Ω � min � (n6 log d)/T, n5 log1/2 d �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since the decision tree complexity of matrix multiplication is Ω(n2), this is Ω((n6 log d)/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For random access machines, the same cumulative memory bound applies without the condition on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 7 Cumulative memory complexity of single-output functions The time-space tradeoff lower bounds known for classical algorithms computing single-output functions are quite a bit weaker than those for multi-output functions, but the bounds we can obtain on cumulative memory for slightly super-linear time bounds are nearly as strong as those for multi-output functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For simplicity we focus on branching programs with Boolean output, in which case, we can simply assume that the output is determined by which of two nodes the branching program reaches at time step T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The general method for bounds for single output functions is based on the notion of the trace of a branching program computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We fix a branching program P computing f : Dn → { 0, 1 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' As in the case of the simple bounds for multi-output functions, we break up P into a sequence of blocks, say ℓ of them, that are separated by time steps 0 = t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , tℓ, tℓ+1 = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' A trace τ in P is a sequence of ℓ nodes of P, one node in the set of nodes Lti at time step ti for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The set of all traces T = Lt1 × · · · × Ltℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 24 A key object under consideration is the notion of an embedded rectangle, which is a subset of R ⊆ Dn with associated disjoint subsets A ⊂ [n] and B ⊂ [n] with |A| = |B| = m(R) = m and assignment σ ∈ D[n]−A−B such that R = RA × RB × σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We write α(R) = min(|RA|, |RB|)/|D|m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1 (Implicit in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2 of [BSSV03]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let P be a branching program of length T computing a function f : Dn → {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Suppose that T ≤ kn for k ≥ 4 and n ≥ ℓ ≥ k22k+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If 0 = t1 < t2 < · · · < tℓ+1 = T are time steps with ti+1 − ti ≤ n/(k2k+6), then there is an embedded rectangle R ⊆ f −1(1) with m(R) = m ≥ n/2k+1 and α(R) ≥ 2−12(k+1)m−2 · |T |−1 · | f −1(1)|/|D|n where T is the set of traces of P associated with time steps t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , tℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let P be a D-way branching program of length T computing a function f : Dn → {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If T ≤ kn for k ≥ 4 and n ≥ k22k+8, then there is an embedded rectangle R ⊆ f −1(1) with m(R) = m ≥ n/2k+1 and α(R) ≥ 2−12(k+2)m−k·2k+9·CM(P)/n−2 · | f −1(1)|/|Dn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Fix a branching program P of length T ≤ kn computing f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We can extend P to length exactly kn by adding a chain of nodes to the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This does not impact the cumulative memory bound of P – a single node per level is 0 space – so we assume that T = kn without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let ℓ = k22k+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We apply the same basic idea for the choice of time steps 0 = t1, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , tℓ+1 = T used in the simple general method for multi-output functions: Namely, we break P into ℓ time segments of length either h = ⌊kn/ℓ⌋ or ⌈kn/ℓ⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We define t1 = 0 and define ti for 1 < i ≤ ℓ to be the time step during the next segment at which the set |Lti| is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Write Si = log2 |Lti|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then the cumulative memory complexity used by P satisfies CM(P) ≥ ℓ ∑ i=1 Si · h = h · log2 |T |, since |T | = ∏t i=1 |Lti|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Clearly each ti+1 − ti is at most 2⌈kn/ℓ⌉ ≤ n/(k2k+6) by definition, since their difference is at most the length of two consecutive time segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore, the conditions of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1 apply and we obtain that there is an embedded rectangle R ⊆ f −1(1) with m(R) ≥ n/2k+1 and α(R) ≥ 2−12(k+2)m−2 · |T |−1 · | f −1(1)|/|Dn| ≥ 2−12(k+2)m−2−CM(P)/h · | f −1(1)|/|Dn| ≥ 2−12(k+2)m−k·2k+9·CM(P)/n−2 · | f −1(1)|/|Dn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' An example of a natural problem that we can apply this to is the Hamming Closeness problem HAM1/8,n,N : [N]n → {0, 1} which outputs 1 iff there is a pair of input coordinates xi, xj ∈ [N] such that the Hamming distance between the binary representations of xi and xj is at most 1 8 log2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 ([BSSV03]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For f (x) = 1 − HAM1/8,n,N(x), and N ≥ n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='39 we have (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='15) | f −1(1)| ≥ Nn/2, and (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='17) there is a constant β > 0 such that any embedded rectangle R ⊆ f −1(1) has α(R) ≤ N−βm(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 25 [BSSV03] apply the above to prove that any [N]-way branching program computing HAM1/8,n,N for N ≥ n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='39 in time T and space S requires T that is Ω(n log � n log n S � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For N ≥ n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='39 any [N]-way branching program computing HAM1/8,n,N in time T that is o(n log n) requires cumulative memory (n2 log n)/2O(T/n) which is n2−o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let P be an [N]-way branching program computing HAM1/8,n,N in time T that is o(n log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We can swap the sink nodes to obtain a branching program P′ computing f = 1 − HAM1/8,n,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Write k = T/n and assume wlog that k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore k is o(log n) and hence k22k+8 is no(1) and hence ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore by Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2, there is an embedded rectangle R ⊆ f −1(1) such that m(R) = m ≥ n/2k+1 and α(R) ≥ 2−12(k+2)m−k·2k+9·CM(P′)/n−2 · | f −1(1)|/Nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore by Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3, for some constant β > 0 we have N−βm ≥ α(R) ≥ 2−12(k+2)m−k·2k+9·CM(P′)/n−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since CM(P) = CM(P′), solving we obtain k · 2k+9 · CM(P) ≥ βnm log2 N − 12(k + 2)mn − 3n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since k + 2 is o(log N) we obtain that k · 2k+9 · CM(P) ≥ δnm log2 N for some constant δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore, plugging in the value of T/n for k, we see that CM(P) is (n2 log n)/2O(T/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This is n2−o(1) by the bound on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Similar bounds can also be shown by related means for various problems involving computation of quadratic forms, parity-check matrices of codes and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For some problems the following stronger lower bound method is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5 (Implicit in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='4 of [BSSV03]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let P be a D-way branching program of length T computing a function f : Dn → {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Suppose that T ≤ (k − 2)n for k ≥ 8 and n ≥ ℓ ≥ 2q5k2 for q ≥ 240k8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If 0 = t1 < t2 < · · · < tℓ+1 = T are time steps with ti+1 − ti ≤ kn/q5k2, then there is an embedded rectangle R ⊆ f −1(1) with m(R) = m ≥ q−2k2n/2 and α(R) ≥ 2−q−1/2m · |T |−1 · | f −1(1)|/|D|n where T is the set of traces of P associated with time steps t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , tℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let P be a branching program of length T computing a function f : Dn → {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If T ≤ (k − 2)n for k ≥ 8 and n ≥ 2q5k2 for q = 240k8, then there is an embedded rectangle R ⊆ f −1(1) with m(R) = m ≥ q−2k2n/2 and α(R) ≥ 2−q−1/2m−q5k2CM(P)/n · | f −1(1)|/|Dn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof Sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The proof is the analog of that of Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2 using Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5 in place of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Define the Element Distinctness function EDn,N on [N]n to be the Boolean function that is 1 iff all values in the input are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='7 ([BSSV03]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For N ≥ n2, 26 (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='11) |ED−1 n,N(1)| ≥ Nn/e, and (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='12) Every embedded rectangle R in ED−1 n,N(1) has α(R) ≤ 2−m(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' [BSSV03] used this to prove that the time T and space S for computing EDn,n2 must satisfy T = Ω(n � log(n/S)/ log log(n/S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We strengthen this to the following theorem using Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Any [n2]-way branching program computing EDn,n2 in time T that is o(n � log n/ log log n) requires cumulative memory n2/(T/n)O(T2/n2) which is n2−o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let P compute EDn,n2 in time T that is o(n � log n/ log log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Write k = T/n + 2 so that T ≤ (k − 2)/n and assume wlog that k ≥ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Write q = 240k8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since T is o(n � log n/ log log n), k is o( � log n/ log log n) and 2q5k2 which is kO(k2) and hence no(1) and therefore ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We can then apply Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 to say that there is a rectangle R ⊆ ED−1 n,n2(1) with m(R) = m ≥ q−2k2n/2 and α(R) ≥ 2−q−1/2m−q5k2CM(P)/n · |ED−1 n,n2(1)|/|Dn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='7, we have 2−m ≥ α(R) ≥ 2−q−1/2m−q5k2CM(P)/n/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Solving, we obtain that q5k2CM(P) ≥ n · m(1 − 1/q1/2) − 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore, since m ≥ q−2k2n/2, we have constant c such that CM(P) ≥ n2/qck2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' As noted above, qck2 is no(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' More precisely, the bound we obtain is CM(P) ≥ n2/(T/n)O(T2/n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 8 Extending the general method to quantum lower bounds Quantum circuit time-space lower bounds have the same general structure as their classical branch- ing program counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The standard method for quantum time-space tradeoff lower bounds Let f : Dn → Rm be a multi-output function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For simplicity, we will assume that the output of f is always m elements in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' To obtain a time-space tradeoff lower bound on f, we must prove a lemma of the following form for some m′, h(k, n), µ, r(n) and constant C: Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1 (Quantum generic property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For all k ≤ m and any quantum circuit C with at most h(k, n) layers, there exists a distribution µ such that when x ∼ µ, the probability that C produces at least k correct output values of f (x) is at most C · |R|−k/r(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Such lemmas have historically been proving using direct product theorems [KŠdW07, AŠdW09] and, more recently, using the recording query technique [HM21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This is the quantum version of condition (B) for the classical general method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In the classical setting, the lemma could be extended 27 to account for the 2S boundary nodes between layers by using a union bound over 2S possible branching programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' However in the quantum setting it is not as obvious how to use a lemma that does not account for initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Aaronson showed in [Aar05] how to do exactly this using the following proposition, which we previously used in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='4 ([Aar05]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let C be a quantum circuit, ρ be any S qubit (possibly mixed) state, and I be the S qubit maximally mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If C with initial state ρ produces some output O with probability p, then C with initial state I produces O with probability at least p/22S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Thus for any problem where we can prove something similar to Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1, we can bound the probability of circuits with S qubits of input-dependent state producing k correct outputs as being at most 22S · C · |R|−k/r(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This idea has been applied in [KŠdW07, AŠdW09, HM21] to bound the probability that blocks produce correct outputs, even when they are given initial state from previous blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' From here we take any circuit C with T layers and S qubits and split it into sub-circuits C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , Cℓ with h = h(k, n) layers each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This makes ℓ = ⌈T/h⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' While Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1 gives us that C1 produces at least k correct outputs with probability at most |R|−k/r(n), sub-circuits C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , Cℓ start with some initial state ρ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' , ρℓ that can depend on the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since ρi has at most S qubits, the probability that Ci produces at least k correct outputs is at most 22S · C · |R|−k/r(n) Assume that: 2S ≤ log2 |R| r(n) m′ − log2(2C) (9) Then we can set k = [2S + log2(2C)] · r(n)/ log2 |R| and get that the probability of producing K correct outputs is at most 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' There must be some block that produces at least m · h(S, n)/T correct outputs, so we must have that m · h(S, n)/T > k = [2S + log2(2C)] · r(N)/ log2 |R| This gives us that TS is Ω �mh(S, n) log |R| r(n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In the event that (9) is not satisfied, we can instead use the bounded-error quantum query complex- ity of f (denoted Q( f )) instead of the decision tree complexity to obtain that TS is Ω �Q( f ) · m′ log2 |R| r(n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Generic quantum cumulative complexity Quantum sorting In Section 4 we are able to exploit some specific structure that leads to a cleaner proof than we can obtain in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Specifically for quantum sorting we have h0(s) = √s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since h0 is not a constant function, we cannot apply arguments like Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 or Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' However h0 is a polynomial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This means that a block with at least s/4 qubits per layer for at least 28 h(s/4, n)/2 layers has a constant fraction of the cumulative complexity of a block with h(s, n) layers that each have s qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This means that we can use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 to upper bound the number of outputs for a block with h(4s, n) layers and 4s initial qubits while obtaining a lower bound on the cumulative complexity of such a block that is within a constant factor of the TS complexity of that block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' To obtain such a bound on the cumulative complexity, we can start with a segment of length h(s, n) when s = 1 that ends at the start of the next block and then repeatedly multiply s by four until we find a block of length h(s, n) where one of the first h(s, n)/2 layers has less than s space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since this is the first such segment, we know that there must be h(s/4, n)/2 layers that each have s/4 qubits, which gives us the asymptotically tight cumulative complexity lower bound for the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The argument we used in our quantum sorting proof can be applied to other classical and quantum time-space tradeoffs where h0(s) is a polynomial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The generic completion In general, h0(s) may not be a polynomial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' When this is not the case, we can observe that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 does not exploit any structure of branching programs that cannot be applied to quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' It depends only on the existence of a lemma that bounds the number of outputs for short computation and a way to apply that lemma to computation with input dependent initial state, which are given by our generic Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We state the quantum versions of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 and Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='8 when α = 1 here for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Note that since Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='4 gives us a bound of p/22S rather than p/2S, the cumulative memory bounds we obtain in the quantum setting are half of those from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Suppose that function f defined on Dn satisfies generic Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1 with m′ that is ω(log2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For s > 0, let h(s, n) = h′(s · r(n)/ log2 |R|, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let h(s, n) = h0(s)h1(n) where h0(1) = 1 and h0 is a differentiable function where s/h0(s) is increasing and concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let S∗ be defined by: S∗ h0(S∗) = m · h1(n) · log2 |R| 12r(n)T Then either T log2(2CTc+1) > m · h1(n) · log2 |R| 12r(n) Which implies that T is Ω( m·h1(n)·log |R| r(n) log n ) or the cumulative memory used by a quantum circuit that computes f in time T with error ε ≤ (1 − 1/(2Tc)) is at least Lh0(n log2 |D|) · min � m · h(S∗, n), 3m′ · h′(m′/2, n) � · log2 |R| 12r(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Additionally if the quantum circuit uses o( m′ log |R| r(n) ) qubits, then the cumulative memory bound instead is Lh0(n log2 |D|) · m · h(S∗, n) · log2 |R| 12r(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 29 9 Quantum applications of the generic method Disjoint Collision Pairs Finding In [HM21] the authors considered the problem of finding k disjoint collisions in a random function f : [m] → [n], and were able to prove a time-space tradeoff that T3S is Ω(k3n) for circuits that solve the problem with success probability 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Specifically, they consider circuits that must output triples (xj2i, xj2i+1, yji) where f (xj2i) = f (xj2i+1) = yji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' To obtain this result, they prove the following theorem using the recording query technique: Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 in [HM21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For all 1 ≤ k ≤ n/8 and any quantum circuit C with at most T quantum queries to a random function f : [m] → [n], the probability that C produces at least k disjoint collisions in f is at most O(T3/(k2n))k/2 + 2−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The above theorem can be extended to a lemma matching Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1 by choosing a sufficiently small constant α and setting T = αk2n to obtain a probability of at most 2S+1−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This is sufficient to obtain a matching lower bound on the cumulative memory complexity using Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Finding ω(log2 n) ≤ k ≤ n/8 disjoint collisions in a random function f : [m] → [n] with probability at least 2/3 requires time T is Ω(kn1/3/ log n) or cumulative memory Ω(k3n/T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1 lets us apply Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2 with m = m′ = k, h′(k, n) = αk2/3n1/3, |R| = m2n − mn, and r(n) = log2 |R|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Thus we have h(s, n) = h′(s, n) and h0 is a differentiable function where s/h0(s) is an increasing and concave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' With these parameters, we have: S∗ is Ω �k3n T3 � By Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2 with the observation that the loss is constant we get that: T is Ω �kn1/3 log n � or the quantum cumulative memory is: Ω � min �k3n T2 , k5/3n1/3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1 we know that any quantum circuit with at most T′ = αk2/3n1/3 layers can produce k disjoint collisions with probability at most 21−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Thus we know that T > T′ and our cumulative memory bound becomes Ω(k3n/T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' On Tradeoffs for Linear Inequalities and Boolean Linear Algebra In this section we consider problems in Boolean linear algebra where we write A • x for Boolean (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' and-or) matrix-vector product and A • B for Boolean matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In [KŠdW07] the authors prove the following time-space tradeoff for Boolean matrix vector products: 30 Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 (Theorem 23 in [KŠdW07]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For every S in o(n/ log n), there is an n × n Boolean matrix AS such that every bounded-error quantum circuit with space at most S that computes Boolean matrix vector product AS • x in T queries requires that T is Ω( √ n3/S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This result is weaker than a standard time-space tradeoff since the function involved is not independent of the circuits that might compute it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In particular, [KŠdW07] does not find a single function that is hard for all space bounds, as the matrix A that they use changes depending on the value of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For example, a circuit using space S′ ≫ S could potentially compute AS • x using o(n3/2/(S′)1/2) queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This means that an extension of their bound to cumulative memory complexity does not follow from our Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2, as blocks with distinct numbers of initial qubits would be computing outputs for different functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In [AŠdW09] the authors use the same space-dependent matrices to prove a result for systems of linear inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='4 (Theorem 19 in [AŠdW09]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let S be in min(O(n/t), o(n/ log n)) and⃗t be the all-t vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' There is an n × n Boolean matrix AS such that every bounded error quantum circuit using space S for evaluating the system ASx ≥⃗t using T queries requires T that is Ω( � (tn3/S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Again this result is not a general time-space tradeoff and hence is not compatible with obtaining a true cumulative memory bound8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' While neither of the above results is a time-space tradeoff for a fixed function, [KŠdW07] leverages the ideas for Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 to compute a true time-space tradeoff lower bound for computing Boolean matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5 (Theorem 25 in [KŠdW07]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If a quantum circuit computes the Boolean matrix product A • B with bounded error using T queries and S space, then TS is Ω(n5/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5, unlike in Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='3 and Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='4, both A and B are inputs to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This allows the lower bound argument to use the properties of the circuit to find matrices A and B for which the circuit will be particularly challenged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' More precisely, to prove the above result, the authors use a lemma matching the form of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1 that we extract from their lower bound argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 (from Theorem 25 in [KŠdW07]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let R ⊆ [n] × [n] be any fixed set of k ∈ o(n) outputs to the function f (A, B) = A • B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then there are constants α, γ > 0 such that for any quantum circuit C with at most α √ kn layers, there is a distribution µC over pairs of matrices such that when (A, B) ∼ µC, the probability that C produces the correct values for R is at most 2−γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Note that, though there are Ω(n2) total output values, Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 only works when k — the number of output values in a block — is sublinear in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' This is not a problem in the time-space tradeoff lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 upper bounds the value of k for a block as O(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since the time T must be Ω(n2) simply to read the input, the bound T2S = Ω(n5) trivially holds when S is Ω(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Thus the time-space tradeoff proof only needs to apply Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 when S (and therefore k) is sublinear in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' We cannot apply such an argument when considering cumulative memory complexity, as a circuit can use Ω(n) qubits for a small number of layers without having an asymptotic effect on the 8The analogous cumulative complexity result would require the matrix A to depend extensively on the structural properties of the circuit, including the number of qubits after each layer and the locations of each fixed output gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' It is unclear whether the TS results also may need the matrix AS to depend on the locations of the output gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 31 cumulative memory complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' However, if we consider o(n) space bounded computation, we can get a matching bound on the cumulative memory complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Any quantum circuit that computes the boolean matrix product A • B requires Ω(n) ancilla qubits, time T that is Ω(n5/2/ log n), or cumulative memory that is Ω(n5/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='6 lets us apply Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2 with m′ being o(n), m = n2, h′(k, n) = α √ kn, |R| = 2, and r(n) = 1/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Thus we have h(s, n) = h′(s/γ, n) = α � sn/γ and h0(s) = √s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore we define S∗ to be S∗ = γα2n5 36T2 Thus by Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='2 we get that T is Ω(n5/2/ log n) or since the space bound is o(n) and the loss function is constant, the cumulative memory is Ω(n5/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Though this is somewhat limited in its range of applicability, it still yields a strict generalization of the time-space tradeoff lower bound of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='5 when S is o(n) and T is o(n5/2/ log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 10 Acknowledgements Many thanks to David Soloveichik for his guidance and contributions to our initial results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Thanks also to Scott Aaronson for encouraging us to consider cumulative memory complexity in the context of quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' References [Aar05] Scott Aaronson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Limitations of quantum advice and one-way communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Theory of Computing, 1(1):1–28, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' [AB16] Joël Alwen and Jeremiah Blocki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Efficiently computing data-independent memory- hard functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In Matthew Robshaw and Jonathan Katz, editors, Advances in Cryptol- ogy – CRYPTO 2016, pages 241–271, Berlin, Heidelberg, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Springer Berlin Heidel- berg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' [ABB21] Mohammad Hassan Ameri, Alexander R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Block, and Jeremiah Blocki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Memory- hard puzzles in the standard model with applications to memory-hard functions and resource-bounded locally decodable codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Cryptology ePrint Archive, Paper 2021/801, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' https://eprint.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=', 20(2):118–132, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' [Yao77] Andrew Chi-Chih Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Probabilistic computations: Toward a unified measure of complexity (extended abstract).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In 18th Annual Symposium on Foundations of Computer Science, Providence, Rhode Island, USA, 31 October - 1 November 1977, pages 222–227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' IEEE Computer Society, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' [Yes84] Yaacov Yesha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Time-space tradeoffs for matrix multiplication and the discrete Fourier transform on any general sequential random-access computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Journal of Computer and System Sciences, 29(2):183–197, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' [Zha19] Mark Zhandry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' How to record quantum queries, and applications to quantum in- differentiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' In Alexandra Boldyreva and Daniele Micciancio, editors, Advances in Cryptology – CRYPTO 2019, pages 239–268, Cham, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Springer International Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' 36 A Optimizations In this section we prove general optimization lemmas that allow us to derive worst-case properties of the allocation of branching program layers into blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The first special case is relevant for our analysis of quantum sorting algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' If ∑i xi ≤ ∑i x2 i then ∑i x3 i ≥ ∑i x2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Without loss generality we remove all xi that are 0 or 1 since they contribute the same amount to each of ∑i xi, ∑i x2 i , and ∑i x3 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore every xi satisfies 0 < xi < 1 or it satisfies xi > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For simplicity we rename those xi with 0 < xi < 1 by yi and those xi with xi > 1 by zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Then ∑i xi ≤ ∑i x2 i can be rewritten as ∑ i yi(1 − yi) ≤ ∑ j zj(zj − 1), and both quantities are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let y∗ be the largest value < 1 and z∗ be the smallest value > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Therefore we have ∑ i (y2 i − y3 i ) = ∑ i y2 i (1 − yi) ≤ ∑ i y∗yi(1 − yi) = y∗ ∑ i yi(1 − yi) ≤ y∗ ∑ j zj(zj − 1) < z∗ ∑ j zj(zj − 1) = ∑ j z∗zj(zj − 1) ≤ ∑ j z2 j (zj − 1) = ∑ j (z3 j − z2 j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Rewriting we have ∑i y2 i + ∑j z2 j < ∑i y3 i + ∑j z3 j , or equivalently ∑i x3 i > ∑i x2 i , as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' The following is a generalization of the above to all differentiable functions p : R≥0 → R≥0 such that s/p(s) is a concave increasing function of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Let p : R≥0 → R≥0 be a differentiable function such that q(x) = x/p(x) is a concave increasing function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' For x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' ∈ R≥0, if ∑i xi ≥ K and ∑i p(xi) ≤ L then ∑i xip(xi) ≥ q−1(K/L) · L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' By hypothesis, ∑ i (xi − Kp(xi)/L) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' (10) 37 Observe that s − Kp(s)/L is an increasing function of s since s/p(s) is an increasing function of s that is 0 precisely when s = q−1(K/L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE5T4oBgHgl3EQfmg_h/content/2301.05680v1.pdf'} +page_content=' Since all xi with xi = q−1(K/L) evaluate to 0 in Equation (10), we can rewrite it as ∑ xi>q−1(K/L) (xi − Kp(xi)/L) ≥ ∑ xi